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Effects of stage structure on coexistence: mixed benefits
Gaël Bardon1,2,3,* and Frédéric Barraquand1,†
1Institute of Mathematics of Bordeaux, University of Bordeaux, CNRS, Talence, France
2Department of Polar Biology, Centre Scientifique de Monaco, MC 98000 Monaco,
Principality of Monaco
3Department of Ecology, Physiology and Ethology, Institut Pluridisciplinaire Hubert Curien
UMR 7178, University of Strasbourg, CNRS, F-67000 Strasbourg, France
Abstract
The properties of competition models where all individuals are identical are relatively well-understood;
however, juveniles and adults can experience or generate competition differently. We study here structured
competition models in discrete time that allow multiple life history parameters to depend on adult or
juvenile population densities. While the properties of such models are less well-known, a numerical
study with Ricker density-dependence suggested that when competition coefficients acting on juvenile
survival and fertility reflect opposite competitive hierarchies, stage structure could foster coexistence. We
revisit and expand those results using models more amenable to mathematical analysis. First, through
a Beverton-Holt two-species juvenile-adult model, we obtain analytical expressions explaining how this
coexistence emerging from life-history complexity can occur. Second, we show using a community-level
sensitivity analysis that such emergent coexistence is robust to perturbations of parameter values. Finally,
we ask whether these results extend from two to many species, using simulations. We show that they
do not, as coexistence emerging from life-history complexity is only seen for very similar life-history
parameters. Such emergent coexistence is therefore not likely to be a key mechanism of coexistence in
very diverse ecosystems.
Keywords: coexistence; stage structure; competition; matrix population models
Correspondence: *gael.bardon2@gmail.com,†frederic.barraquand@u-bordeaux.fr
1
arXiv:2110.00315v1 [q-bio.PE] 1 Oct 2021
1 Introduction
A basic tenet of demography and population ecology is that species vital rates, like survival and fertility,
can vary with age or stage (Leslie,1945;Cushing,1998;Caswell,2001). Surprisingly, little of this rich age-
structured population-level theory influences community-level coexistence theory, that by contrast mostly
builds on unstructured Lotka-Volterra competition or consumer-resource models (Chesson,2000;Barabás
et al.,2016;Letten et al.,2017). There have been, however, some calls to include more demography into
community ecology (e.g., Miller and Rudolf,2011) and coexistence studies in particular, as well as a number
of empirically-driven modelling studies doing so (Péron and Koons,2012;Chu and Adler,2015). They show
in general that the vital rates of the various life-stages do not react in the same way to changes in the densities
of juvenile and adult life-stages. These differences may well create new avenues for coexistence.
Although mainstream coexistence theory often neglects the complexity of life histories, some stage-
structured coexistence theory has been previously developed: several theoretical models, in past decades
(Haigh and Smith,1972;Loreau and Ebenhoh,1994) as well as in more recent literature (Moll and Brown,
2008;Fujiwara et al.,2011) have considered complex life cycles in which ontogenetic changes occur, leading
to models where vital rates can be differentially affected by competition. Such complexity can promote equal-
izing mechanisms (Fujiwara et al.,2011) as well as niche differentiation (Moll and Brown,2008). A main
observation of Moll and Brown (2008) was the occurrence of so-called ‘emergent coexistence’ in 2-species
2-life-stages models, which was defined as a coexistence equilibrium where competition on a single vital rate
would lead to exclusion, but the combination of competition processes affecting two different vital rates leads
to coexistence (e.g., competition affecting juvenile survival leads to species 1 winning, competition affecting
fertility leads to species 2 winning, and yet both species coexist in the model with both types of competition).
This prompted the exciting idea that community models with stage structure may foster more widespread
coexistence than classic Lotka-Volterra theory, suggesting stage structure as a key missing ingredient in our
explanations of diversity maintenance.
However, the results of Moll and Brown (2008) had two limitations that prevented the generalization of
this coexistence-enhancing role of stage structure. First, their investigations were restricted to a particular
life cycle where juveniles and adults occupy separated habitats (i.e., adults can only compete with adults
and juveniles with juveniles). This works well for systems with metamorphosis, such as dragonflies, but may
be more limited for say, birds and mammals where adults can impose strong competition on the juveniles.
Second, the analyses of Moll and Brown (2008) relied entirely on numerical simulations, which means that
while we can observe emergent coexistence in the Ricker-based density-dependent models that they consid-
ered, it is still difficult to understand fully the phenomenon from a theoretical standpoint. Moreover, it is yet
unclear if the emergent coexistence equilibria highlighted by Moll and Brown (2008) are structurally stable
(robust to small deviations of the model framework) and realistic (occurring for a large range of parameter
2
values). This highlights the need to investigate further structured population models with additional model
structures and analytical techniques.
A further key question is whether emergent coexistence can occur in a S-species context (where S >> 2).
Indeed, the emergent coexistence observed by Moll and Brown (2008) resulted from a trade-off across the
life-history stages which allowed each species to the best competitor for a given vital rate. While it is
obvious that such a trade-off may occur with two species, there are multiple ways to generalize such trade-
offs in competitive rankings to the S-species case, and not all of them necessarily lead to more coexistence.
In fact, unstructured Lotka-Volterra models suggest great caution in generalizing results from two-species
competition to many-species competition, as the criteria for coexistence with many species usually correspond
to much broader niche separation between species than required for two species, with intraspecific competition
dominating (Barabás et al.,2016). Similar phenomena might be expected for stage-structured competition
models.
Here, we generalize the results of Moll and Brown (2008) to a new model framework and life-history, using
both numerical methods and an analytical invasion criterion. This criterion allows us to better understand
how and why emergent coexistence occurs. We also use the sensitivity analysis developed by Barabás et al.
(2014a) to explore the robustness of emergent coexistence equilibria. We find that emergent coexistence is
a general feature of structured two-species competition models with two density-dependent vital rates, and
that this equilibrium is not more sensitive to perturbation on parameters than ‘classic’ coexistence equilibria
allowed by niche separation. We finally explore S-species models properties using simulations of diverse
community structures (number of initial species, similarity of species). These many-species models show the
limitation of the emergent coexistence mechanism to explain coexistence in highly speciose communities, as
we do not find a positive effect of the trade-off between competition coefficients on final species richness.
