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Theor Ecol (2015) 8:349–368
DOI 10.1007/s12080-015-0255-y
ORIGINAL PAPER
The influence of host competition and predation on tick
densities and management implications
Christina A. Cobbold ·Jack Teng ·
James S. Muldowney
Received: 15 November 2014 / Accepted: 22 January 2015 / Published online: 22 May 2015
© Springer Science+Business Media Dordrecht 2015
Abstract Host community composition and biodiversity
can limit and regulate tick abundance which can have pro-
found impacts on the incidence and severity of tick-borne
diseases. Our understanding of the relationship between
host community composition and tick abundance is still
very limited. Here, we present a novel mathematical model
of a stage-structured tick population to study the influence
of host behaviour and competition in the presence of het-
erospecifics and the influence of host predation on tick
densities. We examine the influence of specific changes
in biodiversity that modify the competition among and the
predation on small and large host populations. We find
that increasing biodiversity will not always reduce tick
populations, but depends on changes in species compo-
sition affecting the degree and type competition among
hosts, and the host the predation is acting on. With
indirect competition, tick densities are not regulated by
increasing biodiversity; however, with direct competition,
C. A. Cobbold ()
School of Mathematics and Statistics, University of Glasgow,
Glasgow G12 8QW, UK
e-mail: christina.cobbold@glasgow.ac.uk
J. Teng
Resource Management and Environmental Studies, 2202 Main
Mall, University of British Columbia, British Columbia,
Vancouver, V6T 1Z4, Canada
e-mail: jiayang.teng@gmail.com
J. S. Muldowney
Department of Mathematical and Statistical Sciences,
University of Alberta, Edmonton, T6G 2G1, Canada
e-mail: jim.muldowney@ualberta.ca
increased biodiversity will regulate tick densities. Gen-
erally, we find that biodiversity will regulate tick den-
sities when it affects tick-host encounter rates. We also
find that predation on small hosts have a limited influ-
ence on reducing tick populations, but when the predation
was on large hosts this increased the magnitude of tick
population oscillations. Our results have tick-management
implications: while controlling large host populations (e.g.
deer) and adult ticks will decrease tick densities, mea-
sures that directly control the nymph ticks could also be
effective.
Keywords Stage-structured model ·Biodiversity ·
Competition ·Predation ·Tick population management ·
Tick-borne zoonoses
Introduction
Most vectors of zoonotic pathogens, diseases normally
present in wildlife that can be transmitted to humans, are
host generalists (Keesing et al. 2009). Recent studies have
suggested that high biodiversity in the host community can
prevent the emergence and spread of zoonoses (Ostfeld
and Keesing 2000). Hosts form the vector’s food source;
indeed, the presence or absence of particular hosts will
determine vector densities (Sonenshine 2005). However,
the vector’s hosts will be influenced by the other species
present in their ecosystem, which may compete with them
for resources or prey on them. As a result, the presence
of host competition or predation can potentially regulate
the populations of the hosts, and thus regulate vector pop-
ulations. Our objective is to gain an understanding of how
350 Theor Ecol (2015) 8:349–368
host biodiversity and the interactions within a complex host
community may limit and regulate vector abundance. In
this paper, we describe a stage-structured model of vector
dynamics. We use the model to examine how vector abun-
dance is influenced by specific changes in biodiversity that
affect competition among and predation on the vector’s host
population.
We focus our study on the disease vector: ticks. Ticks can
transmit numerous zoonoses (Gratz 1999; Goodman et al.
2005; Jongejan and Uilenberg 2004), of which Lyme dis-
ease is perhaps the most well known. Lyme disease, caused
by the spirochete Borrelia burgdorferi s.l. and present in
North America, Europe and Asia (Barbour and Fish 1993;
Ogden et al. 2008; Sperling and Sperling 2009), is a debil-
itating neurodegenerative disease with over 20,000 new
cases reported per year in the USA (Hanincova et al. 2006).
However, ticks are also the vectors of many other diseases
such as Rocky Mountain fever, Babesiosis, Ehrlichiosis,
Tick-Borne Encephalitis (Walker 1998; Labuda and Nut-
tall 2004). Here, we focus on hard ticks (e.g. ixodes spp.,
Dermacentor spp.), rather than soft ticks (e.g. Argas spp.,
Alveonasus spp.); hard ticks are diurnal and search for hosts
in the environment, while soft ticks are nocturnal and search
for hosts in their nests or burrows (Oliver 1989; Need-
ham and Teel 1991). Depending on the life-stage, ticks
normally feed on either small hosts (e.g. deer mice, Per-
omyscus mainculatus, or chipmunks, Tamias townsendii)
during the larval and nymphal life- stages, and, large hosts
(e.g. deer, Odocoileus hemionus) during the adult life-stage.
However, ticks are opportunistic feeders that attempt to feed
on any animal they encounter, they have been reported on
more than 1000 species of mammals, birds and reptiles
(Ostfeld and Keesing 2000).
One way in which host biodiversity can potentially have
an impact on tick abundance is through tick-host encounter
rates. The hosts provide the blood meal required for ticks
to molt to their next life-stage and the feeding success
and survival of ticks varies between host species (Keesing
et al. 2009). Recent studies have begun to provide evidence
that disease vectors can elicit host choice and preference.
