Evolutionarily stable transition rates in a stage-structured model. An application to the analysis of size distributions of badges of social status

Abstract

This paper deals with the adaptive dynamics associated to a hierarchical non-linear discrete population model with a general transition matrix. In the model, individuals are categorized into n dominance classes, newborns lie in the subordinate class, and it is considered as evolutionary trait a vector eta of probabilities of transition among classes. For this trait, we obtain the evolutionary singular strategy and prove its neutral evolutionary stability. Finally, we obtain conditions for the invading potential of such a strategy, which is sufficient for the convergence stability of the latter. With the help of the previous results, we provide an explanation for the bimodal distribution of badges of status observed in the Siskin (Carduelis spinus). In the Siskin, as in several bird species, patches of pigmented plumage signal the dominance status of the bearer to opponents, and central to the discussion on the evolution of status signalling is the understanding of which should be the frequency distribution of badge sizes. Though some simple verbal models predicted a bimodal distribution, up to now most species display normal distributions and bimodality has only been described for the Siskin. In this paper, we give conditions leading to one of these two distributions in terms of the survival, fecundity and aggression rates in each dominance class.

Full text

Evolutionarily stable transition rates in a
stage-structured model. An application to the analysis
of size distributions of badges of social status
Jordi Ripoll
a
, Joan Salda~
na
a,*
, Juan Carlos Senar
b
a
Dept. d’Inform
atica i Matem
atica Aplicada, Campus de Montilivi, Universitat de Girona, E-17071 Girona, Spain
b
Museu de Ci
encies Naturals de Barcelona, Pg. Picasso s/n, Parc Ciutadella, E-08003 Barcelona, Spain
Received 17 June 2002; received in revised form 8 March 2004; accepted 25 March 2004
Available online 4 June 2004
Abstract
This paper deals with the adaptive dynamics associated to a hierarchical non-linear discrete population
model with a general transition matrix. In the model, individuals are categorized into ndominance classes,
newborns lie in the subordinate class, and it is considered as evolutionary trait a vector gof probabilities of
transition among classes. For this trait, we obtain the evolutionary singular strategy and prove its neutral
evolutionary stability. Finally, we obtain conditions for the invading potential of such a strategy, which is
sufficient for the convergence stability of the latter.
With the help of the previous results, we provide an explanation for the bimodal distribution of badges
of status observed in the Siskin (Carduelis spinus). In the Siskin, as in several bird species, patches of
pigmented plumage signal the dominance status of the bearer to opponents, and central to the discussion on
the evolution of status signalling is the understanding of which should be the frequency distribution of
badge sizes. Though some simple verbal models predicted a bimodal distribution, up to now most species
display normal distributions and bimodality has only been described for the Siskin. In this paper, we give
conditions leading to one of these two distributions in terms of the survival, fecundity and aggression rates
in each dominance class.
Ó2004 Published by Elsevier Inc.
Keywords: Matrix hierarchical models; Adaptive dynamics; Bimodality
*
Corresponding author. Tel.: +34-972 418 834; fax: +34-972 418 792.
E-mail addresses: jripoll@ima.udg.es (J. Ripoll), jsaldana@ima.udg.es (J. Salda~
na), jcsenar@intercom.es
(J.C. Senar).
0025-5564/$ - see front matter Ó2004 Published by Elsevier Inc.
doi:10.1016/j.mbs.2004.03.003
www.elsevier.com/locate/mbs
Mathematical Biosciences 190 (2004) 145–181

1. Introduction
Rohwer proposed in 1975 [1] that the patches of pigmented plumage displayed by several bird
species could be related to dominance signalling. The advantage of the system would be that in
communicating the dominance status of the bearers, contestants could resolve a potential fight
without resorting to costly violence. The so called ‘status signalling hypothesis’, albeit contro-
versial, has been found to work in several birds species [2].
However, we are still far away from a full understanding of the dynamics of a signalling system
as the one proposed above. Great controversy exists in relation to how the system could be
evolutionarily stable and several disparate hypotheses have been proposed [2–5]. It is still unclear
how badges of dominance status should be related to sexual selection processes [2]; although some
authors propose that many sexual ornaments have first evolved as armaments in intrasexual
disputes [6], the relationship between dominance and mate choice is far from clear [7]. Incon-
sistencies between species add additional confusion to the problem [8,9].
Central to the discussion on the evolution of status signalling within the context of social
selection [10] is the understanding of which should be the frequency distribution of badge sizes.
Rohwer and Ewald [11] proposed that bimodal distributions, with many birds either with large or
small badges, but fewer individuals with intermediate badge sizes [12], should be quite common
and could be maintained by negative frequency-dependent selection when individuals of different
appearance and status either played mutually beneficial roles or employed alternate competitive
tactics. However, although the Siskin (Carduelis spinus) has been found to display a bimodal
distribution of badge sizes [13], the analysis of frequency distributions from most species reveals
normality to be the rule [2,12,14]. Similar discussions appear in relation to characters seemingly
evolved under sexual selection, where bimodal distributions are also predicted under certain
mixed ESS [15], but again normal distributions are also quite common [16–18] (see Fig. 1). The
subject can be further complicated by the interaction between social and sexual selection, al-
though this has been rarely explored [2].
The aim of the present work is to use a hierarchical non-linear matrix model for the size dis-
tribution of badges of social status in natural (bird) populations, in order to predict evolutionarily
stable equilibrium distributions under different hypotheses on parameters related to selection.
Therefore, the goal is not to explore the dynamics itself of the signalling system, but to determine
conditions for the evolutionary stability of the transition rates among classes (or ranks) under
asymmetrical competition based on a system of status signalling with the hope that such condi-
tions give clues of how a hierarchical or dominance ranking can be maintained. More precisely,
assuming the honesty of the signalling system, individuals are categorized into discrete dominance
classes according to the size of a given (plumage) trait which acts as a badge of social status. This
size can increase and decrease over the lifetime of an individual by means of transitions between
any pair of dominance classes (including reverse transitions). The dominance hierarchy is intro-
duced by means of a negative class-specific dependence of the demographic rates on weighted
sums of the population members. Such weighted sums are different for each dominance class and
reflect the environmental conditions experienced by individuals in those classes. For instance, one
can assume that subordinate individuals do not affect the environmental conditions experienced
by those in the dominant classes, while dominant (and subdominant) individuals do affect the
environment experienced by subordinates. An example of environmental condition can be the
146 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

Great tit bib size (mm
2
)
No. of birds
0
5
10
15
20
25
30
35
40
45
50
55
300 400 500 600 700 800 900 1000 1100 1200 1300
Siskin bib size (mm
2
)
No. of birds
0
50
100
150
200
250
300
350
400
450
500
550
600
650
-10-5 0 5 10152025303540455055606570
House sparrow bib size (mm
2
)
No. of birds
0
2
4
6
8
10
12
14
16
18
20
22
280 300 320 340 360 380 400 420 440 460 480
(a)
(b)
(c)
Fig. 1. Bib size frequency distribution in three species of birds for which the bib has been found to work as a signal of
social status. Only data for males are used. (a) House sparrow (Passer domesticus); data and bib size measured
according to [52]; Shapiro Wilk W ¼0.99, p¼0:91, N ¼80, 1983–1986. (b) Great tit (Parus major) [53]; bib size
measured according to [51]; Shapiro Wilk W ¼0.99, p¼0:18, N ¼160, 1997–2000. (c) Siskin (Carduelis spinus); data
and bib size measured according [20]; Shapiro Wilk W ¼0.93, p<0:001, N ¼3454, 1990–2000; the class -5 to 0 refers
to birds with a bib size of 0 mm2. The mean and the variance of each fitted normal curve are the same as in the
corresponding data.
J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181 147

average number of aggressive interactions per unit of time which, of course, is a function of the
population composition.
Astrategy in the model is given by a subset of probabilities of moving among classes. In
particular, given the current state/class of an individual, it is assumed that natural selection acts to
change the probabilities of moving from this state to any other state so as to maximize its fitness.
Similarly to what is assumed in many metapopulation models where an optimal distribution of
individuals in a patchy environment is pursued (see, for instance, [19,22]), in this adjustment by
natural selection it is also assumed that individuals do not have information about the population
composition so that they only experience their own environmental conditions in an average way
as it could be, for instance, the total number of aggressive interactions they suffer per time
interval. Other considerations as, for instance, considering an energetic cost of the transitions
which affects the subsequent division of the energy between reproduction and maintenance, or
introducing transition probabilities which are conditional on the actual population composition,
lie outside of the scope of the present paper and will be considered in forthcoming research. In
summary, premises of the present model are, first, to admit that it is convenient to change the
expected time of residence in a given class in order to increase fitness, and, secondly, that this
change is made by natural selection choosing optimal transition strategies and, so, assuming that
individuals behave as if they have a ‘roulette wheel in their heads’ [19] when they randomly decide
the transition for the next time period from a set of possible transitions according to the prob-
abilities given by a transition matrix.
We use the Siskin as a model species, because of the fact of being the only species for which a
bimodal status signalling system has been found [2]. The size of the black bib in this species is
highly correlated to dominance [13], and manipulative experiments have shown that it functions
as a reliable signal of dominance [20]. More importantly, the system seems to be independent of
age, since both yearling and adult birds display either small or large badges and both yearlings
and adults may be dominant or subordinate individuals [13]. Birds moult in autumn contour
feathers, and as a consequence, the size of the black bib may increase or decrease in each moult
(JCS, personal observation). We specifically test on the conditions favouring bimodality and
normality in status signalling characters at equilibrium, taking into account population para-
meters such as survival rates, reproductive rate, and aggression rates. As we have already said, we
do that from an evolutionary point of view taking as evolutionary variables the (vector of)
probabilities of moving to other bib-size classes.
The paper is organized as follows. Section 2 introduces the matrix model, first presenting the
linear problem and related concepts and hypotheses, and, afterwards, dealing with the full
problem and the corresponding assumptions. The section ends with a bifurcation analysis to
establish the existence of a unique positive equilibrium of the model and its (local) stability. A
sufficient condition for the global stability of the extinction equilibrium is also given.
In the first part of Section 3 we obtain the evolutionarily singular transition probabilities for a
general transition matrix by means of maximizing the net reproductive number here denoted by R0.
This procedure allows us to see that the evolutionary stability of such a singular strategy of the
adaptive dynamics is, in fact, neutral since it is not a strict local maximum of R0when adopted by
the resident. In other words, R0turns out to be constant (and equal to 1) for any mutant when the
resident, in demographical equilibrium, adopts the singular strategy. Since the transition matrix in
the model is arbitrary, such a neutrality is also the case for the evolutionarily stable growth rates
148 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

obtained by Abrams in [21] under the assumption of a cost free growth in a size-dependent model.
On the other hand, neutral singular strategies have been also obtained by Lebreton et al. in [22,23]
when dealing with the problem of finding optimal dispersal strategies in discrete metapopulation
models, and by Diekmann in [24,25] for the evolution of the timing of reproduction in semel-
parous species.
A singular strategy has the ability to spread as a mutant, property that is also called invading
potential, when a small population of mutants (or invaders) adopting it can spread in a population
at equilibrium where residents adopt a (nearby) strategy. Moreover, in case of neutral ESS-sta-
bility of the singular strategy, as in the present model, invading potential is equivalent to the so-
called convergence stability of the singular strategy (see [26,27]). In the second part of Section 3 we
numerically analyze such a property of the singular strategy. This study is made under particular
assumptions about the survival and reproduction probabilities, which allows to obtain an explicit
expression of the equilibrium when the parameter values define the evolutionarily singular
strategy, even though the model is non-linear. Such an expression will become very useful for the
simulations of invasion events. In particular, we will see that as long as competition effects within
classes are stronger than those among classes (i.e., like-versus-like competition), the singular
strategy turns out to be an evolutionarily stable strategy.
Section 4 is devoted to analyze which conditions on the parameters guaranteeing the existence
of an evolutionarily singular strategy, also imply bimodality in the shape of the equilibrium. We
recall again that this is one of our main purposes since, in case that such conditions exist, it could
explain when bimodality is observed in nature and under which circumstances it appears instead
of the more frequently observed unimodal equilibrium distributions.
Section 5 considers the particular situation in which only one-step transitions are allowed. In
such a case, the transition matrix is tridiagonal and it is possible to obtain an explicit expression of
R0, the net reproductive number. We use it to directly check the neutral stability of the singular
strategy obtained in Section 3 for a general transition matrix.
Finally, Section 6 contains some remarks on the neutrality of the ESSs in optimization models,
on the role of like-versus-like aggression in the so-called social control hypothesis, and on how
bimodality in the size-distribution of badges can arise in species like the Siskin in which the
plumage trait acts as a badge of status but does not have any role in mate choice.
2. Description of the model
2.1. The linear problem
If NðtÞdenotes the n-class distribution vector at time t, the dynamics of a model as the one
described before in a virgin environment (i.e., when competition effects are not present) is given by
the following linear matrix equation
Nðtþ1Þ¼PNðtÞ;
where, as usual, the nnprojection matrix Pdecomposes as a sum of a transition matrix
T0¼ðtijÞplus a fecundity matrix F¼ðfij Þ, i.e., P¼T0þF.
J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181 149