2 Material and Methods
2.1 Models
We studied a two-species model with two life stages (juvenile and adult) for each species. It is a development
of models presented in Fujiwara et al. (2011), who restricted themselves to the case where only one matrix
parameter per species (either fertility or survival) is affected by densities. Our model extends this framework
to multiple density-dependent parameters per species, which provides a useful springboard to combine two
types of competition, affecting both fertility and survival rates. In our model, competition is generated
by adults because it felt more realistic for many animal species (see e.g., in snowshoe hares populations,
Boutin,1984 or avian scavengers populations, Wallace and Temple,1987). We used Beverton-Holt functions
to model competition—these models are sometimes called Leslie-Gower after Leslie and Gower (1958)—in
order to promote fixed points equilibria, instead of cycles or chaos that can be generated by Ricker functions
3
(Neubert and Caswell,2000), and to foster analytical insights (as in Fujiwara et al.,2011). Our model (model
1) can be written for one species with a projection matrix Ai, for instance with i= 1,
A1(n) =
(1−γ1)φ1
1+β11n1a+β12 n2a
π1
1+α11n1a+α12 n2a
γ1φ1
1+β11n1a+β12 n2as1a
=
(1 −γ1)s1j(n)f1(n)
γ1s1j(n)s1a
(1)
where γi,fi,sij and sia respectively denote maturation rate, fertility, juvenile survival rate, and adult sur-
vival rate of species i. The juvenile survival rates and fertility rates in absence of competition are denoted
φiand πi.αij and βij are the competition coefficients associated to the effect of species jon species ion
fertility and juvenile survival, respectively.
We also considered a model with a structure similar to the one described by Moll and Brown (2008),
where juveniles compete only with juveniles and adults only with adults, but contrary to Moll and Brown
(2008), we chose Beverton-Holt functions to model competition so that it can be compared to model 1. This
model (model 2) can be written for one species with a projection matrix AM B
i, for instance with i= 1 and
the same notations as for model 1:
AMB
1(n) =
(1−γ1)φ1
1+β11n1j+β12 n2j
π1
1+α11n1a+α12 n2a
γ1φ1
1+β11n1j+β12 n2js1a
.(2)
We extended our models to S-species contexts. For model 1, we obtained the following projection matrix
(for species i):
Ai(n) =
(1−γi)φi
1+PS
k=1 βiknk,a
πi
1+PS
k=1 αiknk,a
γiφi
1+PS
k=1 βiknk,a
si,a
.(3)
The extension for model 2 is similar (see Supplement B for details).
2.2 Parameter sets
We simulated dynamics over 3000 time steps with our model 1, using parameter sets chosen to include
scenarios of emergent coexistence as described by Moll and Brown (2008). We chose three scenarios of
coexistence: (1) a classic coexistence that is suggested by both αand βcoefficients (intra >interspecific
competition), (2) an emergent coexistence where the competition coefficients associated with the two vital
rates suggest exclusion by the opposite species, and (3) a coexistence where there is a priority effect suggested
by competition on one vital rate while competition on the other vital rate suggests coexistence. For simplicity,
we name these two last scenarios ‘emergent coexistence’ even if coexistence scenario (3) does not strictly
corresponds to an emergent outcome as defined by Moll and Brown (2008) (because competition coefficients
4
for one of the two vital rates in fact already suggest coexistence).
For parameter sets promoting emergent coexistence, we set a trade-off between αij and βij for both
species (large αij is associated to low βij and vice versa). To be able to compare the three scenarios, we set
intraspecific competition coefficients to a constant and only changed interspecific competition coefficients.
Note that we also chose the values of interspecific competition coefficients such that we obtain the same
densities at equilibrium across the three coexistence scenarios. The competition coefficients for the three
corresponding parameter sets are given in table 1(and full parameter sets in Appendix A).
Set α12 α21 β12 β21
1 0.05 0.06 0.06 0.06
2 0.02 0.112 0.125 0.01
3 0.043 0.035 0.155 0.165
Table 1: Interspecific competition coefficients found to obtain equal equilibrium densities but corresponding
to contrasted scenarios of coexistence. We use intraspecific coefficients αii =βii = 0.1.
We checked that coexistence outcomes were indeed produced by the abovementioned mechanisms through
the computation of invasion criteria in models where only a single vital rate is affected by competition. The
invasion criteria were computed using the expression given by Fujiwara et al. (2011), which was found using
the fact that the dominant eigenvalue of each species’ population matrix must be equal to one at any stable
fixed point equilibrium. As shown in table 2, the three parameter sets do correspond to three mechanisms
mentioned above. Parameter set 1 corresponds to mutual invasibility in models with a single vital rate that
is density-dependent. Parameter set 2 matches emergent coexistence since 1 excludes 2 when competition is
on fertility while 2 excludes 1 when competition is solely on juvenile survival. With parameter set 3, we have
priority effects suggested by the model with competition on fertility only and classical coexistence through
mutual invasibility in the model with competition on juvenile survival only.
Parameter set RαRβOutcome
Set 1 2.29 /1.46 1.76 /1.58 Classical coexistence
Set 2 5.72 / 0.78 0.84 / 9.47 Emergent coexistence
Set 3 2.60 /2.43 0.68 / 0.57 Emergent coexistence
Table 2: Invasion criteria for species 1/species 2, evaluated for models where a single vital rate is affected
by competition. Rαand Rβare invasion criteria respectively for models with competition only on fertility
and juvenile survival. Invasion criteria whose values are larger than 1 are in bold.
2.3 Invasion analysis
We devised invasion criteria for our model combining competition on fertility and juvenile survival generated
by adults (model 1). We were inspired by the work of Cushing (1998,2008) who studied the stability of
the exclusion equilibria (with the focal invading species absent) to describe the ability of the focal species
to invade a community. We therefore looked for conditions equivalent to an unstable exclusion equilibrium,
5
which corresponds to the fact that the absent species can actually invade. All equations are presented in
Appendix B.
We used the aggregated parameters C= (1−γ)φand D=πγ φ
1−sain the single-species model. By considering
the model with a single species (since at the time of invasion, the invader is almost absent), we found an
expression for the inherent net reproductive number, C+D= (1 −γ)φ+πγ φ
1−sa. This number must exceed
1 to have a viable population of the resident species. We then obtained the densities at a stable positive
equilibrium for the single-species model:
n∗
j= (1 −sa)1+βn∗
a
γφ n∗
a
n∗
a=(αC−α−β)+√(−αC +α+β)2−4αβ(−C−D+1)
2αβ
(4)
provided that
C+D= (1 −γ)φ+πγφ
1−sa
>1.(5)
We then evaluated the stability of the exclusion equilibrium in the two-species model: if it is locally asymp-
totically stable, the invader converges to zero density; if not, the invader invades. To do this, we calculated
the eigenvalues of the Jacobian of the 2 species system evaluated at the exclusion equilibrium. We found the
following condition for stability when species 2 is absent:
C2
1 + β21n∗
1a
+D2
(1 + β21n∗
1a)(1 + α21n∗
1a)<1(6)
with C2= (1−γ2)φ2and D2=π2γ2φ2
1−s2aas denoted previously. This last expression (eq. 6) gives us an invasion
criterion, in the sense that if the expression is larger than 1, species 2 is able to invade the community when
rare.