Mosquitos are an example of such a vector, where host pref-
erence is evident from patterns in feeding indices which
could not be simply explained by random foraging and host
availability alone (Lyimo and Ferguson 2009). In ticks, data
suggests a preference for opossums and squirrels over mice
(Keesing et al. 2009). Likewise, there is evidence for hosts
displaying behavioural changes in response to the presence
of heterospecifics, hosts of different species, which also
impacts on vector-host encounter rate. For instance, deer
mice, a host for larval and nymphal stage ticks, have been
found to change their foraging behaviour in the presence
of other competitors (Davidson and Morris 2001). Similarly
deer, a host for adult ticks, changes their behaviour when
other large hosts are present (Hobbs et al. 1996;Latham
1999). Tick-host encounters can also increase if hosts show
more clumping in the presence of heterospecifics (Keesing
et al. 2006). These behaviour modifications are examples
of direct host competition, whereby hosts modify their
behaviour in the presence of heterospecifics. Host biodi-
versity also effects indirect competition between hosts. We
use the term indirect competition to refer to competition
between hosts that simply effects their relative abundance.
Hosts competing indirectly do not modify their encounter
rate with ticks based on the composition of the ecosystem,
which is in contrast to direct competition. There is mixed
evidence for indirect competition. Chipmunks and mice
are both small granivorous rodents so are likely to com-
pete with each other for resources; however, some recent
empirical evidence puts this into question (Brunner et al.
2013), and yet it is indirect competition that is commonly
used in theoretical studies of tick dynamics in multi-host
systems.
Changes to ecosystem biodiversity through the introduc-
tion of predators can also effect the vector-host system.
In particular, high levels of predation can induce popula-
tion fluctuations in the host (Ostfeld and Keesing 2000;
Ostfeld and Holt 2004). Host fluctuations can also be a
result of bottom-up and environmental processes such as
masting events (Giardina et al. 2000). Both the behavioural
responses of hosts to heterospecifics and the fluctuating
host populations associated with high predator abundance
mean that increasing the abundance of any individual host
species may not have a simple additive effect on tick abun-
dance. Indeed, an empirical study by Keesing et al. (Keesing
et al. 2009) found removal of hosts from a community could
increase tick numbers, and that host identity may be key
to understanding this. Here, we address this knowledge gap
and study the relationship between tick abundance and host
behaviour and abundance in the presence of heterospecifics.
Most previous studies, both empirical and theoretical
have focussed on the dynamics of tick-borne pathogens and
given less consideration to how host community composi-
tion influences tick abundance. These works have examined
the transmission dynamics of tick-borne zoonoses using
stage-structured models of the different tick life-stages and
SIR epidemiological models of the disease (e.g. Caraco et
al. 1998;Mwambi2002; Randolph and Rogers 1997;Ros
´
a
et al. 2003; Stanko et al. 2007). Using this approach, stud-
ies have gained insights on how transmission dynamics are
affected by factors such as climate and seasonality (Ghosh
Theor Ecol (2015) 8:349–368 351
and Pugliese 2004; Brownstein et al. 2003; Ogden et al.
2006) or metapopulation and spatial dynamics (Caraco et al.
2002; Gaff and Gross 2007). But, while some works have
examined multi-host systems (Norman et al. 1999;Schmidt
and Ostfeld 2001; Dobson 2004), they were interested in the
abundance of pathogens and did not include predation, nor
did they distinguish between the different hosts of the tick
life-stages (i.e. small hosts vs. large hosts; but see (Ogden
et al. 2005) for the influence of only predation, though
without host competition).
Recent studies have begun to consider the impact of
variation in host-vector encounter rates. Wonham et al.
(2006) demonstrated that the choice of transmission term
in their west Nile virus models, equivalent to a combined
feeding and encounter rate in the tick setting, could both
qualitatively and quantitatively alter predictions. Lou and
Wu (2014) considered the role of frequency-dependent,
density-dependent and Holling type II vector-host contact
rates in their model of the tick life-cycle. Frequency-
dependent contact is independent of host density and as
such tick densities are unaffected by host densities. In con-
trast, density-dependent contact rates depend linearly on
host densities. Density-dependent contact is used in the
majority of tick models in the literature, with a fixed trans-
mission probability per contact (here, transmission means
the transmission of a blood meal from the host to the
vector). Density-dependent contact is a good model when
hosts compete indirectly, such that host abundance is the
soul outcome of competition. However, none of these pre-
vious investigations have taken into account when host
behaviour and host-vector encounter rates are modified by
the presence of heterospecifics. Depending on the effect
of heterospecifics on a particular host species, their pres-
ence may increase or decrease the contact rate between
the host and vector. We refer to this as an effect of direct
competition.
Here, we take a phenomenological approach to exam-
ining how biodiversity can regulate tick density through
ecological processes. Given the variety of hosts that ticks
can feed on and the limited number of studies of host
behavioural responses to heterospecifics that currently exist
in the literature, we chose not to model the host popu-
lation explicitly, but instead we implicitly assume hosts
are either at their demographic equilibrium or fluctuating.
By not explicitly modelling host demographic processes,
but merely describing the final host density, we have the
flexibility to explore consequences of host competition by
varying the choice of function describing vector-host con-
tact rates, accounting for host behavioural responses to the
other species in the ecosystem without explicitly modelling
the complex ecosystem. We can also account for the effects
of biodiversity that result in increased or decreased preda-
tion on tick hosts by describing host densities by fluctuating
functions, but without specifying the detailed biological
interactions that give rise to these fluctuations. In the model,
we consider two types of small host and two types of
large host, which is the minimum needed to model the
behavioural effects of competition between small hosts and
between large hosts. This approach to studying the effects
of biodiversity on tick abundance has the virtue of illuminat-
ing how sensitive tick dynamics may be to these features. In
the discussion, we return to the role of ticks in the spread of
zoonoses, which allows us to more readily relate our find-
ings to those in the literature. We discuss the implications
of our findings on the pathogen basic reproduction number
and on the nymphal infection prevalence, the two measures
of disease risk commonly applied to study the tick-borne
zoonosis, lyme disease.