From now on we will assume the following hypotheses on the model ingredients:
(H1). For each time interval (census time, year), the transition probability tij from class jto a class
iis given by
tij ¼sijsj;ð1Þ
where, for each class j,0<sj<1 is the survival probability of a j-class individual, and sij P0is
the probability of moving from class jto class i. Since, per unit of time, any individual at any class
either moves to another class or remains in the same, it follows that
X
n
i¼1
sij ¼1;
for j¼1;...;n. The special form of the transition probabilities given by (1) allows us to express
T0as product of a stochastic matrix Ttimes a survival matrix Sin such a way that the projection
matrix becomes
P¼TS þF;
with T¼ðsijÞand S¼diagðs1;...;snÞ.
(H2). Newborns are assumed to lie in the subordinate class (1-class), i.e., F¼ðfijÞwith f1j¼fj>0
and fij ¼0 for i¼2;...;n,j¼1;...;n. Reproduction occurs at the beginning of the time period
which implies that the contribution of the j-class to the 1-class is given by the term fjNj.
(H3). The projection matrix Pis primitive. Under hypotheses (H1) and (H2) a sufficient condition
for primitivity of Pis siþ1;i>0 for all i(see [28]).
Under the previous hypotheses, Pis a non-negative matrix and, by means of the Perron–
Frobenius theorem, it has a simple and strictly dominant eigenvalue k1>0 with strictly positive
left and right eigenvectors, uand v, respectively (see [29]). Therefore, the dynamics of the model is
given by the so-called Fundamental Theorem of Demography which says that, at the long run, the
total population grows like kt
1and its normalized class-distribution tends to the so-called stable
distribution which is given by the normalized right eigenvector corresponding to k1,v=kvk(see [28]).
Hence, the population will go extinct if k1<1 and grows exponentially without bound if k1>1.
2.1.1. The net reproductive number R0
Another important quantity which, under certain hypotheses, is equivalent to predict the
asymptotic behaviour of the linear model is the so-called net reproductive number, also called basic
reproduction number, and usually denoted by R0in epidemiology and life history theory. This
number is defined as the individual expected production of offspring per lifetime and, when
newborns belong to a single class, the strictly dominant eigenvalue of the matrix R:¼FðIT0Þ1
corresponds exactly to R0.
The equivalence between k1and R0to describe the long term behaviour of linear matrix models
is given by the fact that, under hypotheses (H1)–(H3), one of the following relationships holds:
k1¼R0¼1 or 0 <R0<k1<1 or 1 <k1<R0(see [28,30]). Therefore, the population will go
extinct if R0<1 and grows exponentially without bound if R0>1.
150 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

The matrix Ris called the next generation matrix since it projects the population from one
generation to the next, and its strictly dominant eigenvalue R0gives the growth rate of the
population from one generation to the next (see [29]). Precisely, since the fecundity matrix Fonly
has entries different from zero in the first row, only the first row of Ris non-zero with r1j>0 for
all j. Hence, Rhas a positive simple strictly dominant eigenvalue R0¼r11, and the other ðn1Þ
eigenvalues are equal to zero. In the model, the existence of the matrix Ris guaranteed because the
elements of T0satisfy Pn
i¼1tij ¼Pn
i¼1sijsj¼sj<18j, which is a sufficient condition for the
existence of ðIT0Þ1(see [28]).
Note that the existence of a positive, simple, and strictly dominant eigenvalue R0of the matrix
Ris guaranteed by the fact that only one row of Fis non-zero and, so, it does not depend on the
primitivity of the projection matrix P, i.e., it does not depend on the existence of a positive,
simple, and strictly dominant eigenvalue k1of P. This implies that it is possible to have genera-
tional stability (R0¼1) but, at the same time, permanent oscillations of the normalized class-
distribution which appear when Pis irreducible but imprimitive and k1¼1 is a dominant but not
strictly dominant eigenvalue (see [30] for a discussion and examples).
On the other hand, since the ði;jÞ-entry in ðIT0Þ1gives the expected number of time steps
spent in class iby an individual starting life in class j, the element r1jof Rgives the expected
lifetime production of newborns of an individual starting life in class j. Hence the dominant
eigenvalue R0of Rhas the interpretation given above, namely, the expected number of offspring
produced by an individual during its lifetime, as long as there is a only offspring class. However,
when this is not the case, i.e., there are more than one offspring classes, this interpretation of R0is
no longer valid, though it remains true that R0corresponds to the generation growth rate (see [29]
for a discussion).
To deal with R0, instead of computing the matrix ðIT0Þ1, which is not easy when the
complexity of T0increases, from now on we will consider the system of linear equations
RðIT0Þ¼F:ð2Þ
2.1.2. Reproductive value
The sum of all the contributions to reproduction from stage ito the (last) stage nis the
reproductive value of class i. In matrix population models, the left eigenvector uof the projection
matrix associated to the dominant eigenvalue k1is interpreted as a measure of these reproductive
values [29]. In fact, it is immediate to see that if uis a left eigenvector of Pcorresponding to k1,
u0P¼k1u0, then uis a left eigenvector of Rk1:¼Fðk1IT0Þ1corresponding to the eigenvalue 1.
Therefore, when k1¼1, Rk1¼Rand R0¼1. In this case, u0P¼u0R¼u0which implies that the
vector of reproductive values is proportional to the first row of R, i.e., u0¼aðr11;...;r1nÞ.In
particular, normalizing uby aand since r11 ¼1, it follows that ~
u¼ð1;r12;...;r1nÞis a left
eigenvector of Pand Rcorresponding to the eigenvalue 1. This fact will be used in the next
sections to give an interpretation of the results we will obtain.
2.2. The non-linear problem
To introduce the effects of population density (i.e., non-linearities) in structured populations
models it is usually assumed a dependence of the vital rates (mortality, survival and reproduction)
J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181 151

on certain population variables in such a way that, when their values are prescribed, models
become linear, that is, individuals are independent from each other. Such variables are sometimes
called interaction variables,/k, and constitute the environmental feedback: environmental con-
ditions are determined by the population distribution and, in its turn, this biotic environment
determine the interactions among individuals (see [31]). When symmetric competition is assumed,
these interaction variables /kare weighted total population sizes and they are the same for any
member of the population. However, when dealing with hierarchical models in which asymmetric
competition among individuals is assumed, the interaction variables can be weighted sums of the
number of individuals of only certain classes, i.e., they can be functionals of the population class-
distribution NðtÞ, and moreover they are different for different classes (see [28] for examples).
In order to introduce the environmental feedback in the population dynamics of the model, we
will consider density effects in both fecundity elements and survival probabilities. In particular we
will assume the following hypotheses:
(H4). The i-class survival probability, si, and the i-class fecundity rate, fi, are continuous and
strictly decreasing functions of /iðNÞwith /iðNÞ 6¼ /jðNÞif i6¼ j, and tending to 0 as /iðNÞ!1.
Moreover, for any class i, the inherent or maximal survival probabilities and the inherent or
maximal fertilities, denoted by s0
iand f0
irespectively, satisfy the inequality
s0
iþf0
i>1;i¼1;...;n:ð3Þ
(H5). The functions /iðNÞ,i¼1;...;n, are weighted sums of the number of individuals at each
class. More precisely, /iðNÞare of the form
/iðNÞ¼X
n
j¼1
wijNj;i¼1;...;n;
with W¼ðwijÞanon-negative invertible matrix.
In terms of the projection matrix, (H4) means that the density effects are included in the
fecundity and survival matrices and the model becomes non-linear and reads
Nðtþ1Þ¼ TSðNðtÞÞ
ðþFðNðtÞÞÞNðtÞ;ð4Þ
where the dependence of Sand Fon Nis by means of U¼ð/1;...;/nÞP0, the vector of
interaction variables.
In turn, the hypothesis on Win (H5) guarantees the existence and uniqueness of a population
distribution ~
Nfor a given Usince then ~
N¼W1U. Roughly speaking, this is equivalent to say
that, given certain environmental conditions as described by a vector of interaction variables U,
there exists only one population distribution causing them.
The values of the weights wij determine what sort of environment is experienced by each
dominance class. For instance, if, for a given i, one assumes that coefficients of the weighted
averages satisfy wi1<<win , this means that contributions of individuals in higher classes to
the density effects experienced by individuals in class i(the environment of the i-class) are more
important than those of individuals in lower classes.
In fact, the choice of the weights wij is a way to introduce dominance hierarchy in the social
structure of the population, i.e., a disproportionate share of the resources by the dominants, in our
152 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

case, the individuals of the biggest bib-size class. To see this fact, let us consider two dominance
classes, iand j, and the corresponding reciprocal contributions to the environment experienced by
them, namely, wij and wji. Whenever wij <wji , the contribution to the experienced environment by
class iby individuals of class jis lower than that of individuals of class ito the environment of
class j. In this case, class iis dominant to class j.
Such a dominance hierarchy appears many times by means of aggressive interactions among
individuals of the same class (intra-class aggressions) and among individuals of different classes
(inter-class aggressions). In such cases, wij can be thought of as the mean number of aggressive
interactions that an individual of class isuffers from an individual of class jper time interval.
Normalizing the number of aggressive interactions by the total number of per capita aggressions
experienced by an i-class individual per time unit (Pjwij), the entry wij of Wwould represent the
probability that an individual of class isuffers an aggression from an individual of class jper time
period and, so, it satisfies 0 6wij 61 with Pjwij ¼1 for i¼1;...;n. In such a case, /iðNÞgives
the expected number of aggressions experienced by an individual of class iduring a time interval
when the class distribution in the population is given by N, which is clearly a measure of the
competition effects (environment) experienced by those members belonging to the i-class. Under
this interpretation of wij, individuals of class idominate individuals of class jif the (mean) number
of aggressions suffered by an i-class individual per time interval and per j-class individual is lower
than the (mean) number of aggressions suffered by a j-class individual per time interval and per
i-class individual.
2.2.1. Stationary solutions
A positive stationary solution or equilibrium of the model is a positive time-invariant solution
of the ecological dynamics governed by (4), i.e., a solution N>0 of the non-linear equation
N¼PðNÞN¼TSðNÞðþFðNÞÞN:ð5Þ
Since, under (H1)–(H3), the projection matrix at the equilibrium, PðNÞ, is assumed to be non-
negative and primitive, to say that N>0 is an equilibrium of (4) is equivalent to say that Nis a
positive right eigenvector of PðNÞcorresponding to the eigenvalue 1. In particular, under (H3),
this implies that k1¼1 has to be a strictly dominant eigenvalue of PðNÞwith a strictly positive
right eigenvector vwhich is proportional to N, and with a strictly positive left eigenvector u,
proportional to the first row of the next generation matrix RðNÞ, which is also a left eigenvector
of RðNÞbelonging to the eigenvalue 1. Hence, it also follows that R0ðNÞ¼1 (see Subsection
2.1.2).
On the other hand, N0is always a solution to (5) and, so, an equilibrium of (4), and it
satisfies the following
Lemma 1. Under hypotheses (H4) and (H5) the extinction equilibrium N0of (4) is always
unstable.
Proof. As siand fiare assumed to be continuous and strictly decreasing functions of /iðNÞ, there
exists e0>0 such that, for all i,siðeÞþfiðeÞ>18e<e0. Let us consider an arbitrary initial
population Nð0Þsuch that /iðNð0ÞÞ <e0for all i. Then, adding up the nequations of (4), it follows
that, for t¼1,
J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181 153