We applied the exact same method to model 2 where juveniles affect juveniles and adults affect adults as
in Moll and Brown (2008).
To observe emergent coexistence outcomes as did Moll and Brown (2008) for their Ricker competition
model, but this time through analytical calculations, we computed analytically the boundaries of the coex-
istence domains over α12 and α21, for values of β12 and β21 suggesting respectively coexistence, exclusion of
species 1 or a priority effect. The expressions for the boundaries of the coexistence domains on α21 and α12
are respectively given by equations 7and 8that are derived directly from the invasion criteria:
α∗
21 =D2
(1 + β21n∗
1a)−C2−11
n∗
1a
(7)
α∗
12 =D1
(1 + β12n∗
2a)−C1−11
n∗
2a
(8)
We computed α∗
12 and α∗
21 for the three parameter sets mentioned in section 2.2. If both conditions
6
α12<α∗
12 and α21<α∗
21 are met, then there is coexistence. If both conditions are violated, there is a priority
effect, and if one condition is met and the other is violated, there is exclusion of one of the species.
2.4 Sensitivity analysis
To examine the sensitivity of the different scenarios of coexistence to perturbations of parameters, we applied
the community-level sensitivity analysis developed by Barabás et al. (2014a) to model 1. The computation
of fixed point densities, required to perform the analysis, was done numerically (these do not have closed-
form expressions in the general case). We sum up here the notations and expressions used to compute the
sensitivity. We use ni=niqiwhere niis the vector of population density of the species iat equilibrium
for the various stages and qia vector of weights given to stages. We considered only the case where qiis a
vector of ones. We also need to define pi=ni
nithe proportion of each stage of species i. We denote Aithe
projection matrix of species iwith its eigenvalues λk
iand its left and right eigenvectors vk
iand wk
i. We also
define the dominant eigenvalue associated with its eigenvector : λiand wi. We normalized the vectors to
have |wi|= 1 and vj
iwk
i=δjk with δj k = 1 if j=kand 0otherwise. Barabás et al. (2014a) used regulating
variables Rµwhich correspond for example for species 1 in our model 1 to Rα,1=α11n1a+α12 n2aand
Rβ,1=β11n1a+β12n2a. Note that the notation for regulating variables of Barabás et al. (2014a) is similar to
that used for invasion criteria in single-species models that we previously introduced, even though ecological
meanings differ greatly. As regulating variables are not used later in the text, there is no risk of confusion
and we simply follow the conventions of Barabás et al. (2014a) in this section.
We then computed the sensitivity of population densities of both species to perturbations on each param-
eter using the following expressions, where Eis the perturbed parameter:
dni
dE =
S
X
j=1
(Mij )−1gj(9)
where the matrix Mij is given by
Mij =X
µ,ν vi
∂Ai
∂Rµ
wiδµν −∂Gµ
∂Rν−1∂Rν
∂nj
pj(10)
and the vector gjis given by
gj=vj
∂Aj
∂E wj+X
σ,ρ vj
∂Aj
∂Rσ
wjδσρ −∂Gσ
∂Rρ−1∂Gρ
∂E .(11)
The function Gµ(Rν, E)introduced by Barabás et al. (2014a) to simplify notations, is given by
Gµ(Rν, E) =
S
X
j=1 nj
qjwj
∂Rµ
∂nj
sj
X
k=2
1
λj−λk
j× wk
j−qjwk
j
qjwj
wj!⊗vk
j!Aj(Rν, E)wj.(12)
7
With these expressions, we were able to compute the sensitivity to perturbations of parameter values for
the three parameter sets / contrasted scenarios of coexistence. These have near-equal densities at equilibrium
(see section 2.2), to allow for meaningful comparisons.
2.5 S-species simulations
We simulated the dynamics of larger communities (5, 10 and 40 species) with parameters chosen to favour
emergence of coexistence through life-history complexity, using the extension of model 1 in S-species contexts
(eq. 3). Our idea to extend the mechanism behind emergent coexistence from two to an arbitrary number
of species was to create opposite competitive hierarchies in the competition coefficients related to the two
density-dependent vital rates, fertility (α) and juvenile survival (β). This was done through a negative
correlation between the coefficients αij and βij. The negative correlation was parameterized using a bivariate
normal distribution and a covariance between the two variable equals to −0.9σ2where σ2is the variance of
both αij and βij .
To draw the full set of parameter values, we have used:
•Log-normal fertilities πi∼Log-N(µπ, σπ)
•Beta-distributed survival and transition probabilities φi∼Beta(aφ, bφ),sa,i ∼Beta(asa, bsa)and γi∼
Beta(aγ, bγ)
•Normally distributed intraspecific competition coefficients αii ∼ N(µα, σα)and βii ∼ N(µβ, σβ)
•As mentioned above, when i6=jwe use interspecific coefficients drawn as (αij, βij )∼ N2(µαβ ,Σαβ )
with Σαβ =
σ2ρσ2
ρσ2σ2
and ρ=−0.9. We have µαβ = 0.8
µα
µβ
to ensure a strong interspecific com-
petition, which seemed necessary to see possible effects of trade-off between competitive coefficients. Meta-
parameter values determining the abovementioned parameter distributions are given in full in Appendix D.
We chose 3 sets for the distribution-level parameters (µπ, σπ, aφ, bφ, ...), corresponding to 3 different levels of
variance in the vital rates that allowed us to have either species with very close vital rates (i.e., low variance
across species of the parameter distributions) or more variable vital rates (medium and high variation, see
Appendix D).
We simulated the dynamics and computed the number of extant species after 3000 generations, which
allowed to reach an equilibrium state (see Appendix D). Then, for each of the three main parameter sets,
we created new, replicated parameter sets where we permuted all the inter-specific coefficients in order to
remove the correlation between αij and βij , while keeping the same inter-specific competition values. We
performed these permutations 200 times and represented the histogram of the number of species that coexist
after 3000 generations for the 200 permutated datasets. This allowed us to compare the scenarios where αij
8
and βij are negatively correlated to the null hypothesis of no correlation between them (no emergence of
coexistence through life-history complexity possible).