Stage-structured tick model
We develop a stage-structured model of the tick life-cycle
based on their ecology and life history, using empirical
field data from the previous works (Ogden et al. 2005;
Caraco et al. 2002; Gaff and Gross 2007; Perkins et al.
2006; LoGiudice et al. 2003; Giardina et al. 2000). The
model accounts for the questing life stages of the tick
that are dependent on obtaining a host blood meal: larva
(x1), nymph (x2)and adult (x3). Questing ticks are not
attached to a host, but are dormant or searching for one
to attach to. Once a tick finds and attaches itself to
a host, it may take up to 1–3 days to obtain a blood
meal, after which it detaches from the host and molts
into the next stage (Needham and Teel 1991). As our
focus is on hard ticks (e.g. Ixodes spp. and Dermacen-
tor spp), we model ticks such that they require only a
single blood meal to molt to the next stage (Sonenshine
2005). To study the effects of small host biodiversity, we
make the assumption that tick larvae and nymphs can
parasitise either their most common small host, H1(i.e.
deer mice or white-footed mice), or an alternative small
host, H2(e.g. chipmunks and birds). Similarly, we make
the assumption that adult ticks can either parasitise their
most common host H3(i.e. deer) or an alternate large
host H4(e.g. raccoons, cattle and horses). We make the
simplifying assumption that ticks do not explicitly regu-
late the density of their hosts, while the hosts can regulate
the tick numbers. A resent study on white-footed mouse
survival supports this assumption (Hersh et al. 2014).
352 Theor Ecol (2015) 8:349–368
Our assumption allows us to consider the host dynam-
ics independently of the ticks and so we do not
explicitly model the hosts. Instead, we assume host
density is either a constant or an oscillating func-
tion of time. Oscillations describe the temporal effect
of predation pressure or environmental conditions on
host density.
We describe the tick population with three ordinary
differential equations corresponding to the three tick life-
stages:
(1)
(2)
(3)
The three tick stages undergo natural mortality at rate μi.
After each blood meal, a new tick stage is produced: new
larvae are produced from the eggs laid by adult ticks;
new nymphs are produced from larvae, and new adults
are produced from nymphs; βiis the number of stage i
ticks produced by a tick in the previous stage. For exam-
ple, β1is the number of new larvae produced per fed and
mated adult. Hence, development from one stage to the
next gives rise to a loss and production term for each equa-
tion. The transition from a larvae to nymph and nymph to
adult produce at most a single new stage, while the adult
is the only stage that can produce eggs and hence multi-
ple larvae. The final term in Eq. 3accounts for the fact
that adult ticks can only produce one batch of eggs in
their lifetime. Once an adult tick has found a blood meal
and mated the females will overwinter and lay their eggs
the following spring and will no longer be contributing
to the numbers of questing adults (Ostfeld and Keesing
2000).
Each tick stage iobtains its blood meal from its preferred
or alternate host, Hn, where a host can carry an average of
λi,n stage iticks per time unit. Since each host can carry
a maximum number of ticks the production terms saturate
with a type II functional response in tick density (Brunner
and Ostfeld 2008), where aiis half the maximum number
of stage iticks per hectare. The full list of parameters and
their values are summarised in Table 2.
Implicit modelling of host competition The probability that
larvae find and feed on host nis given by σn(y), where
yis the proportion of small (large) hosts that are of type
Hnand host nis a small (large) host. Models in the lit-
erature often make the simplifying assumption that σn(y)
is a constant (e.g. Norman et al. 1999; Sandberg and
Awerbuch 1992). We, however, include the effects of host
competition by relaxing this assumption and allowing the
feeding probability to depend on the relative abundance
(y) of a given host. By varying the relative abundance
of either small (large) host and keeping the total num-
ber of small (large) hosts fixed, we simulate the effects
of competition. We summarise our choices of σn(y ) in
Table 1.
When σn(y) is a constant (σn(y ) =1), the hosts do
not affect each others behaviour and only complete for
resources, this implies indirect competition, such that the
difference in abundance between hosts reflect their ability
to forage (Wooton 1994). In this case, the ticks’ proba-
bility of finding and feeding on a host is not affected by
Theor Ecol (2015) 8:349–368 353
Table 1 Functional forms of σn(y),whereyis the proportion of small (large) hosts that are of type Hn, where host nis a small (large) host. Thus,
for n=1,then y=ps, while for n=2,3and4thenyis 1 −ps,pLand 1 −pL, respectively
Competition Probability of finding and feeding on host nNotes
Indirect σn(y) =1
Direct σn(y) =φny
φny+(1−φn)(1−y) =yφ
1=1−φ2and φ3=1−φ4
(no behavioural difference among hosts, φn=0.5)
Direct σn(y) =φny
φny+(1−φn)(1−y) σ1(y) =1−σ2(1−y) and
(behavioural difference among hosts, φn= 0.5) σ3(y) =1−σ4(1−y)
the relative composition of the host community and the
term describing tick-host interaction is simple mass action
(density-dependent contact). Finding the host is only condi-
tional on its presence and so if the tick encounters the host
it feeds with probability 1.