X
i
Nð1Þ¼X
i
½siðNð0ÞÞ þ fiðNð0ÞÞNið0Þ>X
i
Nið0Þ;
which means that the total size of small enough populations increases with tas long as
their weighted sizes /iðNðtÞÞ are lower than e0for all i. Hence the instability of 0follows, i.e.,
k1ð0Þ>1. h
This fact allows us to prove the following
Theorem 1. Under hypotheses (H1)–(H5) there exists an equilibrium N>0of (4).
Proof. Under (H4) all the entries of PðNÞare strictly decreasing functions of /ifor some i, and
tending to zero as /i!1for all i. Moreover, under (H5), such entries tend to zero as Ni!1for
all i. Hence, from the Perron–Frobenius theorem, it follows that the spectral radius of PðNÞ,
qðPðNÞÞ ¼ k1ðNÞ, decreases (strictly) with /ifor any i2[1] with k1ðNÞ!0asNi!1for all i
since then kPðNÞk ! 0 and the spectral radius satisfies qðPÞ6kPk(see [30]). Finally, from the
previous Lemma, k1ð0Þ>1 and, hence, the existence of a value N>0 such that k1ðNÞ¼
1<k1ð0Þfollows. h
Theorem 2. If (3) is replaced by s0
iþf0
i<18iin (H4), the extinction equilibrium is globally stable
under hypotheses (H1)–(H5).
Proof. The stability of the extinction equilibrium trivially follows because
k1ð0Þ<kPð0Þk1:¼max
jX
i
ðsijs0
jþf0
jÞ¼max
jfs0
jþf0
jg<1:
Moreover, using similar arguments as in the proof of the previous theorem, it follows the non-
existence of an equilibrium N>0 since there is no vector N>0 for which k1ðNÞ¼1. In fact,
monotonicity hypotheses in (H4) and the condition on the parameters of the lemma imply global
extinction, i.e., NðtÞ!0as t!1for all Nð0Þ>0, since k1ðNÞ<k1ð0Þ<1 for all N>0 (see [28]
for details). h
If s0
iþf0
i>1 for some i, while s0
jþf0
j<1 for j6¼ i, the instability of the extinction equilibrium
is not guaranteed and a detailed analysis is needed to establish conditions for the existence of an
equilibrium N>0. In particular, note that f0
1>1 is a sufficient condition for the instability of
the trivial equilibrium since qðPð0ÞÞ >qðFð0ÞÞ ¼ f0
1.
As usual in applications of matrix models in population dynamics, we can use R0ð0Þas a
bifurcation parameter since it follows that 0 <R0ð0Þ<k1ð0Þ<1 if s0
iþf0
i<18i, and
1<k1ð0Þ<R0ð0Þif s0
iþf0
i>18i. In fact, for continuity with respect to the parameters of the
model, R0ð0Þattains the critical value Rc
0¼1 for s0
iþf0
i¼1(i¼1;...;n), even though other
combinations of parameters giving R0ð0Þ¼1 are possible.
Note that, under (H4) and (H5), the entries of PðNÞare non-increasing functions of Ni,
i¼1;...;n, which are strictly decreasing for at least some i. Hence, from bifurcation theory in
discrete dynamical systems (see Theorems 1.2.5 and 1.2.6 in [28]), the next result follows:
154 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

Theorem 3. (Supercritical bifurcation): Under hypotheses (H1)–(H5) and assuming that si¼
siðs0
i;NÞand fi¼fiðf0
i;NÞare Ck-functions, kP2, there exists a continuum Cþof equilibrium pairs
ðR0ð0Þ;NÞwith ð1;0Þ2Cþand containing only positive equilibria for R0ð0Þ>1. Moreover, in a
sufficiently small neighbourhood of the bifurcation point ð1;0Þ, there exists a unique equilibrium
N>0for each value R0ð0Þ>1which is locally asymptotically stable, whereas the extinction
equilibrium is unstable.
Corollary. Under the hypotheses of Theorem 3, the equilibrium N>0of (4) is (locally) asymp-
totically stable for s0
iþf0
i>1near 1 for all i, whereas the extinction equilibrium is unstable.
Remarks
(i) According to the theory of positive matrices (see, for instance, [29], Chapter 4), if there exists a
non-negative stationary solution Nwhich is not strictly positive (i.e., N
j¼0 for some j), this
means that (H3) does not hold and, in particular, that PðNÞis reducible and, so, the existence
of a strictly dominant eigenvalue is not guaranteed. It is well-known that this fact depends on
the pattern of PðNÞand, since the pattern of FðNÞis given and SðNÞis diagonal, it only
depends on the pattern of T. In such a situation, however, the vector of reproductive values,
which is the left eigenvector of Rwhich belongs to R0¼1, is still strictly positive as R0is a
strictly dominant eigenvalue of R. When this is the case, it follows that qðPÞ¼R0¼1 or
1<qðPÞ6R0or 0 6R06qðPÞ<1 (see [30]).
(ii) One can introduce R0explicitly in the model (4) by scaling the fecundity matrix to R0. In such
a case, we have fij ¼R0wij and P¼TS þR0Wwith W¼ðwijÞ. With this normalized fecundity
matrix, it follows that qðTS þWÞ¼1 if R0>0 (see [28,30]).
3. Adaptive dynamics
3.1. Preliminaries
From an evolutionary point of view, we consider that individuals are characterized by a (multi-
dimensional) variable specifying relevant aspects of the life history (according to our interests) in
which they may differ. Consequently, the values of this variable are submitted to natural selection
and, for this reason, it is called evolutionary trait or variable. A particular value of it is called type
or, more generally, strategy. In our case, this evolutionary variable is given by a vector gof
dimension less than or equal to nðn1Þand formed by a subset of transition probabilities sij .
Due to the existence of a linear restriction on the elements of each column of T(namely,
Pisij ¼18j) and in order to have a well-posed optimization problem (see below), the set of
possible strategies has to be restricted to a subset of admissible strategies.
Definition. A vector gof dimension l6nðn1Þ, is said to define an admissible strategy if
(a) The whole of the (non-zero) entries of any column of T¼ðsij Þis not included among the com-
ponents of g,
J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181 155

(b) For every i2f1;2;...;ngthere exists at least one j2f1;2;...;ngsuch that sij >0 and either
sij is a component of gor there is another element skj,k6¼ i, of the same column of Twhich is
a component of g.
The first condition excludes linear dependencies among the elements of gand, hence, it allows
us to do without Lagrange multipliers in the optimization procedure below. The second condition
assures that all the (dominance) classes are affected by the transitions defining the strategy vector
g. For instance, this is the case when the strategy is given by the growth rates in discrete size-
structured models [21], or when the strategy is given by the dispersal rates in discrete metapop-
ulation models with a ‘full’ dispersal matrix [22,23]. In general, however, one has to be precise
with respect to the choice of the transitions in order to have a well-defined optimization problem
(see the proof of Theorem 4 below).
More precisely, in a standard size-structured matrix model we have the following transition
matrix
T¼
1g100... 00
g11g20..
.
00
0g2
..
...
.
00
00
..
...
.
00
.
.
..
.
..
.
...
.
1gn10
000... gn11
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
;
with g¼ðg1;...;gn1Þthe only admissible strategy. In fact, if nP3, the strategy g¼ðg1;...;gn2Þ
is not admissible because there is no component of the strategy vector gdirectly affecting the class
n. Another example of an admissible strategy is given by g¼ðs1j;...;sn1;jÞwhere T¼ðsij Þis a
nntransition matrix without zero entries, since then the column jis also ‘full’ and, so, its entries
define transitions from class jto any other class. Finally, another example is given by the case we
have called ‘the dilemma of the subdominants’ in Subsection 3.3. In this case, Tis a 3 3
primitive matrix with the second column being the only full column and with g¼ðs12;s32 Þ.In
both examples, any change in gdirectly affects all classes in the life histories of the individuals.
In the following, we will make a rather restrictive assumption on the (ecological) dynamics of
the model, namely, the existence of a globally asymptotically stable equilibrium N>0 for all
positive initial condition Nð0Þ. This assumption allows for supposing that, before the arrival of a
mutant, the resident population has come to a steady state, which is the usual setting in most of
the analysis in adaptive dynamics [24,27,31–33]. In fact, though it is well-known that discrete
models usually show complicated dynamics under a suitable selection of parameters [28], espe-
cially when exponential non-linearities are considered as in Subsection 3.3, the previous
assumption of global stability of Nis in agreement with what we have observed in all the
numerical simulations as long as (3) holds with its LHS close enough to 1 for all i(recall that we
already saw in Section 2 that, in this case, N>0 appears as a consequence of a supercritical
bifurcation from the extinction equilibrium). Otherwise, assuming high enough fecundity rates, N
becomes unstable and periodic orbits appear.
Since the resident population is assumed to be at equilibrium, we can use R0as a fitness measure
to analyze the success of invasion events. For a fixed l6nðn1Þ, let Xbe the trait space, i.e., the
156 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