3 Results
3.1 Invasion analysis
First of all, the analytical expressions for the boundaries of coexistence domains obtained thanks to the
invasion analysis (eq. 7and 8) can be interpreted to understand how the trade-off between competition
coefficients actually promotes coexistence. In equation 7(and similarly in equation 8), as n1a, C2, D2are
independent of inter-specific competition coefficients, a decrease of β21 (the effect of species 1 on species 2’s
juvenile survival) on the right hand side will mechanically increase the value of α∗
21 on the left-hand site (the
boundary value for α21, the effect of species 1 on species 2’s fertility). What does this mean ecologically?
When the effect of species 1 on species 2’s juvenile survival β21 decreases, α21 can increase without species 2
getting extinct. This means that as one type of competition is lowered, the other type of competition can be
increased while having both species coexisting. Thus we can have coexistence with low β21 and high α21, and
reciprocally, using the other equation, high β12 and low α12. That is, we have coexistence through opposite
competitive hierarchies.
To have a better overview of coexistence properties implied by our invasion analysis, we produced co-
existence outcome domains for model 1. These are similar to Moll and Brown (2008)’s but are computed
analytically. They are produced using the newly derived invasion criterion (eq. 6). Figures 1A and 1B are
used for reference (with dotted lines corresponding to the domains boundaries of figure 1A). We found, as
shown in figures 1C, 1D and 1E, that emergent coexistence outcomes can be observed in our model. This
is best seen in figure 1D, where the coexistence domain (in light grey) is extended far to the right of the
reference dashed line, well beyond what would be suggested based on αcoefficients only (Fig. 1A). Thus
we have coexistence with competition on both vital rate when competition on a single vital rate (α) would
instead have suggested exclusion of species 2, while βsuggests exclusion of species 1.
In figures 1C and 1E, we do not find strictly emergent coexistence, in the sense that coexistence is not
suggested by either αor βbut is obtained by combining both types of competition. We do find other emergent
outcomes though (e.g. in figure 1C: exclusion of one of the species when αor βwould suggest priority effect
and coexistence, respectively).
9
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5
a21
a12
Competition on fertility only (α = f(N))
A
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5
a21
a12
Competition on both vital rates with β = α
B
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5
a21
a12
β suggests coexistence
C
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5
a21
a12
β suggests exclusion of species 1
D
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5
a21
a12
β suggests a priority effect
E
Outcome predicted by the invasion criteria
Coexistence
Priority effect
Species 1 excludes species 2
Species 2 excludes species 1
Figure 1: Outcomes of the model predicted by invasion analysis depending on the coefficients of the
inter-specific competition on fertility (α12 and α21)with βcoefficients suggesting coexistence, exclusion of
species 2 and priority effect. The dotted lines correspond to the outcome boundaries of panel A which is
used as a reference
We also reproduced figure 1for model 2 where habitats of juveniles and adults are separated so that
they cannot compete with each other as in the life-history cycle described by Moll and Brown (2008) (see
Supplement A for details).
3.2 Sensitivity analysis
We represented elasticity E
ni
dni
dE , i.e., a relative response of the density niof species ito a relative perturbation
on E. The elasticity values obtained from the sensitivity analysis performed on our three parameter sets are
given in figure 2.
10
-2
0
2
p1p2g1g2f1f2s1a s2a a11 a12 a21 a22 b11 b12 b21 b22
Elasticity
Parameter set
1
2
3
Elasticities of the parameters for the species 1
-2
0
2
4
6
p1p2g1g2f1f2s1a s2a a11 a12 a21 a22 b11 b12 b21 b22
Elasticity
Parameter set
1
2
3
Elasticities of the parameters for the species 2
Figure 2: Elasticities for the 3 parameter sets leading to equal densities at equilibrium but corresponding to
contrasted mechanisms of coexistence.
We found here that the emergent coexistence that occurs when there are opposite competitive hierarchies
on fertility and juvenile survival (parameter set 2) is as sensitive as the ‘classical’ (niche-based) coexistence
that occurs when the competition coefficients on both vital rates suggest coexistence (parameter set 1).
However, when the coexistence stems from a competition on one vital rate suggesting a priority effect and
the other classical coexistence (parameter set 3), the resulting equilibrium is more sensitive to perturbation,
according to our elasticity analysis. A coexistence regions representation (Barabás et al. 2014b, presented in
Appendix C) confirmed the results of figure 2 by showing similar regions for the ‘classical’ and the ‘emer-
gent’ coexistence, together with a smaller domain for the third scenario (priority effect + coexistence →
coexistence). This is likely due to the destabilizing influence of priority effects.
3.3 S-species simulations
The species richnesses at the end of the simulations for the extended model 1 with Sspecies (see eq 3) are
given in figure 3. Each panel displays a histogram of the number of species remaining in the community after
11
3000 generations, for the original parameter set including a negative correlation between αij and βij, as well
as for the 200 permutated parameter sets where this correlation is set to zero.
5 species, low variance
Number of species at equilibrium
Frequency
012345
0 20 40 60 80
A10 species, low variance
Number of species at equilibrium
Frequency
0 2 4 6 8 10
0 20 40 60
B40 species, low variance
Number of species at equilibrium
Frequency
0 10 20 30 40
0 10 20 30 40
C
5 species, mid variance
Number of species at equilibrium
Frequency
012345
0 40 80 120
D10 species, mid variance
Number of species at equilibrium
Frequency
0 2 4 6 8 10
0 20 40 60
E40 species, mid variance
Number of species at equilibrium
Frequency
0 10 20 30 40
0 20 60 100
F
5 species, high variance
Number of species at equilibrium
Frequency
012345
0 20 60 100
G10 species, high variance
Number of species at equilibrium
Frequency
0 2 4 6 8 10
0 20 60 100
H40 species, high variance
Number of species at equilibrium
Frequency
0 10 20 30 40
0 10 30 50
I
Figure 3: Histograms of the number of species persisting at equilibrium for 200 permutations of
inter-specific competition coefficients. The bar in gray indicates the number of species persisting at time
t=3000 generations for the original αij and βij with Corr(αij , βij )<0. From left to right, columns
correspond the communities with initially 5, 10 and 40 species. From top to bottom, the lines correspond to
situations with high, average and low variance on the vital rates.