When we have direct competition between hosts, tick-
host encounter rates can depend on the relative abundance
of the hosts in a nonlinear manner and the probability of
finding and feeding on host nis given by
σn(y) =φny
φny+(1−φn)(1−y),(4)
where φnis the probability of encountering host nas
opposed to encountering the alternative host of the same
type (small/large). Note φ1=1−φ2and φ3=1−φ4.
The parameter φnreflects behavioural differences of host n
in response to heterospecifics. Deer mice have been found
to change their foraging behaviour in the presence of other
competitors (Davidson and Morris 2001). When more hosts
of one type are present, the probability of encountering
the other hosts may be much lower or higher than pre-
dicted by relative abundance alone, since the behaviour of
one host may change in the presence of the more abundant
competitor; hence, differences in the hosts’ relative abun-
dances would lead to a nonlinear relationship in the ticks’
host-finding probability, similar to the relationship proposed
by Ros´
aetal.(2003). Equation 4describes direct com-
petition, since the hosts’ behaviour and ability to forage
for resources, and hence contact ticks, are affected by the
presence or absence of the other host and competitor. In
particular, σ1+σ2=1 (similarly σ3+σ4=1) so the prob-
ability of feeding on host 1 is determined by the probability
of feeding on host 2, which is in contrast to indirect compe-
tition. When φn=0.5, there is an equal probability of the
ticks encountering either small (large) hosts and σn(y) =y.
When φn>0.5 ticks have a higher probability of encoun-
tering Hnthan would be suggested by relative abundance
alone; with a value of φn<0.5 ticks have a lower proba-
bility of encountering Hn. We denote ps=H1/(H1+H2)
as the proportion of small hosts that are of type H1.Sim-
ilarly, we denote pL=H3/(H3+H4), the proportion of
large hosts of type H3. Thus, in Eq. 4,yis chosen to be ps,
(1−ps),pLor 1 −pLfor the cases n=1,2,3 and 4,
respectively.
Implicit modelling of predation or seasonality To include
the effects of predation on the tick hosts, we account
for the presence of predators through the dynamics of
the host term Hn(t). Assuming Lotka—Volterra predator-
prey interactions in the ecosystem, we model the host
population, Hn(t), as a constant or a periodic func-
tion depending on the predator, prey (host) species and
the environment we wish to describe. When predation
pressure is low, the host population is constant, and
when predation pressure is high, the host population is
assumed to be oscillating, where the amplitude of the
oscillations reflects the intensity of predation. For tem-
porally oscillating host populations, we use a cosine
function.
Since we consider small and large hosts separately, when
we discuss biodiversity of small hosts, we are referring to
a population of two ‘species’, H1and H2and so diver-
sity is maximised when ps=0.5. As psmoves away
from 0.5, in either direction, diversity declines and reaches
a minimum at ps=0 or 1. The same argument applies
to large host diversity where pLis used as a proxy for
diversity.
Analytical results
In this section, we consider the case when Hnis constant
for all n. We calculate general analytical conditions for
tick population persistence and cyclic dynamics. These both
have management implications that are further examined in
our numerical analysis: persistence criteria can be used to
identify conditions where ticks may be eradicated, while
354 Theor Ecol (2015) 8:349–368
criteria for cyclic behaviour can be used to identify con-
ditions where tick population dynamics are unstable and
potentially vulnerable to interventions. To analyse the
model, we introduce some simplifying notation. Let the
coefficients in front of the second term of equation ibe
denoted by αiand the coefficients in front of the third term
be denoted by γi. Thus (1)–(3) can be rewritten as:
˙x1=−μ1x1+α1x3
a3+x3
−γ1x1
a1+x1
,(5)
˙x2=−μ2x2+α2x1
a1+x1
−γ2x2
a2+x2
,(6)
˙x3=−μ3x3+α3x2
a2+x2
−γ3x3
a3+x3
.(7)
We note that
γ1=α2/β2≥α2and γ2=α3/β3≥α3,(8)
because each larvae produces at most one nymph and each
nymph produces at most one adult. Similarly,
γ3=α1/β1≤α1
as each adult female successfully produces of the order of
350 female eggs.
As the model is stage-structured, it can be easily seen that
the only axial equilibrium is the trivial (0,0,0)equilibrium.
To examine the stability of this equilibrium and address the
question of population persistence, we consider the equation
for the total tick population, x=x1+x2+x3. Thus,
˙x=−μ1x1−μ2x2−μ3x3+(α2−γ1)x1
a1+x1
(9)
+(α3−γ2)x2
a2+x2
+(α1−γ3)x3
a3+x3
.
All of the terms in Eq. 9are negative except the final term,
which saturates for sufficiently large x3. As the final term
is bounded while the first three terms grow linearly with x,
for all sufficiently large x,wehave ˙x<0 and hence the
population is bounded and the system is dissipative, in other
words the tick population is self-regulated. The necessary
condition for persistence is α1>γ
3. This is equivalent to
requiring that, on average, adult ticks produce more than one
surviving offspring. A sufficient condition for persistence
can be obtained from standard stability analysis and apply-
ing the Routh-Hurwitz criteria (Murray 1989). Persistence
occurs if
(μ1a1+γ1)(μ2a2+γ2)(μ3a3+γ3)−α1α2α3<0.(10)
Equation 10 is a local stability condition. Rearranging (10)
gives us an expression for Rtick, the average number of
female adult ticks produced by a single female during her
lifetime,
Rtick =α1
(μ3a3+γ3)
α2
(μ1a1+γ1)
α3
(μ2a2+γ2).(11)
The first term in Eq. 11 corresponds to the average
number of larvae produced by an adult female over her
lifetime and the next two terms correspond to the respec-
tive probabilities that a larvae will survive to become a
nymph and a nymph will survive to become an adult.