set of all possible non-negative values of the strategy vector g. The adaptive dynamics (AD) of the
model will be given by trajectories in Xdefined by trait substitution sequences. Steady strategies of
the AD are called evolutionarily stable (or steady) strategies (ESS) [25,26]. By assuming infini-
tesimal mutational steps, the AD is described by a deterministic limit equation called canonical
equation [27,33,34]. This equation accounts both for the mutational process – described by a
covariance (or mutational) matrix – and for the action of natural selection – by means of the so-
called fitness gradient. Rest points of this equation correspond to singular strategies.
It is well-known from the modern theory of adaptive dynamics that the (local) ESS-property of
a singular strategy, i.e., immunity of the resident adopting it in front of any new (nearby) strategy,
called mutant or invader, is not a dynamic concept. The reason is that its definition does not imply
its evolutionary attractiveness, i.e., it does not imply that the substitutions of types (strategies)
occurring during the course of evolution lead to the establishment of such a singular strategy. This
fact, indeed, is related to the concept of convergence stability (a singular strategy is said to be
(locally) convergence stable if a resident type having a (nearby) strategy can be invaded by mu-
tants adopting an even closer strategy). In other words, convergence stable singular strategies
correspond to asymptotically stable equilibria of the canonical equation of the AD [27,34]. This
property and the (local) ESS-property are two totally independent stability concepts that can
occur in any combination. In fact, a convergence stable strategy which is also evolutionarily stable
is called continuously stable strategy [26].
Another property that a singular strategy can have is the ability to spread in populations
adopting slightly different strategies, a property that is also called invading potential. In particular,
a singular strategy that is evolutionarily stable and that has the ability to spread as mutant
strategy is also necessarily convergence stable, i.e., such a singular strategy is an attractor of the
corresponding adaptive dynamics.
In a one-dimensional trait space, this equivalence of properties is easily seen from the so-
called pairwise invasibility plot [26]. For multidimensional traits and using R0as a fitness
measure, a sufficient condition for a singular strategy gss to be convergence stable is that the
following matrix
o2R0ðginv;gres Þ
oginv
ioginv
j
"#
ginv¼gres ¼gss
o2R0ðginv;gres Þ
ogres
iogres
j
"#
ginv¼gres ¼gss
ð6Þ
is negative definite, where gres denotes the strategy adopted by the resident which is assumed to be
at population dynamical equilibrium, and ginv denotes the strategy adopted by the invader [27,34].
Therefore, since evolutionary stability of the singular strategy implies that the first matrix is
negative definite, and the invading potential implies that the second matrix has to be positive
definite, the condition for the convergence stability is fulfilled when the singular strategy has both
previous properties (see [27,34] for details).
In matrix games, for instance, a restriction appears in the combination of these properties as the
fitness function is linear in the mutant strategy. A similar situation occurs in population dynamics
if the fitness function becomes constant for any mutant strategy when the resident, in demo-
graphical equilibrium, adopts the singular strategy. In both cases, the singular strategy does not
define a strict maximum of the fitness measure. Hence, it follows that the condition for having an
invading potential implies the convergence stability of the singular strategy as the first matrix in (6)
J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181 157

is equal to the null matrix and, then, invading potential is a sufficient condition for convergence
stability.
Note that, since the elements of the transition matrix T0are given by (1), at the equilibrium N,
T0is a function of both the strategy gand the distribution population vector at equilibrium N,
which is itself a function of the resident strategy. More precisely, the dependence of T0on ginv and
gres has the form
T0¼TðginvÞSðNðgresÞÞ:
Therefore, as R0is the dominant eigenvalue of R¼FðIT0Þ1, it follows that the dependence
of R0on the second argument in (6) is through N, i.e.,
R0ðginv;gres Þ¼R0ðginv;Nðgres ÞÞ:
3.2. Singular strategies
The success of an invasion process can be analyzed by means of the ESS concept. The fitness of
an invader adopting ginv at a demographic equilibrium of a resident adopting gres is given by
R0ðginv;Nðgres ÞÞ where, as before, NðgÞdenotes the equilibrium distribution of a resident
adopting the strategy g. A strategy gess is said to be an ESS if the mapping
g!R0ðg;NðgessÞÞ
is maximal at g¼gess (see [31]). That is, a (local) ESS is a strategy such that, when it is adopted by
the resident, no mutant or invader with another (nearby) strategy can spread.
In general, if a fitness measure is differentiable with respect to g, we say that a strategy
gss 2intXis a singular strategy if the gradient of this fitness measure evaluated at the demographic
equilibrium of resident adopting it is equal to zero at this strategy (in our case,
rgR0ðg;NðgssÞÞjg¼gss ¼0). Therefore, since by definition of ESS the fitness measure (here R0) has a
local maximum at such an strategy, the fitness gradient at an interior ESS is zero, which means
that an interior ESS is a singular strategy of the adaptive dynamics. When this maximum is not
strict, then there exists a neighbourhood of the singular strategy UðgssÞsuch that all strategies in it
render mutants (invaders) with the same fitness as the resident, i.e., R0ðg;NðgssÞÞ ¼ 1 for all
g2UðgssÞ. When this is the case, the singular strategy presents what is called a neutral evolu-
tionary stability.
Of course, one can ask about the existence of strategies at the boundary of the trait space, oX,
for which the fitness gradient needs not to be zero but, even so, the fitness measure has a local
maximum, i.e., boundary ESSs. To answer this question one has to compute the signs of the
components of rgR0ðg;NðgbÞÞjg¼gbwith gb2oXin order to see if oXacts as an attractor or as a
repellor of the AD (see [25] for an example in a one-dimensional trait space). However, numerical
simulations of the two-dimensional AD of the model show that, as long as an interior singular
strategy has invading potential, such a property holds for any (nearby or not) resident strategy.
This fact, plus the global and neutral evolutionary stability of the singular strategy (see below),
amounts to the global attractiveness of such a strategy. Otherwise, it would exist a one-dimen-
sional path in Xalong which the singular strategy is able to spread but is not convergence stable,
which would contradict the equivalence between such properties in one-dimensional AD when the
singular strategy is globally evolutionary neutral [26]. Therefore, conditions for an invading po-
158 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

tential will guarantee the global convergence stability of the interior ESS. For this reason, from
now on we will only consider interior fixed points of the AD.
Theorem 4. Under hypotheses (H1)–(H5) and assuming the existence of a globally stable positive
equilibrium of the non-linear model (4), there exists a set of evolutionarily singular strategies of the
adaptive dynamics associated to the model which define a unique equilibrium population Nðgss Þ.
Moreover, all these singular strategies are characterized by a neutral evolutionary stability.
Proof. Let skm be a component of an admissible strategy gand consider another element of the
same column slm 6¼ 0 with l6¼ k. We know that such an element exists since, otherwise, skm ¼1.
Moreover, let NðgssÞbe the equilibrium defined by the singular strategy gss , and s
i¼siðNðgssÞÞ
and f
i¼fiðNðgssÞÞ (i¼1;...;n). Using that slm ¼1Pq6¼lsqm , and differentiating (2) with re-
spect to skm, one obtains the following linear system in or1i
oskm,i¼1;...;n:
X
n
i¼1
ðdim timÞor1i
oskm
¼s
mðr1kr1lÞ;ð7Þ
X
n
i¼1
ðdij tijÞor1i
oskm
¼0ðj6¼ mÞ;ð8Þ
where dij is the Kronecker delta.
Since, at the singular strategy, or11=oskm ¼0, (8) is a homogeneous system of n1 equations
with n1 unknowns. On the other hand, since ITis non-singular and the matrix of (8) is
obtained by deleting row mand column 1 of the transpose of IT, it follows that (8) is a non-
singular homogeneous system and, so, it only has as a solution the trivial one, that is,
or1i
oskm
¼0;i¼2;...;n:ð9Þ
Hence, in order to fulfill (7), it follows that r1k¼r1lfor all l6¼ ksuch that slm 6¼ 0.
Computing the derivative of (2) with respect to the rest of the components of gand using the
condition of singular strategy, namely,
for all sij in the strategy g;oR0
osij
¼or11
osij
¼0;
(7) and (8) imply that, at the equilibrium determined by the singular strategy, the elements of the
first row of Rsatisfy
r11 ¼r12 ¼¼r1n¼1:ð10Þ
Notice that the fact that gis an admissible strategy guarantees that this condition holds for all
r1i. Otherwise, some of the elements of the first row of Rwould remain undetermined. This means
that, if gwere not admissible, there would be unaffected reproductive values by the maximization
of the fitness measure or, in other words, there would be unaltered stages of the life history of the
individuals when we change the strategy g, which is clearly unsatisfactory.
Substituting (10) in (2) it follows immediately that, to be a singular strategy, ghas to render an
equilibrium Nfulfilling the condition
s
iþf
i¼1;i¼1;...;n:ð11Þ
J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181 159

Note that (H4) and (H5) guarantee the existence of a unique equilibrium N>0 satisfying (11),
and that the number of strategies rendering such an equilibrium depends on the dimension of
strategy. In particular, if the dimension of gis less than nthen the singular strategy will be un-
iquely determined from (5) and (11).
To see whether g!R0ðg;NðgssÞÞ is maximal at g¼gss or not, we can differentiate (7) and (8)
with respect to skm,skm a component of the strategy vector g, as many times as we need. Using (9),
it follows directly that
oiR0
osi
km
ðgss;Nðgss ÞÞ ¼ 0;i¼2;3;...
for any skm in the strategy g. That is, the singular strategy does not define a strict maximum of the
mapping g!R0ðg;NðgssÞÞ. In fact, we can easily obtain more information about this mapping by
computing R0ðginv;Nðgss ÞÞ with ginv 6¼ gss.
Eq. (2) corresponding to an invader adopting a strategy ginv , which determines a transition
matrix Tinv, when the resident adopts gss is
RinvðITinv SÞ¼F;
where Sand Fare the survival and the fecundity matrices at the equilibrium NðsssÞ, respec-
tively. However, since Tinv ¼ðsinv
ij Þis a stochastic matrix, it follows that, for all j,Pisinv
ij s
j¼
sjðNðgssÞÞ. Therefore, in order to fulfill (2) and since s
iand f
isatisfy (11), Rinv must be equal to
R, the matrix Rof the resident adopting the singular strategy, which implies that
R0ðginv;Nðgss ÞÞ ¼ 1;
for any (nearby or not) admissible strategy ginv adopted by the invader. That is, the mapping
g!R0ðg;NðgssÞÞ is constant and equal to 1. h
In other words, we have seen that under the equilibrium conditions set by a resident population
adopting a singular strategy gss, any mutant population of small size adopting a (nearby or not)
strategy ginv will be also in equilibrium, i.e., all mutant strategies perform equally well.
3.2.1. Reproductive value at the singular strategy
According to the characterization of the equilibrium defined by the singular strategy and since
the sum of the ith column of Pat the singular strategy is equal to s
iþf
i, it follows that 10P¼10.
So, the normalized vector of reproductive values ~
uis equal to 1, which is equivalent to say that
r1i¼1 for all 1 6i6n(see Section 2). That is we again obtain the same conclusion as before: at
the equilibrium imposed by the singular strategy, all the dominance classes have the same
reproductive value which is equal to 1.
Another way to look at the neutrality among dominance classes is by rewriting (11) as
f
i
1s
i
¼1;i¼1;...;n;ð12Þ
which says that, at the equilibrium defined by a singular strategy gss , the expected life-time pro-
duction of offspring of an individual that remains always in the same class is equal to 1 for any
160 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

class. In other words, there exists equality of fitness among all dominance classes, which is equal
to the fitness corresponding to the optimal strategy given by gss . From a biological point of view,
this explains the neutral evolutionary stability of the singular strategy: under the ESS conditions,
all dominance classes are equally suitable for staying alive and reproduce.
Remarks. The neutrality of mutant strategies when the resident plays the singular strategy already
obtained is completely analogous to the one obtained in [22] for evolutionarily stable dispersal
strategies in the context of metapopulation theory when there is no cost of dispersal. In this case,
the reproductive value of every local population (habitat patch) is the same when the evolutionary
stable dispersal strategy is adopted in the whole metapopulation and gives rise to the so-called
ideal free distribution among habitat patches (see [19]). In a different context, namely, the evo-
lution of the timing of reproduction in a semelparous individuals, a neutrality result is also found
in [24,25]. In particular, in [25] it is claimed that neutral singular strategies is what one should
expect when a model contains more than one interaction variable, as in our case, and, so, that
neutrality has to be used as ‘a guiding principle for the search of singular points’. In fact, this
could be the key point for overcoming the apparent incompatibility between matrix games and
(density-dependent) optimization models regarding the outcome of evolutionary processes that
has been pointed out by some authors as, for instance, in [27] (see Section 6 for a discussion).
3.3. Spread of the (mutant) singular strategy and convergence stability
Let us now analyze by means of numerical simulations the conditions for the ability to spread
as a mutant, of a singular strategy of the adaptive dynamics of the model. First of all, since we
have already proved that, at the demographical equilibrium Ndetermined by the singular
strategy gss,R0ðg;Nðgss ÞÞ ¼ 1 for all g, we are in the situation in which the ability to spread is
equivalent to being convergence stable. Therefore, we only need to determine the conditions for
the invading potential of the singular strategy to have also guaranteed its convergence stability.
More precisely, for the case of having three dominance classes (n¼3), we will see numerically
under which conditions gss has the ability to spread as mutant under the following choices of siðNÞ
and fiðNÞ,i¼1;2;3,
siðNÞ¼s0
iexpðci/iðNÞÞ;fiðNÞ¼f0
iexpðci/iðNÞÞ;ð13Þ
where, for each class, s0
iand f0
isatisfy hypothesis (H4), and ci>0 denotes the sensitivity coef-
ficient to competition (sometimes also called competition coefficient). Such a form is sometimes
called overcompensatory relation (see [28] for other non-linearities in the vital rates). Moreover, we
will consider the following weighted averages of the population distribution /iðNÞ:
/1ðNÞ¼ w1N1þ1
2ð1w1ÞðN2þN3Þ;0<w161;
/2ðNÞ¼ w2N2þð1w2ÞN3;0<w261;
/3ðNÞ¼ ð1w3ÞN2þw3N3;0<w361:
According to (H5) in Section 2.2, Whas to be an invertible matrix and, hence, in this case this
means that detðWÞ¼w1ðw2þw31Þ 6¼ 0. So, we assume w2þw36¼ 1 and w1>0.
From this choice of the matrix W¼ðwijÞit could seem that we are restricting ourselves to a
very particular case, namely, a case in which Pjwij ¼1. However, it is easy to see that, in our case
J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181 161