We checked that the negative correlation between αij and βij indeed induces opposite hierarchies of com-
petition in sets of coefficients associated to fertility and juvenile survival (i.e., a negative correlation between
rank(αij )and rank(βij ), see Appendix D). In spite of this, the parameter sets with the above-mentioned
opposite competitive hierarchies do not systematically lead to richer communities than the corresponding
permutated parameter sets (where Corr(αij , βij )=0). For very close parameters between species (low vari-
12
ances in parameter distributions, Figs. 3A, 3B and 3C), opposite competitive hierarchies seem to have a
positive effect on the species richness, when compared to the null hypothesis of zero correlation between αij
and βij . However, the assumption of a community composed solely of very similar species for all life-history
parameters (as opposed to similar averaged birth and death rates, as in neutral theory) is rather unrealistic.
To sum up, we showed that the negative correlation between αij and βij, even if it indeed induces opposite
competitive hierarchies on the two vital rates, does not seem to allow, for reasonably variable distribution
of parameter values, an increase of the number of species coexisting at the equilibrium. In other words,
two-species coexistence through life-history complexity does not scale up to many-species.
4 Discussion
In this article, we studied competitive interactions between species with structured life-cycles, with multiple
vital rates that can depend on population densities. Using structured models for two competing species,
Moll and Brown (2008) described a form of coexistence whereby opposite competitive hierarchies in the
competition coefficients associated to two vital rates could yield coexistence in the model where these two
density-dependent vital rates combine—while models with a single density-dependent vital rate would predict
exclusion of the inferior competitor on that vital rate. To show the existence of such emergent coexistence,
they used Ricker functions for density-dependence and a particular interaction setup where competition be-
tween adults and juveniles is forbidden, as in systems with metamorphosis, where adults and juveniles occupy
separated habitats. This brought into question whether these results could be generalized to other functional
forms and model structures, which we do here. We considered as a baseline an alternative and perhaps
more frequent life-history setup, where adults can affect both adults and juveniles, so that both fertilities
and juvenile survival rates can depend upon adult densities of both species. We first found, using structured
two-species models, that emergent coexistence can be generalized to Beverton-Holt functional forms and our
new model structure (as opposed to scenarios where juveniles and adults interact solely within a life-stage,
although we have considered this as well and emergent coexistence remains possible, see Supplement A). We
provided an in-depth explanation of how emergent coexistence can occur, using analytical invasion criteria
(facilitated by the use of Beverton-Holt functions). Then, we explored the robustness of such emergent coex-
istence equilibria with a community-level sensitivity analysis and we found that a perturbation on parameters
had not more effect on equilibrium densities in the case of emergent coexistence than in a ‘classical’ coexis-
tence allowed by niche separation. Finally, we numerically explored the possibility for such mechanisms of
emergent coexistence to manifest in S-species models (with S >> 2). We observed that the opposition of
competitive hierarchies acting on fertility vs juvenile survival, allowed by a negative correlation between the
two sets of competition coefficients, did not seem to generate substantial increases in species richness in our
simulated communities.
13
As we succeeded to reproduce the numerical results of Moll and Brown (2008) with a general analyt-
ical invasion framework and another, more frequently observed model structure, we think that emergent
coexistence may be a general feature of two-species two-life-stages models, which reinforces the importance
of structured models to study interactions between species and their role in shaping community dynamics
(Miller and Rudolf,2011). The robustness of those findings has been confirmed by the community-level
sensitivity analysis. To our knowledge, there has been little implementation in practice of the discrete-time
sensitivity analysis for coupled nonlinear systems developed by Barabás et al. (2014a). Here, it has allowed us
to explore in more depth the properties of equilibria, as we could not easily find an expression for equilibrium
densities (they are defined by the intersection of two conic sections, and parameters are so interwoven in
those expressions that even if analytical solutions were to be found, these might be too complex to interpret).
As our results showed that the emergent coexistence equilibrium with opposite competitive hierarchies was
not sensitive to external perturbations, we think that such coexistence may be likely when environmental
stochasticity is added to two-species two-stages models.
However, two-species coexistence, even with many stages, may be quite different from many-species coex-
istence. Our study of S-species communities did not show a positive effect of opposite competitive hierarchies
in the two sets of competition coefficients, except in communities where species were very similar in all their
parameters (but these communities are probably not very lifelike). We have done additional explorations
of many-species models, using slightly different correlation structures, such as negative correlations between
interspecific αand βcoefficients within each pair of species (see Appendix D), rather than simply a negative
correlation between pooled αij and βij values for all pairs of species. This stronger condition on negative
correlations between competitiveness on the two vital rates has also shown a null effect on the species richness
of the final community. In sum, we showed that the properties promoting emergence of coexistence through
life-history complexity in the 2-species models could hardly be extended to many-species models where all
species interact together (albeit sometimes weakly). These results resonate with those of simpler, unstruc-
tured models (Barabás et al.,2016): criteria needed to promote S-species coexistence with S >> 2are much
more stringent than those for two species, and require a more important separation of niches (manifesting in
the ratio between intra and interspecific competition coefficients).
Finally, we should keep in mind that even if our results are both robust and supported by analytical for-
mulas, some assumptions that we (and most modellers) have made could be debated. Moll and Brown (2008)
and ourselves considered a modified Lotka-Volterra framework, with Lotka-Volterra competition coefficients
modulating the effect of adult or juvenile densities on several vital rates. However, we have no mechanistic
derivations for such models (a derivation through MacArthur-style consumer-resource model with separation
of time scales could perhaps be attempted). Structured mechanistic models of resource-based competition or
competition for space might inform on the realism of the αij and βij correlation structures considered here
14
(Goldberg and Landa,1991;Loreau and Ebenhoh,1994;Tilman,1982;Qi et al.,2021). That said, we expect
that even with a mechanistic perspective, our main finding will likely be robust: while coexistence emerging
from life-history complexity is quite easily seen in two-species models, it is much more difficult to obtain in
many-species models, where some degree of niche separation is usually required to prevent species to inter-
act too much with each other. This could occur either through generally weak interspecific interactions, or
potentially strong yet modular interactions, which has showed potentiality to better understand coexistence
(e.g. Kinlock,2021).
Acknowledgements We thank Sam Boireaud for an exploratory numerical study of some two-species
models, as well as Olivier Gimenez for discussions and Coralie Picoche for comments on the manuscript.
Funding for GB’s internship was provided by the French National Research Agency through ANR Democom
(ANR-16-CE02-0007) to Olivier Gimenez.
Code accessibility Computer codes written for analyses are available at GitHub:
https://github.com/g-bardon/StructuredModels_Coexistence
Contributions GB wrote the computer code, produced the figures, and performed numerical analyses.
Analytical formulas were derived by FB and GB. The first draft was written by GB with help from FB, both
authors contributed equally to subsequent versions.