Biologically, this is not particularly informative as it
involves all of the model parameters and it is diffi-
cult to discern the relative importance of any particular
process.
By a novel application of compound matrix theory
and constructing Lyapunov functions (see Appendix A
for details) global stability criteria can be found which
establish when population persistence is not possible.
These conditions involve less parameters and are there-
fore biologically more informative. In a similar man-
ner, we can construct criteria for when the system does
not exhibit periodic orbits. Table 3summarises these
results.
Criteria A and B describe conditions for the persis-
tence of tick populations. Criteria A can be rearranged to
see that it corresponds to the average number of larvae
produced by an adult female over her lifetime being less
than 1. It can be reexpressed in the original parameters as
μ3>β
1[σ3(pL)H3λ3,3+σ4(1−pL)H4λ3,4]/a3. From
this, reducing numbers of large hosts (H3and H4) can lead
to tick eradication; however, the reduction would need to
be of the order of a thousand fold reduction in deer den-
sity to around 0.0002 deer per hectare. On the other hand,
if we consider the alternative criteria B, then of the three
inequalities μ1>α1
a3is the most difficult inequality to
satisfy due to the high larval production by adults (α1).
But, interestingly, if larval mortality is sufficiently high and
small host density is low, the three conditions that con-
stitute criteria B can be satisfied and tick eradication is
possible. This finding is in line with Loguiudice (LoGiudice
et al. 2008) who suggest there are frequent tick extinc-
tions in small habitat fragments where one might expect
tick mortality to be high and hosts to be present in low
densities.
Criteria C and D give the conditions for when tick
population cycles are absent and the dynamics are sta-
ble. As with criteria A, criteria C is difficult to satisfy
and would require an extreme reduction in the number of
large hosts. Criteria D describes more practical conditions
for stable tick populations, which is achieved when either
small hosts are reduced or by increasing larval and nymph
mortality.
In addition to the extinction equilibrium, the model has
a coexistence equilibrium. However, it is not possible to
derive an explicit analytical expression for this equilibrium;
it can be found by numerically solving an implicit equation.
Theor Ecol (2015) 8:349–368 355
00.2 0.4 0.6 0.8 1
0.6
0.8
1
1.2
1.4
1.6
1.8 x 104
pL
x (Ticks per ha)
(c)
00.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
7
8x 104
pL
(d)
00.2 0.4 0.6 0.8 1
0.6
0.8
1
1.2
1.4
1.6
1.8 x 104
ps
x (Ticks per ha)
(a)
00.2 0.4 0.6 0.8 1
0
0.5
1
1.5
2
2.5 x 104(b)
ps
Hs=150
Hs=100
Hs=50
HL=0.2
HL=10
HL=1
HL=0.1
Fig. 1 The effect of direct and indirect host competition. In aand b,
total tick density is plotted as a function of ps=H1/Hs, the propor-
tion of small hosts that are of type H1.Ina,wefixσ3(y ) =σ4(y) =1
and vary the functional form of σn(y),n=1,2. The solid line is
indirect competition σ1(y) =σ2(y ) =1; the remaining lines repre-
sent direct competition, where the dotted line represents no preference
(φ1=0.5), the dashed line represents preference for H1(φ1=0.8),
and the dash-dot line represents preference for H2(φ1=0.2). In b,we
fix σn(y) =1, for each nand vary Hs.Incand d, total tick density is
plotted as a function of pL=H3/HL, the proportion of large hosts that
are of type H3.Inc,wefixσ1(y) =σ2(y ) =1 and vary the functional
form of σn(y),n=3,4, as with athe solid line is indirect competi-
tion; the dotted line is φ3=0.5, the dashed line is φ3=0.8andthe
dash-dot line is φ3=0.2. dWe fi x σn(y ) =1 for each nand vary the
total large host density, HL. Unless otherwise stated parameters are a
giveninTable2and Hs=H1+H2=100, HL=H3+H4=0.2,
σn(y) =1andps=pL=0.5
Thus, in the next section, we numerically examine the coex-
istence equilibrium and how it is impacted by changes in
host competition and predation.
Numerical results
The influence of host competition
Throughout this section, we assume the host popula-
tion is at equilibrium, such that Hnis a constant. We
focus on changes in biodiversity that modify host com-
petition and hence tick-host contact rates, via the func-
tion σn(y), and examine the effects on equilibrium tick
densities.
Indirect competition between hosts In Fig. 1a, we explore
how psaffects total tick density. The total density of small
hosts (HS) is fixed, thus H1=psHsand H2=(1−ps)Hs.