as well as in many non-linearities appearing frequently in structured population models (see [28]
for examples), this is not a restrictive assumption since the competition coefficient ciin the
expressions of siðNÞand fiðNÞallow us to consider /iðNÞas normalized weighted sums, i.e.,
weighted averages of the population, without loss of generality. More precisely, if Pjwij ¼
w0
i>0, i¼1;2, then we can write
ci/iðNÞ¼ciX
j
wijNj¼ciw0
iX
j
wij
w0
i
Nj¼~
ciX
j
~
wijNj¼~
ci
~
/iðNÞ
where Pj~
wij ¼1 for all i. Note that, in this normalized case, the competition coefficient ~
cihas a
nice biological interpretation since is proportional to the number of per capita aggressive inter-
actions of an i-class individual per time period, w0
i.
On the other hand, in the previous choice of wij it is assumed a sort of hierarchy in the social
structure since the weight of the subordinates in the weighted averages of the other classes is zero
(w21 ¼w31 ¼0), while all classes appear in /1ðNÞ. In terms of aggressive interactions, this choice
assumes that the expected number of aggressive encounters in the subdominant and dominant
classes is independent of the subordinates while the expected number of aggressive encounters
suffered by the latter depends on the dominant and subdominant classes. Moreover, according to
the description of the dominance hierarchy given in Section 2.2, if class 3 has to be the dominant
one then w2<w3. Finally, w1>0 says that subordinates compete among themselves for the
available resources ‘left’ by the dominant class.
Note that, under the choice of the survival rates and fecundities given by (13), and even though
the model is non-linear, the equilibrium ðN
1;N
2;N
3Þdetermined by the singular strategy can be
explicitly obtained because, from (11), it follows that Nis the unique solution of the linear system
w1ð1w1Þ=2ð1w1Þ=2
0w21w2
01w3w3
0
@1
A
N1
N2
N3
0
@1
A¼
c1
c2
c3
0
@1
Að14Þ
with cibeing positive constants given by
ci¼1
ci
lnðs0
iþf0
iÞ:ð15Þ
In particular, from this system it follows that survival probabilities and fecundities at this equi-
librium are given by
s
i¼s0
i
s0
iþf0
i
;f
i¼f0
i
s0
iþf0
i
;i¼1;2;3:
Hence, the equilibrium is given by
N
1¼c11
2ð1w1ÞðN
2þN
3Þ
w1
;
N
2¼c2w3c3ð1w2Þ
w2w3ð1w2Þð1w3Þ;
N
3¼c3w2c2ð1w3Þ
w2w3ð1w2Þð1w3Þ;
ð16Þ
162 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

and it is well-defined since w1>0 and w2þw36¼ 1 implies w2w3ð1w2Þð1w3Þ 6¼ 0 (in fact,
w2þw3<ð>Þ1 if and only if w2w3<ð>Þð1w2Þð1w3Þ).
At this point, it is important to realize three important facts. First, the assumption of a dom-
inance hierarchy in the structure of the population and, so, to consider /1ðNÞ 6¼ /2ðNÞ 6¼ /3ðNÞ,is
anecessary condition for the existence of a unique positive solution to (14). In other words, the
existence of an optimal transition strategy essentially depends on the fact that (11) defines a
unique equilibrium N>0 and, under the present choice of siand fi, this is only possible if each
dominance class has its own environmental conditions. Otherwise, when the dimension of the
environment (i.e. the number of interaction variables) is less than the number of dominance
classes, it is not possible to find out a singular strategy according to our procedure because there
exists a continuum of solutions to (14) and, hence, the hypothesis of the theorem about the
existence of a globally stable equilibrium of the model is not fulfilled.
Secondly, the equilibrium Ncorresponding to the singular strategy gss is independent of the
particular choice of the strategy vector g. In particular this means that, when the dimension of gis
less than n, then the singular strategy is uniquely determined from (5) once Nis replaced by (16).
Otherwise, i.e. when the dimension of gPn, then there exists a continuum of strategies defining
the same equilibrium Nand, so, there does not exist a unique singular strategy but a (infinite) set
of singular strategies defining the same environmental conditions f/ig.
Thirdly, according to (16), it is possible to have a component of the equilibrium defined by a
singular strategy equal to zero. However, the corresponding values of the (internal) singular
strategy are non-admissible for some of its components (see, for instance, expressions (18) and
(19) for gin the examples below when N
2¼0). The only possibility of having N
2¼0 (or N
3¼0)
is that s21 ¼s23 ¼0 (or s31 ¼s32 ¼0) in T, but this amounts to a reducible projection matrix and,
so, (H3) is not fulfilled. Indeed, any strategy with these values lies on the boundary of the trait
space and, hence, the fitness gradient needs not be zero at such a strategy.
On the other hand, in order to get some rough feeling about possible conditions for the spread
of gss as a mutant strategy, notice that ðN
2;N
3Þis given by the intersection point of the straight
lines defined by the second and third equation in (14). Clearly, their relative position in R2
þis
determined by the relationships among the parameters s0
i,f0
i, and wi(i¼2;3) since, depending on
them, the lines may or not may intersect in the interior of R2
þ. If they do, then for N>0itis
required that
1w2
w3
<ð>Þc2
c3
<ð>Þw2
1w3
;if w2þw3>ð<Þ1:ð17Þ
Although the (ecological) stability of Nis the same in both cases as the bifurcation from ð1;0Þ
does not depend on the condition w2þw3>1, it seems likely that the convergence stability of the
singular strategy depends on satisfying or not this condition.
3.3.1. Two case studies
Having in mind that our final goal is to give an explanation of the observed bimodal distri-
butions of badges of social status, we have concentrated our study on what we have called the
dilemma of the subdominants. The idea is to consider the following tridiagonal transition matrix
T¼ðsijÞof order 3
J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181 163

1s21 s12 0
s21 1s12 s32 s23
0s32 1s23
0
@1
A
and to focus our analysis on a strategy ghaving as components the transition probabilities of the
second class, the subdominant class, to see which are their optimal values. Precisely, we will
consider as evolutionary variable the vector g¼ðs12;s32 Þ. Notice that this is an admissible strategy
since si26¼ 0 for i¼1;2;3. However, it is important to realize that the equilibrium Ncorre-
sponding to a singular strategy gss is independent of the particular choice of gsince Nis obtained
from condition (11) which does not depend on g. To illustrate other possibilities of evolutionary
variables and due to its own importance, we also consider the standard size-structured model with
the probabilities of growth defining the evolutionary variable, i.e., with g¼ðs21;s32 Þ.
For the case of the dilemma of the subdominants, it follows from the equilibrium equations
with a projection matrix of order 3 that
sss
12 ¼ð1s11Þs
1N
1ð1s
2ÞN
2ð1s
3ÞN
3
s
2N
2
;sss
32 ¼ð1s33s
3ÞN
3
s
2N
2
;ð18Þ
where s11 and s33 are fixed beforehand and s22 ¼1s12 s32.
Similarly, for the standard size-structured model, it follows
sss
21 ¼ð1s
2ÞN
2þð1s
3ÞN
3
s
1N
1
;sss
32 ¼ð1s
3ÞN
3
s
2N
2
;ð19Þ
since, in this case, s12 ¼0 and s33 ¼1.
3.3.2. Explanation of the simulations
Once the equilibrium distribution N
ss determined by the singular strategy is obtained from (16)
for a given choice of the parameter values ðwi;ci;s0
i;f0
iÞ, the value itself of the singular strategy gss
is computed, depending on the analyzed case, from (18) or from (19). Then it is computed, firstly,
the equilibrium distribution Nof a resident population adopting a strategy gres different from gss,
and, secondly, the value of R0of the gss-mutant in such a resident population (i.e., the dominant
eigenvalue of Rðgss;Nðgres ÞÞ). Although other forms for siand fihave been also considered, no
significant differences have been observed with respect to the results below.
A dominant eigenvalue Rss
0of the next generation matrix Rðgss;Nðgres ÞÞ greater than 1 indicates
an initial spread of the mutant adopting the singular strategy gss under the conditions set by the
resident equilibrium population Nadopting gres 6¼ gss (i.e., an initially successful invasion).
Conversely, if Rss
0<1, the number of gss-mutants decreases and, hence, the strategy can not spread
under such environmental conditions. Notice that use of Rss
0as a measure of the success of an
invasion is completely equivalent to use of the dominant eigenvalue of the projection matrix
Pðgss;Nðgres ÞÞ under the hypotheses of the model (see [28], Theorem 1.1.3).
Figs. 2 and 3 show some results for the spread of mutants adopting the singular strategy gss
defined by the transition probabilities s12 and s32 from the middle class (subdominant class), and
with s11 and s33 being fixed. In this case, there is the restriction s12 þs32 61, the equality being
fulfilled when s22 ¼0. Figs. 4 and 5 show the corresponding results of the spread of gss-mutants
for the standard size-structured model. In this case, g¼ðs21;s32 Þ. In all the simulations, ci¼0:01
164 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

and the parameters of the vital rates are specified. All the numerical experiments follow one-
dimensional paths in a two dimensional g-space. Along these paths, one of the components of the
(resident) strategy gres
i(i¼1;2) remains constant and is equal to the corresponding one of the
singular strategy gss
i.
As it can be immediately seen, the sign of w2þw31 or, equivalently, the sign of w2w3
ð1w2Þð1w3Þ, is the factor that determines the success of the spread of gss-mutants when they
Ro
1.000
1.001
1.002
1.003
1.004
0.15 0.30 0.45
1st component of the resident strategy
Ro
1.000
1.002
1.004
1.006
1.008
1.010
1.012
1.014
0.15 0.30 0.45 0.60 0.75 0.90
2nd component of the resident strategy
(b)
(a)
Fig. 2. Spread of the singular strategy gss when w2þw3>1 in the case of the dilemma of the subdominants with a
tridiagonal transition matrix. The success of the spread (invasion) of gss is measured in terms of R0¼R0ðgss ;NðgÞÞ with
g¼ðs12 ;s32Þand such that s12 þs32 61. Parameter values: w¼ð0:3;0:4;0:8Þ,s0¼ð0:810;0:775;0:770Þ,
f0¼ð0:39;0:39;0:39Þ,ðs11 ;s33Þ¼ð0:3;0:9Þ,gss ¼ð0:113557;0:526599Þ. The circles in the plot correspond to numerical
outputs of the simulations with: (a) g¼ðs12 ;gss
2Þ, and (b) g¼ðgss
1;s32Þ. In both cases, R0>1 for g6¼ gss .
J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181 165