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17
Appendix A Parameter sets used to simulate contrasted scenarios
of two-species coexistence
The parameter sets chosen are the following :
Set π1π2γ1γ2φ1φ2s1as2aα11 α12 α21 α22 β11 β12 β21 β22
1 30 25 0.7 0.8 0.5 0.4 0.5 0.6 0.01 0.05 0.06 0.01 0.01 0.06 0.06 0.01
2 30 25 0.7 0.8 0.5 0.4 0.5 0.6 0.01 0.02 0.112 0.01 0.01 0.125 0.01 0.01
3 30 25 0.7 0.8 0.5 0.4 0.5 0.6 0.01 0.043 0.035 0.01 0.01 0.155 0.165 0.01
Table A1: Full parameter sets chosen to have contrasted scenarios of coexistence
The three parameter sets led to the same densities at equilibrium with n1= 200 ±1and n2= 130 ±1.
To check what coexistence outcomes were suggested by the competition coefficients associated to fertility
and juvenile survival, we used the invasion criteria given by Fujiwara et al. (2011), valid for models with a
single density-dependent vital rate affected by competition (either fertility or juvenile survival). For example,
the invasion criteria of the model with only juvenile survival affected by competition (i.e., with only α
coefficients) is given by:
Rα(i) = R0
i
R0
j
αjj
αij
(A1)
where
R0
i=πi
f(c)
i−1(A2)
and
f(c)
i=1
γisj
(1 −sa) (1 −sj+γisj).(A3)
These criteria reflect invasion outcomes if we only had density-dependence on fertility.
Appendix B Invasion analysis
We develop here the method to obtain invasion criteria in two-species two-life stages models including com-
petition on two vital rates, such as model 1 combining competition on fertility and juvenile survival.
We first place ourselves at the exclusion equilibrium where species 1 lives without species 2:
˜n=
n∗
1j
n∗
1a
0
0
.(A4)
To compute the equilibrium densities n∗
1jand n∗
1a, we have to solve the following system corresponding
18
to the model with a single species (we dropped here species-specific indices for simplicity):
nj= (1 −γ)sj(n)nj+f(n)na
na=γsj(n)nj+sana
(A5)
with the fertility f(n)and the juvenile survival sj(n)affected by only the intra-specific competition according
to the expressions
f(n) = π
1 + αna
and sj(n) = φ
1 + βna
.(A6)
We now express njaccording to na:
nj= (1 −sa)na
1
γsj(n)(A7)
= (1 −sa)1 + βna
γφ na(A8)
and then, as we have by assumption na6= 0 and nj6= 0, we obtain
nj= (1 −γ)sj(n)nj+f(n)na
1 = (1 −γ)φ
1 + βna
+π
1 + αna
na
nj
1 = (1 −γ)φ
1 + βna
+πφγ
(1 −sa)(1 + βna)(1 + αna)
(1 + βna)(1 + αna) = (1 −γ)φ(1 + αna) + πγφ
1−sa
that leads to the polynomial of degree 2:
αβn2
a+ (α+β−α(1 −γ)φ)na+ 1 −(1 −γ)φ−πγφ
1−sa
= 0.(A9)
We denote C= (1 −γ)φand D=π γφ
1−saand we search then for the solutions of the polynomial equation:
n∗
a=(αC −α−β)±p(−αC +α+β)2−4αβ(−C−D+ 1)
2αβ .(A10)
We search for a positive solution and we have C= (1 −γ)φ < 1and then αC −α−β < 0. So we have to
take the positive square root and using the additional conditions α > 0and β > 0, we find:
(αC −α−β) + p(−αC +α+β)2−4αβ(−C−D+ 1) >0(A11)
⇐⇒ −4αβ(−C−D+ 1) >0(A12)
⇐⇒ C+D > 1.(A13)
19
We define the inherent net reproductive number as the term C+D= (1 −γ)φ+πγ φ
1−sathat must exceed
1 to have a viable species. We finally obtain the densities at a stable positive equilibrium for the model with
a single species:
n∗
j= (1 −sa)(1+βn∗
a
γφ )n∗
a
n∗
a=(α(1−γ)φ−α−β)+p(−α(1−γ)φ+α+β)2−4αβ(−(1−γ)φ−πγ φ
1−sa+1)
2αβ
(A14)
provided that
C+D= (1 −γ)φ+πγφ
1−sa
>1.(A15)
We have then obtained the densities at the exclusion equilibrium. We have now to evaluate the stability
of this equilibrium: if it is locally asymptotically stable, the excluded species cannot invade the community
starting from very small density. We have to compute the eigenvalues of the Jacobian of the 2 species system
evaluated at the exclusion equilibrium. The full system iterated over a time step reads:
n1j(t+ 1) = (1 −γ1)φ1
1+β11n1a(t)+β12 n2a(t)n1j(t) + π1
1+α11n1a(t)+α12 n2a(t)n1a(t)
n1a(t+ 1) = γ1φ1
1+β11n1a(t)+β12 n2a(t)n1j(t) + s1an1a(t)
n2j(t+ 1) = (1 −γ2)φ2
1+β22n2a(t)+β21 n1a(t)n2j(t) + π2
1+α22n2a(t)+α21 n1a(t)n2a(t)
n2a(t+ 1) = γ2φ2
1+β22n2a(t)+β21 n1a(t)n2j(t) + s2an2a(t).
(A16)
We place ourselves at the abovementioned exclusion equilibrium where species 1 dominates the community
and species 2 is absent. The Jacobian evaluated at this point is a 4×4matrix and it has the following
triangular block form:
B1B2
0B3
.(A17)
Therefore, we only need to know the eigenvalues of the 2×2matrices B1and B3. The B1matrix corresponds
to the Jacobian of the system in the absence of species 2, so we directly know that the eigenvalues are smaller
than 1 (since species 1 is at the stable resident equilibrium). The B3matrix is given by:
B3=
(1 −γ2)φ2
1+β21n∗
1a
π2
1+α21n∗
1a
γ2φ2
1+β21n∗
1a
sa2
.(A18)
The B3matrix corresponds to the projection matrix of species 2, which implies that the exclusion equilibrium
is not stable when species 2 has a growth rate larger than 1 (i.e., the modulus of B3’s leading eigenvalue is
larger than 1). The stability of this two-dimensional system can be investigated thanks to the Jury conditions,
20
given by the equations (A19,A20,A21):
1−tr J(n) + det J(n)>0(A19)
1 + tr J(n) + detJ(n)>0(A20)
1−det J(n)>0(A21)
with J(n)the Jacobian matrix of the system evaluated at the exclusion equilibrium. If these three conditions
hold, the system is stable. If the first condition (equation (A19)) is violated, one of the eigenvalues of J(n)
is larger than 1, which means that the excluded species can invade the community. The violations of the
conditions (A20) and (A21) correspond to the creation of limit cycles (Neubert and Caswell,2000). We then
check these conditions on the B3matrix. We have:
tr B3= (1 −γ2)φ2
1 + β21n∗
1a
+sa2
det B3= (1 −γ2)φ2
1 + β21n∗
1a
sa2−γ2
φ2
1 + β21n∗
1a
π2
1 + α21n∗
1a
.