Under indirect competition, σn(y) =1 for each n,andthe
equilibrium total tick density is found to depend on psin a
monotonic fashion, decreasing with higher densities of H1,
(ps→1). H1is a less suitable host for the nymphal class,
but more suitable for the larval class. However, since the
parameters in the production terms in the nymph equation
are smaller than those in the larval equations, the nymph
production determines the rate of total tick production. Con-
sequently, as host H2can support the highest number of
nymphs (compare λ21 and λ22 in Table 2), maximising the
density of H2hosts maximises tick densities and hence we
see tick densities in Fig. 1a are maximised when ps=0and
356 Theor Ecol (2015) 8:349–368
Table 2 Description and typical parameter values used in numerical simulations
Parameter Description (units) Val u e
μ∗
1Natural mortality of larvae (yr−1)1.1
μ∗
2Natural mortality of nymph (yr−1)0.73
μ∗
3Natural mortality of adult (yr−1)0.037
β∗∗
1Number of female eggs laid by an adult which survive to be larvae (larval ticks/ adult tick) 350
β∗∗
2Number of nymphs produced by a larvae (nymph ticks/ larvae tick) 1
β∗∗
3Number of adults produced by a nymph (adult ticks/nymph tick) 1
λ∗∗∗
1,1Average tick load of larvae on H1(ticks/deer mice/yr) 28.95
λ∗∗∗
1,2Average tick load of larvae on H2(ticks/chipmunk (birds)/yr) 12.57 (3)
λ∗∗∗
2,1Average tick load of nymphs on H1(ticks/deer mice/yr) 0.68
λ∗∗∗
2,2Average tick load of nymphs on H2(ticks/chipmunk (bird)/yr) 4.90 (9.67)
λ∗∗∗∗
3,3Average tick load of adults on H3(ticks/deer/yr) 201.84
λ∗∗∗∗
3,4Average tick load of adults on H4(ticks/racoon/yr) 69.54
a∗∗∗∗∗
iHalf the maximum number of stage iticks per hectare 650
H∗∗∗∗
1Number of mice per hectare 0-100
H∗∗∗∗
2Number of chipmunks (birds) per hectare 0-50 (31)
H∗∗∗∗
3Number of deer per hectare 0.075-0.4
H4∗∗∗∗ Number of raccoon per hectare 0.2
psProportion of small hosts of type H1H1
H1+H2=H1
Hs
pLProportion of large hosts of type H3H3
H3+H4=H3
HL
The data refers to Ixodes scapularis ticks. Average tick loads λi,n account for successful molting of the tick which is typically around 50 %
(LoGiudice et al. 2003). β1assumes an average of 1000 eggs produced per engorged adult, of which 70 % survive (Gaff and Gross 2007)ofwhich
half are female, giving the 350 adopted in the paper. In the absence of data on alternative large hosts (H4), we used raccoon data, a common
tick host with large tick burdens. We convert ticks per host into ticks produced per host per year by multiplying by the reciprocal of the average
duration of the tick stage. We estimated tick mortality assuming that larvae are the most sensitive stage (Ogden et al. 2005; Caraco et al. 2002;
Gaff and Gross 2007)
∗(Ogden et al. 2005), ∗∗(Gaff and Gross 2007), ∗∗∗ (Giardina et al. 2000), ∗∗∗∗ (LoGiudice et al. 2003), ∗∗∗∗∗(Perkins et al. 2006)
minimised when ps=1. These results hold for a range of
values of Hs(see Fig. 1b). So, we conclude that under indi-
rect host competition maximising biodiversity in small hosts
(ps=0.5) does not reduce tick densities; instead, reducing
or completely eliminating the abundance of the host which
is more suitable for nymphs (H2) would most effectively
reduce tick densities.
To study indirect competition between the large hosts, we
vary pL, while fixing the total density of large hosts, HL,
thus H3=pLHLand H4=(1−pL)HL. The results are
similar to those found under indirect competition between
small hosts. In Fig. 1c, we see that increasing biodiversity
in large hosts (pL→0.5) does not decrease tick densi-
ties. Tick densities are maximised when there are only large
hosts of type H3(pL=1); this is because H3hosts can carry
the largest burden of adult ticks, and only the adult tick feeds
on the large hosts.
Direct competition between small hosts. Under direct com-
petition host finding probability depends on relative host
abundance. Here we assume σ3(y) =σ4(y) =1 while
σ1(y) and σ2(y ) are given in Eq. 4. We varied the proba-
bility of ticks encountering H1from φ1=0.5, 0.8 to 0.2.
Under direct competition, we find that increasing biodiver-
sity in small hosts, away from the extremes of ps=0or
1, can lower tick densities, such that there is a minimum
in tick densities at an intermediate value of ps(Fig. 1a). In
accordance with the previously described role of nymphs
as a rate limiting life-stage, when φ1=0.5, the minimum
tick density lies to the right of ps=0.5 and so higher
proportions of H1hosts lead to the lowest tick density,
because the H1host is the less suitable hosts for the nymphs.
However, the minima can shift closer towards ps=0or
ps=1, when ticks have a respectively higher probabil-
ity of encountering host H1or H2(φ1equal to 0.8 or 0.2).
We can estimate the location of the minima be considering
the rate limiting step in tick production. In Eq 5,γ2deter-
mines the rate nymphs become adults. Taking φ1=0.5,
thus σ1(ps)=psand σ2(1−ps)=1−ps,wehave
γ2=psH1λ2,1+(1−ps)H2λ2,2. Applying H1=Hsps
and H2=Hs(1−ps),thenγ2is minimised, with respect
to ps,when2psHsλ2,1−2(1−ps)Hsλ2,2=0, that is,
Theor Ecol (2015) 8:349–368 357
when ps=λ2,2/(λ2,1+λ2,2)=0.88. The minimum
of the dotted curve for φ1=0.5 in Fig. 1a is located at
ps≈0.85, in good agreement with our estimate. Notice
that this estimate of the location of the minimum is inde-
pendent of total small host density and is determined only
by average nymph loads on the hosts. So, the identity of
the host species and the tick loads they carry is important
to determining whether high biodiversity will lower tick
densities.