appear in a resident population at equilibrium adopting g6¼ gss . From Figs. 2 and 4, it follows
that, as far as w2þw3>1 (or w2w3>ð1w2Þð1w3Þ), gss-mutants spread in resident popula-
tions playing another (nearby or not) strategy. In contrast, when w2þw3<1, there are always
nearby strategies to the singular strategy that can not be invaded (when adopted by the resident)
by any gss-mutant (see Figs. 3 and 5). More precisely, when w2þw3<1, the mapping
g!R0ðgss;NðgÞÞ has a saddle point at the singular strategy gss , which means that the second
Ro
1.0000
1.0002
1.0004
1.0006
1.0008
1.0010
1.0012
1.0014
0 0.15 0.30 0.45
1st component of the resident strategy
Ro
1.00000
0.99995
0.99990
0.99985
0.99980
0.99975 0.65 0.70 0.75 0.80
2nd component of the resident strategy
(a)
(b)
Fig. 3. Spread of the singular strategy gss when w2þw3<1 in the case of the dilemma of the subdominants with a
tridiagonal transition matrix. The success of the spread (invasion) of gss is measured in terms of R0¼R0ðgss;NðgÞÞ
with g¼ðs12 ;s32Þand such that s12 þs32 61. Parameter values: w¼ð0:3;0:4;0:4Þ,s0¼ð0:810;0:775;0:770Þ,
f0¼ð0:39;0:39;0:39Þ,ðs11 ;s33Þ¼ð0:3;0:9Þ,gss ¼ð0:216192;0:698261Þ. The circles in the plot correspond to numerical
outputs of the simulations with: (a) g¼ðs12 ;gss
2Þ(where R0>1), and (b) g¼ðgss
1;s32Þ(where R0<1).
166 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

matrix in (6) has (two) real eigenvalues of opposite signs. This implies that there is some sym-
metric, positive definite covariance matrix for which the singular strategy is an unstable equi-
librium of the corresponding canonical equation of the AD (see [34] for details).
The difference w2w3ð1w2Þð1w3Þis a measure of the balance of an average of competition
effects within (w2w3) and between (ð1w2Þð1w3Þ) intermediate and dominant classes, and its
Ro
1.000
1.015
1.030
1.045
1.060
0.25 0.50 0.75 1
1st component of the resident strategy
Ro
1.00
1.02
1.04
1.06
1.08
1.10
1.12
0.25 0.50 0.75 1
2nd component of the resident strategy
(a)
(b)
Fig. 4. Spread of the singular strategy gss when w2þw3>1 in a standard size-structured model. The success of the
spread (invasion) of gss is measured in terms of R0¼R0ðgss;NðgÞÞ with g¼ðs21;s32Þ. Parameter values:
w¼ð0:3;0:6;0:9Þ,s0¼ð0:810;0:775;0:770Þ,f0¼ð0:39;0:39;0:39Þ,gss ¼ð0:595688;0:477555Þ. The circles in the plot
correspond to numerical outputs of the simulations with: (a) g¼ðs21;gss
2Þ, and (b) g¼ðgss
1;s32Þ. In both cases, R0>1
for g6¼ gss.
J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181 167

sign determines the success of the initial spread of singular strategy. Notice that such a difference,
which appears in the expression of the gss-equilibrium, is assumed to be non-zero to guarantee
that the matrix W¼ðwijÞis invertible.
In summary, only when the average competition within groups is bigger than the average com-
petition between groups for the dominant and intermediate classes, i.e. when w2w3>
Ro
1.00
1.04
1.08
1.12
1.16
1.20
1.24
1.28
0 0.25 0.50 0.75
1st component of the resident strategy
1
Ro
1.000
0.99
0.98
0.97
0.96
0.25 0.50 0.75 1
2nd component of the resident strategy
(a)
(b)
Fig. 5. Spread of the singular strategy gss when w2þw3<1 in a standard size-structured model. The success of the
spread (invasion) of gss is measured in terms of R0¼R0ðgss;NðgÞÞ with g¼ðs21;s32 Þ. Parameter values:
w¼ð0:2;0:3;0:4Þ,s0¼ð0:810;0:775;0:770Þ,f0¼ð0:39;0:39;0:39Þ,gss ¼ð0:477187;0:556177Þ. The circles in the plot
correspond to numerical outputs of the simulations with: (a) g¼ðs21 ;gss
2Þ(where R0>1), and (b) g¼ðgss
1;s32Þ(where
R0<1).
168 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

ð1w2Þð1w3Þ, a singular strategy has the ability to spread in any resident population adopting
a different (nearby or not) strategy and, hence, is convergence stable for any covariance matrix
due to its neutral evolutionary stability. Note that, in terms of the classical definition of ESS [19],
neutral evolutionary stability plus invading potential implies that a singular strategy is, in fact, a
(classical) ESS. Therefore, from now on, we denote ESS such a singular strategy.
In terms of animal behaviour, the conclusion is that the presence of a like-versus-like aggression
within dominant and subdominant (or intermediate) classes is a sufficient condition for the
convergence stability of the singular strategy.
4. Evolutionary stability of bimodality
One aspect of the model that concerns us is the shape of a non-trivial equilibrium distribution
of sizes of badges and its relation with the dominance role of the plumage trait (colour patch(es)).
More precisely, from the field observations (see Introduction), it follows that a bimodal distri-
bution with respect to a given colour patch has to be an output of the model – under some suitable
choices of the parameter values – when this colour patch is only related to dominance status.
Conversely, when the colour patch has also a sexual role, the equilibrium distribution becomes
unimodal with its maximum in the middle classes. Our aim is to propose an explanation of how to
pass from one situation to the other by means of changes in the values of suitable parameters of
the model and, moreover, to give a biological interpretation of these changes. Therefore, let us
turn our interest towards the shape of the equilibrium of a population adopting an ESS and see
when bimodality is evolutionary feasible or, in other words, under which conditions bimodality
can be observed in nature according to the present model.
Assuming, as in the previous section, that survival probabilities and fecundities are given by
(13) for i¼1;...;n, the equilibrium Ndetermined by the ESS satisfies the linear system
WN¼Cwith C¼ðc1;...;cnÞ0and cigiven by (15). So, an explicit expression of Nis easily
obtained although the model is non-linear. This fact allows us to establish analytical conditions
on the parameters s0
i,f0
iand w0
ifor Nto be positive and bimodal. In particular, bimodality will be
obtained whenever N
i>N
iþ1in low ranked classes (subordinate classes) while N
i<N
iþ1in high
ranked classes (dominant classes). If the maxima of the equilibrium are in both extremes of the
class distribution, then it is obviously required that N
1>N
2and N
n1<N
n.
4.1. Bimodality in models with n-dominance classes
In order to get an insight into the features of bimodality in terms of the parameters of the
model, let us start with the simplest matrix Wsatisfying the hypothesis of like-versus-like
aggression which guarantees the convergence stability of the singular strategy. Clearly, such a
matrix is the diagonal matrix W¼diagðw1;...;wnÞwith wi>0 for all i. In this case, the ESS
equilibrium is simply given by N
i¼ci=wifor i¼1;...;n. That is, the abundance of the i-class at
the ESS equilibrium increases with its potential survival probability (s0
i) and fecundity (f0
i), and
decreases with its aggression rate (wi). Therefore, as long as s0
iþf0
iare similar for all dominance
class, those that are more aggressive (usually the dominant classes) will have higher values of wi
and, hence, a lower number of individuals at equilibrium, with respect to those that are less
J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181 169

aggressive. On the contrary, high enough values of s0
iþf0
ican compensate the aggressions in
dominant classes, whereas low aggression rates allow for increasing the number of individuals of
subordinate classes although their inherent survival and fecundities are lower than those of the
dominant classes. Consequently, when such a balance among different components of the life
history occurs, it may result in bimodal ESS equilibria.
In the previous choice of W, the simplest one, all classes are uncoupled. In particular, there is
no dominance (or ranking) hierarchy between classes. So, let us consider the simplest matrix W
reflecting a dominance hierarchy between any pair of consecutive classes, namely,
W¼
w11w100... 0
0w21w20..
.
0
00 ..
...
.
00
.
.
..
.
..
.
...
...
.
0
00 ... 0wn11wn1
00 ... 00 1
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
;
with 0 6wij 61 and where, to reduce the number of involved parameters and without loss of
generality, it is assumed that wii þwi;iþ1¼1 for all i.
This choice of Wis a sort of generalization of the hypothesis of like-versus-like aggression in
which class iþ1 is dominant to class ias wi;iþ1¼1wi>wiþ1;i¼0. Again, the expression of N
is easily obtained and it is given by
N
i¼ci
wi
þX
n
j¼iþ1
ð1ÞiþjY
j1
k¼i
1wk
wk
!
cj
wj
;i¼1;...;n1;
and N
n¼cn, where wn:¼1 in the expression of N
ifor convenience of notation. Note that, if one
assumes wi>1=2 for all ito impose within-class aggressions to be more frequent than between-
class aggressions, the product in the second term tends to zero as the number of factors increases
since ð1wkÞ=wk<1 for all k. The latter means that the contribution of high ranked (or dom-
inant) classes to the equilibrium values of low ranked (or subordinate) classes tends to be neg-
ligible as the number of dominance classes increases.
Using the previous expression of N
i, the condition N
i>N
iþ1(for low ranked classes) amounts
to
ci>ciþ1
wiþ1
þX
n
j¼iþ2
ð1Þiþjþ1Y
j1
k¼iþ1
1wk
wk
!
cj
wj
;i¼1;...;n2;ð20Þ
which is equivalent to ci>N
iþ1. This result is in agreement with what we already knew about ci
because the latter is a convex linear combination of N
iand N
iþ1and, so, N
iþ16ci6N
ifor all
wi2½0;1under the hypothesis N
i>N
iþ1.
Note that, from the condition N
i>08i, it follows that the second term of the RHS of (20) is
non-positive for all wi2½0;1. So, a sufficient condition for N
i>N
iþ1(in low ranked classes) is
ci>ciþ1
wiþ1
:
170 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

Similarly, the condition N
i<N
iþ1for high ranked classes is equivalent to ci<N
iþ1, the
opposite inequality to (20). That is,
ciþX
n
j¼iþ2
ð1ÞiþjY
j1
k¼iþ1
1wk
wk
!
cj
wj
<ciþ1
wiþ1
;i¼1;...;n2:
As before, it was known beforehand that N
i6ci6N
iþ1for all wi2½0;1if one assumes
N
i<N
iþ1. For the two highest ranked classes, N
n1<N
nis equivalent to
cn1<cn:
Therefore, in order to get bimodal equilibria it is enough (but not necessary) for subordinate
classes that each class has a potential fitness (measured by ci) greater than that of the next ranked
class times the factor 1=wiþ1which is bigger than 1. In particular, this means that the difference of
potential fitnesses between class iand class iþ1 required for bimodality can be relaxed by raising
wiþ1, the aggression rate of class iþ1. On the other hand, in dominant classes, this relationship
between potential fitnesses is just the converse one. More precisely, a necessary and sufficient
condition is to have an increase of the ratio of potential fitness to aggression rate enough to
compensate the potential fitness of the previous class plus an average of ratios of potential fitness
to aggression rate ðcj=wjÞfor all higher ranked classes.
Less simple nnmatrices Wmay be considered to obtain the corresponding conditions for
bimodality as, for instance, upper triangular matrices with Pjwij ¼1 for all i. The environmental
variable of i-class, /iðNÞ, under these matrices is determined by all the higher ranked classes in
addition to itself. In this particular case, it is easily seen along the same lines as before that the
same sufficient conditions for bimodality as in the previous choice of Wstill hold. Note that, also
in this case, ciare convex linear combinations of Njwith jPi, and that the corresponding
expression of N
iis also given by a first positive term (ci=wii) plus a second term which is negative
under the hypothesis N
i>0 for all i.
In general, however, more general nnWmatrices give rise to complicated (although explicit)
expressions for the equilibrium Nwhich make cumbersome (if not impossible) to arrive at simple
and illustrative conditions for bimodality. So, let us return to the examples considered in the
previous section with a matrix Wthat is not an upper triangular matrix and which are more
closely related to the field observations.
4.2. Bimodality in the two case studies
First of all note that, from the expression of the equilibrium at the ESS given by (16) and
assuming w2þw3>1, N>0 implies the following conditions on the parameter values of the
model
N
1>0() c1>ð1w1Þc2ð2w31Þþc3ð2w21Þ
2ðw2þw31Þ;ð21Þ
N
2>0() c2
c3
>1w2
w3
;ð22Þ
J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181 171