We start with the third condition (A21):
1−(1 −γ2)φ2
1 + β21n∗
1a
sa2+γ2
φ2
1 + β21n∗
1a
π2
1 + α21n∗
1a
>0.(A22)
This equation will always be satisfied with biologically meaningful parameters and strictly competitive inter-
action between species:
•The third term of (A22) is positive because each parameter is positive,
•(1 −γ2)φ2
1+β21n∗
1a
sa2<1because 0≤γ≤1and 0≤φ≤1.
Since tr B3>0, the first Jury’s condition implies the second ((A19)⇒(A20)). Then, we only have to check
the first Jury condition to describe the stability of the system:
1−(1 −γ2)φ2
1 + β21n∗
1a
+sa2+ (1 −γ2)φ2
1 + β21n∗
1a
sa2−γ2
φ2
1 + β21n∗
1a
π2
1 + α21n∗
1a
>0
(1 −γ2)φ2
1 + β21n∗
1a−(1 −γ2)φ2
1 + β21n∗
1a
s2a+γ2
φ2
1 + β21n∗
1a
π2
1 + α21n∗
1a
<1−s2a
(1 −γ2)φ2(1 −s2a)<(1 −s2a)(1 + β21n∗
1a)−γ2φ2π2
1 + α21n∗
1a
(1 −γ2)φ2
1
1 + β21n∗
1a
+γ2φ2π2
1−s2a
1
(1 + β21n∗
1a)(1 + α21n∗
1a)<1.
We finally have the following condition for the stability of the equilibrium:
C2
1 + β21n∗
1a
+D2
(1 + β21n∗
1a)(1 + α21n∗
1a)<1(A23)
21
with C2= (1 −γ2)φ2and D2=π2γ2φ2
1−s2a.
This last expression (eq. 6) gives us an invasion criteria that is larger than 1 if species 2 is able to invade
the community when rare.
Note that if we remove the competition on one vital rate by setting all the competition coefficients
associated to this vital rate to 0, we re-obtain the invasion criteria given by Fujiwara et al. (2011) for their
model where only one vital rate was affected by competition. This ensures some internal coherence to the
analytical results.
Appendix C Coexistence regions obtained through sensitivity anal-
ysis for the three scenarios of coexistence
To have another, complementary representation of the sensitivities, we estimated coexistence regions for the
three parameter sets using the definition given by Barabás et al. (2014b). Note that we use the term region in
place of domain to separate the regions found with this sensitivity analysis from the domains of coexistence
found through invasion analysis. The method consists in using the sensitivities at the equilibrium for each
parameter in order to determine the region of parameter space where species both persists despite changes
in parameter values. It is calculated by finding the smallest (positive and negative) perturbation that would
lead to the extinction of one of the species from the value of sensitivities and densities at equilibrium. The
coexistence regions for the three parameter sets are presented in figure A1.
22
0
10
20
30
40
50
p1p2
Fertility
Parameter value
A
0.00
0.25
0.50
0.75
1.00
g1g2f1f2s1a s2a
Maturation rate, juvenile and adult survival
0.0
0.1
0.2
0.3
0.4
a11 a12 a21 a22 b11 b12 b21 b22
competition parameters
0
10
20
30
40
50
p1p2
Fertility
Parameter value
B
0.00
0.25
0.50
0.75
1.00
g1g2f1f2s1a s2a
Maturation rate, juvenile and adult survival
0.0
0.1
0.2
0.3
0.4
a11 a12 a21 a22 b11 b12 b21 b22
competition parameters
0
10
20
30
40
50
p1p2
Fertility
Parameter value
C
0.00
0.25
0.50
0.75
1.00
g1g2f1f2s1a s2a
Maturation rate, juvenile and adult survival
0.0
0.1
0.2
0.3
0.4
a11 a12 a21 a22 b11 b12 b21 b22
competition parameters
Figure A1: Parameter values and coexistence regions of the three parameter sets chosen to promote
different situations of coexistence.
We see here smaller coexistence regions in the third scenario, consistent with this equilibrium being more
sensitive (less robust) to perturbations.
Appendix D Additional simulations with Sspecies, S > 2
The meta-parameter sets, used in the parameter distributions from which the vital rate parameters for each
species have been drawn, are the following:
Distribution variance µπiσπia b µαii σαii µβii σβii µij σij
high 3 0.5 5 5 0.05 0.01 0.05 0.01 0.04 0.02
average 3 0.05 50 50 0.05 0.01 0.05 0.01 0.04 0.02
small 3 0.005 500 500 0.05 0.001 0.05 0.001 0.04 0.02
Table A2: (Meta)-Parameter sets used to draw parameter values to simulate community dynamics with
(a, b)parameters of the beta distribution for φ,saand γ(a=aφi=asa,i =aγiand b=bφi=bsa,i =bγi).
The last two columns correspond respectively to the mean of inter-specific coefficient (for both αij and βij )
and to the variance of the inter-specific coefficients (with the co-variance between αij and βij equal to
−0.9×σij ).
For initially 40 species, a small variance of parameters across species, and opposite competitive hierarchies,
we obtained the dynamics illustrated in figure A2.
23
0 500 1000 1500 2000 2500 3000
0 20 40 60 80
Generations
Individual per species
Population dynamics of 40 species with opposite competitive hierarchies
Figure A2: Population dynamics of a community with initially 40 species and opposite competitive
hierarchies on fertility and juvenile survival. All species have very similar vital rates due to a low variance
in the distribution used to draw the parameter. Each line corresponds to one species.
The population dynamics highlighted in figure A2 demonstrates that 3000 time steps are enough to reach
a stable number of coexisting species.
We verified our hypothesis that a negative correlation between αij and βij lead to opposite hierarchies in
sets of coefficients associated to fertility and juvenile survival by plotting the rank of αij against the rank of
βij . For 40 species, these ranks are plotted in figure A3.