Direct competition between large hosts Here, we assume
σ1(y) =σ2(y ) =1 while σ3(y) and σ4(y) are given by Eq.
4. Under direct competition between large hosts tick den-
sities are minimised when there is high biodiversity among
the large hosts, that is, at intermediate values of pL(Fig.
1c). When the ticks have no host preference (φ3=0.5),
the minimum is located to the left of pL=0.5, that is
majority of large hosts are of type H4, which is a less suit-
able host for the adult ticks. When there is a preference
for H3(e.g. φ3=0.8) or H4(e.g. φ3=0.2), the min-
ima shifts to the right or left, respectively. Tick densities are
minimised when γ3, the rate adults produce larvae, is min-
imised. In an analogous calculation to the small host case,
we ask what value of pLminimises γ3in the simple case
that φ3=0.5. We find pL=λ3,4/(λ3,3+λ3,4)=0.26,
which is a good approximation to the minimum in Fig. 1c
and is determined only by adult tick densities per host. The
identity of the large hosts in the ecosystem will be impor-
tant in determining if high biodiversity will reduce tick
densities.
Both the indirect competition results and the direct com-
petition results hold for a range of values for HL(see Fig.
1d). However, we find that, when large-host densities are
very high (HL=10), changing pLhas little effect on the
equilibrium tick density and there is no longer a minimum.
In other words, here biodiversity has an effect only when
large hosts are at low enough densities, that is, at eco-
logically relevant levels for deer and other wildlife; while,
in the presence of high densities of large hosts (e.g. cat-
tle farms), changing biodiversity (pL) has a minimal effect
on tick numbers. At high values of HL, the availability of
large hosts no longer limits larval production. So, although
increasing biodiversity of large hosts lowers larval num-
bers slightly, larval densities are so high that γ1x1/(a1+
x1)≈γ1, and we thus see virtually no effect from a
change in the biodiversity of large hosts when they are
abundant.
Fig. 2 The effect of varying the
total number of small hosts and
tick mortality. Total tick density,
and its break down into larval,
nymph and adult tick densities,
is plotted as a function of ptot,
the scaling factor multiplying the
baseline number of small hosts
Hs, e.g. a scaling of ptot =1.5
implies Hs=1.5×100 =150.
In a, the parameters are as given
in Table 2.Inb, larval tick
mortality is high, μ1=11, c
nymph tick morality is high,
μ2=7.3andindadult tick
mortality is high, μ3=0.37. In
all cases σn(y) =1 for each n,
ps=pL=0.5andH3=0.1,
H4=0.1
00.5 11.5 2
0
5000
10000
15000
20000
(d) High adult mortality
ptot, scaling of Hs
Total
Larvae
Nymphs
Adults
00.5 11.5 2
0
5000
10000
15000
20000
ptot, scaling of Hs
Ticks per ha
(a) Base parameters
00.5 11.5 2
0
5000
10000
15000
20000
ptot, scaling of Hs
Ticks per ha
(c) High nymph mortality
00.5 11.5 2
0
5000
10000
15000
20000
ptot, scaling of Hs
(b) High larval mortality
358 Theor Ecol (2015) 8:349–368
Comparing the effects of large and small hosts Next, we
aim to compare the differing effects of small and large host
density on tick densities. To focus the study, we only con-
sider indirect competition and fix σn(y) =1 for each nand
ps=pL=0.5. We vary Hs/HL, by introducing ptot as a
factor to scale this ratio up or down. When ptot =1then
Hs/HL=500 and host densities are at their base line val-
ues. Initially, we vary Hs/HLby fixing HLand varying only
Hs. In this case, a scaling factor ptot <1 reflects a decrease
in small hosts, and a scaling factor ptot 1 reflects an
increase in the number of small hosts.
Figure 2a illustrates a biphasic pattern in tick densities
as a function of the scaling factor ptot.Whenptot <1 tick
densities are low and increase rapidly with small additions
to the densities of small hosts; while, when the populations
of small hosts are higher and ptot 1 a small increase
in the density of the small hosts has a smaller effect on
tick densities. At low densities of small hosts (ptot 1),
the availability of small hosts acts as a rate limiting step
in the tick life-cycle. In contrast, when densities of small
hosts become large (ptot 1), they are no longer rate lim-
iting, and further increases in small hosts have significantly
less effect on the total density of ticks. The breakdown of
total tick density into the different life stages reveals that
changes in larval densities are the source of the biphasic
pattern in total tick density. Larval densities follow a hump
shape graph and eventually decrease as ptot increases. The
decrease is due to the fact that larval production from adults
is a saturating function of adult tick densities. Even though
densities of adult ticks are increasing as Hsis increased,
there is little change in the rate larval ticks are produced
once the production term saturates. The rate at which lar-
vae become nymphs (γ1) is a linear function of Hsand does
not saturate leading to the observed net decrease in larval
densities.
The biphasic pattern, we observe when we change
small host densities is also present when we change
large host densities (results not shown). Much like with
small hosts, large hosts can cause a rate limiting step in
the tick life-cycle. Larval densities increase with increas-
ing the density of large hosts; however, in this case,
because the population of small hosts are fixed, the rate of
nymph production saturates and becomes the rate limiting
step.