N
3>0() c3
c2
>1w3
w2
:ð23Þ
In particular, (22) and (23) imply that the RHS of (21) is non-negative and is equal to 0 only
when w1¼1.
Imposing the first condition for having a bimodal ESS equilibrium distribution, namely,
N
2<N
3, it follows
N
2N
3¼c2c3
w2þw31<0;
and, assuming the ESS-condition w2þw3>1, the inequality amounts to
c2<c3
where ci, given by (15), involves all the components of the fitness in our model, namely, the
inherent capabilities of the i-class to survive (s0
i) and reproduce (f0
i), and the competition coef-
ficient (ci).
In particular, if one assumes the same competition coefficient for all (bib-size) classes, i.e.,
ci¼c>0 for all i, then c2<c3is equivalent to s0
2þf0
2<s0
3þf0
3. This implies that for an ESS-
bimodality it is needed that the sum of the maximum probability values of survival and reproduction
(i.e., potential fitness) for the dominant class is greater than the corresponding sum for the inter-
mediate class.
The second condition for having a bimodal ESS equilibrium distribution, N
1>N
2, implies
2c1>N
2þN
3w1ðN
3N
2Þ. Again, assuming w2þw3>1, this condition is equivalent to
c1>c2ð2w31Þþc3ð2w21Þðc3c2Þw1
2ðw2þw31Þ:ð24Þ
Note that, under the assumption c2<c3,
c2ð2w31Þþc3ð2w21Þðc3c2Þw1
2ðw2þw31Þ<c3ðc3c2Þw1
2ðw2þw31Þ<c3;
and, so, the minimum value of c1required for having bimodal ESS-equilibria is always less than c3
for any 0 <w161 whenever c2<c3and w2þw3>1. On the other hand, since condition (24) and
the positivity condition (21) are almost equivalent for (very) small values of w1, an ESS-equi-
librium Nwill be bimodal for a (very) wide range of those values of c1that guarantee N
1>0.
In conclusion, as far as the parameter values of the model verify c2<c3and (24) with cigiven by
(15), any ESS equilibrium N>0of (4) corresponds to a bimodal distribution of dominance
classes.
Under the assumption ci¼cfor all classes, and if the inherent fecundity in the dominant class
is equal to the one in the subdominant class, i.e., f0
3¼f0
2, equilibrium bimodality can not occur
unless an increment of the inherent survival probability in the dominant class (s0
3) takes place. On
the other hand, recalling that ciis proportional to the number of per capita aggressive interactions
of an i-class individual per time period (w0
i¼Pjwij) when working with normalized /iðNÞ,it
follows that values of w0
3lower than values of w0
2can compensate lower survival probabilities and
lead to bimodal equilibria. In other words, another way of obtaining bimodality comes from the
assumption of different competition coefficients among dominance classes. In particular, if the
172 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

sum s0
iþf0
iis similar for all classes, a necessary condition for bimodality is that the competition
coefficient in the dominant class (c3) is small enough compared with that of the subdominant class
(c2). However, this last possibility seems not to be the case in the Siskin due to the high number of
aggressions between individuals of the dominant class.
5. An example: the tridiagonal transition matrix
The decompose of the projection matrix as T0þFis actually the case for most of life cycles
and, of course, it is also our case. However, while the general expression for R0is well-known for
discrete size-structured population models with one offspring class [28], its explicit expression and
interpretation are not so well-known when reverse transitions are allowed. The main reason could
be that, in standard size-structured models, only transitions from the i-class to the ðiþ1Þ-class are
considered because shrinking is not usually considered. In our case, since we are concerned about
transitions among dominance classes, it would be convenient to have an expression of R0that
includes reverse transitions to model properly the dynamics of a rank-structured population.
The case where only one-step transitions (per time interval) are allowed is the simplest one with
reverse transitions. The corresponding transition matrix is tridiagonal and it extends the examples
we have previously considered in Section 3 to ndominance classes.
Note that, from the definition of R, it follows that RðIT0Þ¼Fwhich allows us to obtain an
expression for the elements r1jand, in particular, for R0since R0¼r11 (recall that only the first row
Ris non-zero if there is only one class of newborns). Here we will obtain an expression of R0when
one-step transition are allowed, i.e., for a tridiagonal transition matrix.
5.1. Explicit computation of R0
Let us assume that the transition matrix T0is tridiagonal, i.e., only transitions to adjacent
classes are allowed. Then ðIT0Þis also tridiagonal and the following recurrence among the
elements r1jof the matrix Rholds:
fk
1tkk
¼r1kr1;k1
tk1;k
1tkk
r1;kþ1
tkþ1;k
1tkk
;16k6n;ð25Þ
where, for convenience of notation, r10 ¼r1;nþ1:¼0. This expression says that the expected off-
spring per visit at stage kof an individual is equal to the expected offspring of an individual
starting life in the class kminus two terms: the expected offspring of an individual starting life in
the previous class times the probability of the (reverse) transition to this class from class k, and the
expected offspring of an individual starting life in the next class times the probability of the
transition to this class from class k.
Now, solving (25) for an arbitrary number nof classes and since R0¼r11, it follows
R0¼X
n
i¼1
fiY
i
j¼1
tj;j1
1tjj
1
1pj
;ð26Þ
J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181 173

where t10 :¼1 and pksatisfies the recurrence
pk1¼tk1;k
1tkk
tk;k1
1tk1;k1
1
1pk
;k¼n;n1;...;2;ð27Þ
with pn:¼0 and pk2½0;1Þfor k¼1;...;n1. Notice that, if reverse transitions are not allowed
(ti;iþ1¼0, 1 6i6n1), then pk¼0 for all kand it immediately follows the well-known expres-
sion of R0for standard size-structured population models.
For instance, in case of having three dominance classes (n¼3), which is the one we have
considered in the numerical simulations in Section 3, it follows that the net reproductive number is
given by
R0¼
f1
1t11 þ1t32
1t22
t23
1t33

1t21
1t11
f2
1t22 þt21
1t11
t32
1t22
f3
1t33
hi
11t32
1t22
t23
1t33

1t21
1t11
t12
1t22
;
where 1=ð1tiiÞ¼P1
n¼0tn
ii is the expected number of time steps in class iper visit, ð1t32
1t22
t23
1t33Þ1
is equal to 1 plus the expected number of visits to class 2 from class 3, and, finally, ð1ð1
t32
1t22
t23
1t33Þ1t21
1t11
t12
1t22Þ1is the expected number of visits to the class 1 during the lifetime. In
particular, note that in case that t12 ¼0, the last expected number is equal to one since it is not
possible to return to stage 1 from any other stage.
5.2. Neutral evolutionary stability of the ESS
Now, let us see in a straightforward manner the neutral stability of the ESS when Tis a nn
tridiagonal matrix, i.e., that R0ðg;NðgssÞÞ ¼ 1 for all admissible g. First let us write R0given by
(26) as
R0¼X
n2
i¼1
fiY
i
j¼1
tj;j1
1tjj
1
1pj
þY
n1
j¼1
tj;j1
1tjj
1
1pj
fn1
þfn
tn;n1
1tnn :
At the equilibrium determined by the singular strategy gss,Nðgss Þ,wehavethatf
i¼1s
i8i.
So, since sn¼tnn þtn1;nand sj¼tj;jþtj1;jþtjþ1;j(2 6j6n1), it follows, using the recurrence
(27), that the net reproductive number R0of any mutant population with g6¼ gss at NðgssÞis given
by
R0ðg;NðgssÞÞ ¼ P
n2
i¼1
f
iQ
i
j¼1
tj;j1
1tjj
1
1pjþQ
n2
j¼1
tj;j1
1tjj
1
1pj
tn1;n2
1pn11pn1tn2;n1
1tn1;n1

¼P
n3
i¼1
f
iQ
i
j¼1
tj;j1
1tjj
1
1pjþQ
n2
j¼1
tj;j1
1tjj
1
1pjf
n2þtn1;n21tn2;n1
1tn1;n1
1
1pn1
hi
¼P
n3
i¼1
f
iQ
i
j¼1
tj;j1
1tjj
1
1pjþQ
n3
j¼1
tj;j1
1tjj
1
1pj
tn2;n3
1pn21pn2tn3;n2
1tn2;n2

¼P
n4
i¼1
f
iQ
i
j¼1
tj;j1
1tjj
1
1pjþQ
n3
j¼1
tj;j1
1tjj
1
1pjf
n3þtn2;n31tn3;n2
1tn2;n2
1
1pn2
hi
:
174 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

Repeating the same arrangements as before for the rest of the terms of the sum until the first
one, it follows that the net reproductive number of a mutant adopting any strategy gwhen the
resident population adopts the singular strategy gss and is in ecological equilibrium is
R0ðg;NðgssÞÞ ¼ t10
1t11
1
1p1
f
1
þt21 1
t12
1t22
1
1p2
¼1
1p1
1
t12
1t22
t21
1t11
1
1p2¼1;
where, for convenience of notation, t10 :¼1, and it is used that s1¼t11 þt21.
Remark. As the previous computation does not depend on the values of sij appearing in the
expression of R0, it also proves that R0ðg;0Þ¼1 when s0
iþf0
i¼1, i.e., it proves that this con-
dition on parameters s0
iand f0
iactually defines a bifurcation point from the extinction equilibrium
N¼0when the transition matrix is tridiagonal (see Section 2).
6. Concluding remarks & biological implications
In this paper we have considered a stage-structured model with a general transition matrix and
no cost when moving among classes. The species we have in mind as a model species is the Siskin
(C. spinus). Since males of this species display a plumage trait such that its size has been shown to
act as a reliable badge of status, such individuals are categorized into (discrete) dominance classes
according to it. On the other hand, the motivation of considering a general transition matrix is
due to observed transitions to lower ranks occurring in individuals remaining in captivity under
poor nutritional conditions (JCS, personal observation).
As evolutionary trait, we have considered a set of probabilities of moving among classes
and, to assess invasibility, we use R0, the net reproductive number. From an evolutionary point
of view, the first result we have obtained is that evolutionarily singular values of this trait
determine demographic equilibria where the reproductive value of all classes is the same and
equal to one (see Eq. (11)). This situation implies our second main result: the neutral evolu-
tionary stability of the singular strategy. This means that, when a singular strategy is adopted
by the resident, any small population of mutants with a (nearby or not) different strategy is
also in equilibrium. Both results are, in fact, analogous to the one obtained in [22] for evo-
lutionarily stable dispersal rates in metapopulations under the assumption of no cost of dis-
persal, and to those obtained in [24,25] for the evolutionarily stable timing of reproduction in
semelparous organisms.
One can look at the neutrality result obtained in this paper, as well as in other papers as the
previously cited, from the point of view of the evolutionary game theory and merely in terms of
animal behaviour. In this case, a mixed strategy is a specification of the probabilities of the actions
that an individual can do in any situation in which it may find itself [19]. So, any interior strategy
vector gis a mixed strategy because, given the current class of an individual, it gives the prob-
abilities of moving to any other class in the next period of time. In turn, pure strategies are of the
form ‘when being in class i, move to class k(with probability one)’ and, therefore, they lie at the
boundary of the trait space.
J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181 175