24
0 500 1000 1500
0 500 1000 1500
Rank of aij
Rank of bij
Figure A3: Rank of βij versus rank of αij under a negative correlation of βij and αij corresponding to
φ= 0.9σ
The negative correlation between ranks in figure A3 confirmed that the negative correlation between αij
and βij was efficient to create the expected opposite competitive hierarchies. However, opposite competitive
hierarchies may not be sufficient to promote reciprocal exclusion (e.g., exclusion of 1 by 2 in the fertility
competition model and of 2 by 1 in the juvenile survival competition model) of species within each or most
pairs of species. To make the comparison between 2 and S-species model, we have to check that we consider
situations that are fully comparable, i.e., where species would exclude each other out in a two-species contest
on either αor βcompetition. We therefore checked if the trade-off between being competitive on αand β
could promote such reciprocal exclusion or priority effects (in models with a single parameter that is density
dependent) within pairs of species. We computed the invasion criteria of such simple models (where only one
vital rate is affected by competition) for each pair of species composing the communities. For a community
of initially 40 species and a small variance in parameters across species, we found the coexistence outcomes
in simpler models highlighted in figure A4.
25
We found that for a community of 40 species, the proportions of pairs of species where a priority effect or
a reciprocal exclusion are suggested (by the simpler models with density-dependence on one vital rate) are
low when we assume only a negative correlation between αij and βij .
0
250
500
750
1 2 3 4
1=Coexistence, 2=Priority effect, 3=Opposite exclusion, 4=Other case
count
Outcomes suggested by invasion criteria of pairs of species
Figure A4: Proportions of situations given by pairwise invasion criteria for a parameter set with 40 species
with very close parameters and a negative correlation between αij and βij with φ= 0.9σ
In fact, when we simply use a negative correlation to create opposite competitive hierarchies, we draw our
couple of values (αij,βij ) in a Gaussian distribution, and then most of the values are close to their means,
which does not allow to obtain pairs of species where each species is strongly competitive on a different vital
rate, and exclude the other if the competition was on this vital rate only. Therefore, even if there are opposite
competitive hierarchies within pairs of species modelled, this setup is unlikely to generate situations where
species exclude each other when considering single-vital-rate-competition.
To consider S-species scenarios where species can exclude each other out if competition was solely on
αor solely on β(hereafter referred to as ‘reciprocal exclusion’), we changed our structure of correlation to
generate situations where within most pairs of species, each species is strongly competitive on a different
26
vital rate (e.g. species 1 excludes species 2 in the model with only fertility competition and vice-versa in the
model with only juvenile survival competition).
Our method to achieve reciprocal exclusion of pairs (for single-vital-rate-competition) consisted in drawing
pairs (aij , bij )from a bivariate normal distribution with µaij<µbij and Corr(aij , bij )<0. We then always
assign (aij , bij )to (αij , βij )and (bji , aji )to (αji , βji). In this way, for each pair of species (i, j ), species iis
competitive on αand species jis competitive on β. We acknowledge that this may be hard to grasp, and
suggest to the reader to use the code to create an example if required.
We computed the invasion criteria for each pair of species and counted how often each invasion scenario
in single-density-dependence models, which we represented in the histogram of figure A5.
0
500
1000
1500
3 4
1=Coexistence, 2=Priority effect, 3=Opposite exclusion, 4=Other case
count
Outcomes suggested by invasion criteria of pairs of species
Figure A5: Proportions of situations given by pairwise invasion criteria - for Fujiwara et al. style models -
for a parameter set with 40 species with very close parameters and bimodal distribution of interspecific
coefficients drawn to promote situations of opposite exclusion within each pair of species.
We succeeded to generate reciprocal exclusion within most pairs of species. Moreover, we showed that
a permutation in the competition coefficient was sufficient to remove the structure of reciprocal exclusion
27
(figure A6).
0
300
600
900
1 2 3 4
1=Coexistence, 2=Priority effect, 3=Opposite exclusion, 4=Other case
count
Outcomes suggested by invasion criteria of pairs of species
Figure A6: Proportions of situations given by pairwise invasion criteria for a parameter set with 40 species
with very close parameters and permuted competition coefficient that initially promoted situations of
reciprocal exclusion within each pair of species.
However, as for our simple negative correlation between competitive ranks for αand β, this new parameter
set where reciprocal exclusion occurs within most pairs of species does not allow to significantly increase the
number of extant species at t= 3000 time steps (figure A7).
28
40 species, low variance
Number of species at time t
Frequency
0 10 20 30 40
0 10 20 30 40 50 60
Figure A7: Histograms of the number of species persisting at equilibrium with initially 40 species for 200
permutations of inter-specific competition coefficients. The bar in gray indicates the number of species
persisting at equilibrium for the original αij and βij with Corr(αij , βij )<0and the opposite exclusion for
each pair of species.
Therefore, even if competition is highly structured in a way that makes species exclude each other out when
a single vital rate is density-dependent, which should greatly promote situations of emergent coexistence, we
did not observe a positive effect of this structure on species richness in a community larger than S= 2. We
therefore conclude that emergent coexistence is highly unlikely in many-species context.
29
Supplement A Invasion analysis for a model with Moll & Brown
(2008)’s life history structure
We performed our invasion analysis on the model with a life-history structure similar to Moll and Brown
(2008). The projection matrix of this model was given in equation 2.1. The exact same method as in section
2.3 was used to compute the invasion criteria with M&B’s life-history structure.
We obtain the coexistence outcome domains given in figure S1 with the βcoefficients given in table S1.
The results are qualitatively similar to those presented in the main text with adults →adults + juveniles.
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5
a21
a12
Competition on fertility only (β = 0)
A
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5
a21
a12
Competitions on both vital rates with β = α
B
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5
a21
a12
β suggests coexistence
C
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5
a21
a12
β suggests exclusion of species 1
D
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5
a21
a12
β suggests a priority effect
E
Outcome predicted by the invasion criteria
Coexistence
Priority effect
Species 1 excludes species 2
Species 2 excludes species 1
Figure S1: Outcomes of the model with a structure similar to M&B model predicted by invasion analysis
depending on the coefficients of the inter-specific competition on fertility (α12 and α21)with βcoefficients
suggesting coexistence, exclusion of species 2 and priority effect. The dotted lines correspond to the
outcome boundaries of figure A which is used as a reference
Figure β12 β21
C 0.06 0.06
D 0.11 0.05
E 0.12 0.12
Table S1: βcompetition coefficients used to obtain contrasting situations of coexistence.
30