In the context of evolutionary matrix games, the process of substitution of strategies often leads
to mixed ESSs and, as a consequence of the theorem of Bishop and Cannings, each pure strategy
in the support of the mixed ESS, as well as any mixed strategy different from the ESS but con-
taining pure strategies also in its support, must have the same fitness as the mixed ESS itself
[19,27]. In other words, fitness equality of all strategies contained in a mixed ESS, once the latter is
established in the population, is the prediction resulting from matrix games as a result of an
evolutionary process. As it is pointed out in [27], such a prediction seems to be ‘qualitatively
incompatible’ with that of optimization models, namely, that under evolutionary equilibrium
conditions the strategy widespread in a natural population should have a higher fitness than any
mutant strategy that could be present in the population.
However, the previous incompatibility is only apparent. Neutrality or fitness equality among
different actions as the output of an evolutionary process has been also predicted, in the context of
density-dependent optimization models, by means of the well-known concept of ‘ideal free dis-
tribution’ in metapopulation theory (see [19,22]) or, more recently, by the so-called ‘principle of
indifference’ in [25] (or condition (12) in the present paper). The first essential feature of all these
optimization models leading to neutral ESSs is that the dimension of the environment, i.e., the
number interaction variables considered to introduce non-linearities in the dynamics, exceeds one.
For instance, in metapopulation models, usually each local population experiences its particular
environmental conditions and, in hierarchical models as the one we present in this paper, there are
as many interaction variables as classes in the dominance hierarchy. The second essential feature
is that, under the assumptions of most of these models, there does not exist a direct cost of a given
action (for instance, to move to another patch in a metapopulation model) or, at least, the cost is
always the same regardless of the action. Otherwise, neutrality may not be the output of the
evolutionary process even though high dimensional environments are considered (see, for in-
stance, [21] for a size-dependent model with a cost in the growth and a non-neutral ESS). Even if
there is a cost associated to an action (for instance, to migrate to a given patch) but this cost is the
same for any individual, neutrality arises again as a result of adaptation because, indeed, such a
neutrality was already implicit in the assumption of a constant cost (see [23] for an example of an
ideal free distribution in a metapopulation model with a cost of dispersal which is the same for
any immigrant and only depends on the patch where an organism settles down – and not on the
patch from which it comes from).
On the other hand, when (11) holds and the strategy vector gis defined by more than n1
probabilities, then there exists an infinite set of strategies satisfying the condition of singular
strategy. To see that note that, when the nequations given by (11) hold, it is always possible to
express one of the nequations of the model in terms of the other n1 equations. Therefore, there
are 2n1 linearly independent equations for ncomponents of the equilibrium Nplus the
components of the vector g. In other words, we do not have to expect that the condition for a
strategy to be singular defines a unique strategy when the evolutionary trait has more than n1
components. In any case, however, environmental conditions determined by any of the singular
strategies will be the same as long as (11) defines a unique demographic equilibrium.
Under our choice of the vital rates and assuming a dominance hierarchy among members of the
population belonging to different classes, such an equilibrium is, certainly, unique and it is ob-
tained from the linear system of equations resulting from (11), which does not depend explicitly
on s. Numerically, when three dominance classes are considered, it follows our third main result,
176 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

namely, that any singular strategy turns out to be able to spread as invading strategy only when
the average competition within the dominant classes is bigger than the average competition between
dominant classes. This invading potential of the singular strategy in addition to its neutral evo-
lutionary stability implies that the singular strategy is convergence stable. We conjecture that this
is a general fact for this sort of hierarchical models which does not depend on the dimension of the
trait space. This is, indeed, a generalization of what was conjectured in [21] for the particular case
of size-dependent matrix models.
Therefore, measuring competition effects in terms of aggressive interactions and considering
three dominance classes, to attain ESS we necessarily need intra-class aggression to be higher than
inter-class one, especially in relation to dominant and subdominant classes (i.e., w2w3>
½1w2½1w3). Subordinates, in turn, have to be engaged in some intra-class aggression
ðw1>0Þbut not too much necessarily. That is equivalent to say that the condition for hierarchical
systems as the one assumed in this paper to be evolutionary stable is like-versus-like aggression
between dominant individuals.
Like-versus-like aggression has been described for several species [2], including the Siskin [35],
but not in others, where aggression mainly takes place from dominant to subordinate individuals
(i.e., despotic aggression) [2,9]. Inter-specific differences may be explained by differences in social
behaviour patterns between species [36]. However, we have to point out that most of the examples
of despotic aggression so far described refer to species where plumage coloration is highly age or
sex dependent [8,9,37], and this should not be regarded as true status signalling [2]. This is why we
stated age independence as one of the assumptions for our approach, hypothesis that is reflected
in the model by assuming a general transition matrix T.
The need for like-versus-like aggression between dominant individuals for dominance hierarchy
reaching ESS lends support to the ‘social control hypothesis’ [38,39], which states that social
aggression can maintain the evolutionary stability of status signalling systems. In [38] and [39] it is
suggested that a subordinate would encounter relatively more aggression from true dominants as
a cheat than as a honest signaller, simply by the fact that dominants are normally fighting each
other. As the intrinsic fighting abilities of subordinates on average are lower than those of true
dominants, the heightened aggression cheats would receive would be a cost that would outweigh
any benefits arising from increased dominance status. According to this hypothesis, and in order
to be evolutionarily stable, the heightened aggression should not result from the ‘persecution’ of
cheat by true dominants in the population, but should be the result of dominant individuals
interacting more with other dominant birds than with subordinate ones (i.e.: like-versus-like
aggression between dominant individuals) [9,39,40]. The social control hypothesis, albeit con-
troversial [9], has been tested in several species and at least for some of them it has been shown to
work [8,9,11,38–42]. This view of the social control hypothesis and the presence of like-versus-like
aggression suggests signals of status as a form of handicap (see [43]), where only the individuals
having a better quality can afford costly badges to demonstrate their high status; the main dif-
ference to traditional handicaps is that the cost of a signal of dominance is not direct but an
indirect one, mediated by social interactions: dominant individuals are involved in more fights
than cheaters (not true dominant birds) would be able to bear [2].
In [11,12] it is suggested that within the context of the social control hypothesis and like-versus-
like aggression, both dominants and subordinates have benefits and costs. Subordinates (i.e.,
small badge size) have no preferential access to food sources, but can enjoy the advantages of their
J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181 177

subordinate status by not receiving too much aggressions from the dominant class [35,44], and
dominants (i.e., large badge size), on the other hand, clearly enjoy a preferential access to food,
but are engaged in many aggressive interactions [12,35,44]. According to their verbal model, such
dichotomization of roles should give two distinct plumage classes [12]. However, our mathe-
matical model suggests that because of interactions with sexual selection, this does not necessarily
need to be the rule.
Assuming equal competition coefficients (ci) and like-versus-like aggression between dominant
and subdominant individuals, the sum of potential survival and fecundity of dominant individuals
(s0
3þf0
3) should be higher than that of subdominant birds (s0
2þf0
2) in order to have N
3>N
2,a
necessary condition for bimodality of the frequency distribution of badges of social status. For
instance, in the case of a species with a bimodal badge of status as the Siskin, for which dominant
individuals do not enjoy a pairing advantage [2], their potential survival should compensate for
that and be high since then f0
1¼f0
2¼f0
3. In fact, this is what it seems to happen, with dominant
individuals showing higher body mass and lower metabolic rate than birds of lower social status
[45]. In Red-billed queleas (Quelea quelea), for which plumage coloration has also been found to
be bimodally distributed and not to have any role in mate choice [46], we should also predict a
higher survival potential for the highly ornamented birds, although we have to point out that for
this species no role of plumage coloration in social status signalling has yet been described and our
model only applies to species showing this status signalling. With respect to the second condition
for frequency distribution of badges to be bimodal, N
1>N
2, the model predicts that it will be
fulfilled for most of positive equilibria when subordinates are not engaged in many intra-class
aggressions (w11). In any case (i.e., for 0 <w161), there is no need for subordinates to have a
value of the sum of potential survival and fecundity (s0
1þf0
1) higher than that of dominants
(s0
3þf0
3) in order to have a bimodal equilibrium.
Bimodality can also appear in the case of species in which badges of social status additionally
have a prominent function as sexual ornaments (i.e., ambivalent characters, [6]), raising in this
way the potential reproductive success of dominant individuals (f0
3>f0
2>f0
1). In this case po-
tential survival of dominant individuals is of minor importance. However, when potential survival
for those dominant ornamented individuals is low, frequency distribution of the ornament may
shift to a normal shape. This is in fact what seems to happen to most ornamented species, for
which the ornament may act as a handicap reducing potential survival [47,48] and consequently
shaping their frequency distribution to be normal (Fig. 1). Similarly, when fecundity is inde-
pendent of the size of the badge of dominance, a high predation (i.e., low survival) for individuals
displaying these badges of status (e.g. [9,37,42,49,50]) could also select for normal distributions.
Nevertheless, it is important to distinguish between potential (or inherent) fecundities and
survival rates, which are values in a virgin environment without competition among members of
the population (i.e., f0
iand s0
i, respectively), and the values that these rates reach at equilibrium,
when competition effects among members of the dominance classes are present (i.e., f
iand s
i,
respectively). For instance, our model predicts that, at the equilibrium determined by the evo-
lutionarily stable strategy, whenever the dominance classes have the same fecundities (f1¼f2¼
f3¼f), as it is the case of the Siskin [2], the survival rates at the equilibrium for the three classes
have also to be equal (s
1¼s
2¼s
3¼1f), and this is in fact what it seems to happen (JCS,
personal observation). A possible reason for the larger decrease of the survival probability of the
dominant individuals at equilibrium with respect to that of the subdominant individuals (s0
3>s0
2
178 J. Ripoll et al. / Mathematical Biosciences 190 (2004) 145–181

while s
3¼s
2) is the higher number of aggressive interactions among members of the dominant
class. This similar survival for the dominant and subordinate classes would support the view of
Maynard Smith of status signalling as a mixed ESS [19], which could additionally explain why
cheaters pretending a higher dominance status of the one they have, do not invade the population
[2].
Summarizing, we have that the frequency distribution of badges of status may be predicted by
the inter-relationship between survival, reproductive and aggression rates of the different domi-
nance classes. This has been nicely exemplified with the Siskin, our model species, for which the
bimodality of bib size distribution may be explained by the fact that Siskin females do not choose
mates based on this character, but dominant individuals potentially enjoy a survival advantage.
Hence our model may allow to understand interspecific variation in frequency distribution of
badges of status, and more importantly, helps to understand the roles that shape the evolutionary
stability of the system.
Acknowledgements
We are most thankful to Pedro Cordero for allowing us to use his data on House sparrows. JS
is grateful to Odo Diekmann and Hans Metz for their helpful e-discussions. This work was funded
by DGICYT BOS 2000-0141 research project from the Spanish Research Council, Ministerio de
Ciencia y Tecnolog
ıa to J.C.S., and DGI BFM2002-04613-C03-03 to J.S. and J.R.
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