We tested the robustness of our findings to changes in
tick mortality. The biphasic pattern in total tick density per-
sisted (Fig. 2b–d). Increasing adult tick mortality (d) led
to the largest reduction in total tick densities, followed by
nymph mortality (c) leading to the next largest reduction.
This is an unsurprising result, as adult ticks are responsible
for a very large production of larvae (i.e. 350), while larvae
and nymph individuals produce at most one other individ-
ual. Increasing adult mortality shortens the duration of the
adult life stage available to reproduction.
We also explored the effects of changes to average tick
loads per host, λi,n. Reducing nymph loads on H2led to
the biggest reduction in tick densities, while changing larval
loads had little effect (Fig. 3a). Generally, changes to nymph
Fig. 3 The effect of varying the
total number of small hosts and
host tick loads. Total tick density
is plotted as a function of ptot,
the scaling factor multiplying the
baseline number of small hosts
Hs=100. In aand b, large host
density is fixed at HL=0.2and
in aλi,n is reduced 90 %, while
in bλi,n is increased by a factor
of 10. Finally, in c, we consider
large host densities with HL=5
and λi,n is increased by a factor
of 10. Unless otherwise stated
the parameters are as given in
Tab l e 2and σn(y) =1 for each
nand ps=pL=0.5
00.5 11.5 2
0
0.5
1
1.5
2
2.5 x 105
Ptot, scaling of Hs
(c)
x (Ticks per ha)
00.5 11.5 2
0
0.5
1
1.5
2
2.5 x 105
ptot, scaling of Hs
(b)
x (Ticks per ha)
Base
λ1,1
λ2,1
λ1,2
λ2,2
λ3,3
λ3,4
00.5 11.5 2
0
0.5
1
1.5
2
2.5 x 104
ptot, scaling of Hs
x (Ticks per ha)
(a)
Theor Ecol (2015) 8:349–368 359
tick loads caused the most significant change in tick den-
sities and was most noticeable at high small-host densities
(Fig. 3c).
The influence of host oscillations
Changes in biodiversity can also influence predation, we
consider this by assuming that predation is at a sufficiently
high level that it leads to Lotka-Volterra oscillations in the
host populations. We allow periodic temporal cycling in the
small or large host densities. Cycling in the small hosts,
H1(t),isgivenbypsHs(1+Acos(2πt/10)), where the
average density is psHs,andwhereAis the amplitude,
with increasing Aanalogous to increased predation pressure
on H1(t), and similarly for H2(t). Cycling in large hosts,
H3(t),isgivenbypLHL(1+Acos(2πt/10)) where the
average density is pLHL, and where Ais the amplitude, or
predation pressure on H3(t), likewise for H4(t).Forsim-
plicity, we assume indirect host competition, σn(y) =1for
each n.
Oscillations in either H1or H3have little effect on aver-
age tick densities (Fig. 4). Predation induced oscillations in
small hosts are damped out with essentially no correspond-
ing oscillations in tick densities (Fig. 4a, c). For large hosts,
predation induced oscillations lead to corresponding large
oscillations in the tick population (Fig. 4b a, d). Upon exam-
ining the tick time series, we found that the oscillations in
H1(t) are damped by the differing responses of nymph and
larvae to H1(t) densities. While larvae and nymph oscilla-
tions have the same period as the host, they are out of phase
with one another (Fig. 5a), their effects essentially can-
cel one another out resulting in no oscillations in total tick
density. Oscillations in adults have extremely small ampli-
tude so do not result in oscillations in total tick density
either.
We can understand the relative phases of the larval and
nymph oscillations by considering how H1(t) effects the
flow into and out of the larval and nymph classes. When
H1(t) is high, there is a large flow out of the larval class, and
the flow out of the larval class oscillates in line with H1(t).
We, thus, see a minimum in larval density when H1(t ) is at
its peak and a maximum in larval density when H1(t) is at
a minimum. Nymphs, on the other hand, have a flow in and
out of the nymph class that oscillates in line with H1(t),but
the flow in is greater because H1(t) hosts support more lar-
vae (λ1,1λ2,1). The net effect is that nymph cycles are in
phase with H1(t), while larvae cycles are out of phase. Fur-
thermore, because λ1,1λ2,1, the oscillations in nymph
Fig. 4 The effect of host
oscillations. The solid lines in
all the plots correspond to the
average tick density over the
period of the attractor, and the
dashed lines correspond to the
maximum and minimum tick
densities over the period of the
attractor. aand bIllustrate total
tick density plotted as a function
of ps.aIllustrates the effect of
oscillations in H1(t), while b
illustrates the effect of
oscillations in H3(t).cand d
Illustrate total tick density
plotted as a function of pL.c
Illustrates the effect oscillations
in H1(t), while dillustrates the
effect of oscillations in H3(t).In
all plots, the bold lines
corresponds to A=1, a 100 %
fluctuation in H1(t) or H3(t)
around the average, while the
lighter lines correspond to
A=0.1, a 10 % fluctuation
about the average. Unless
otherwise stated Hnis a
constant, Hs=100, HL=0.2,
ps=pL=0.5andσn(y) =1
for each n
00.2 0.4 0.6 0.8 1
0.4
0.8
1.2
1.6
2
2.2 x 104
ps
x (Ticks per ha)
(a)
00.2 0.4 0.6 0.8 1
0.6
1
1.4
1.8
2.2
2.6 x 104
pL
x (Ticks per ha)
(c)
00.2 0.4 0.6 0.8 1
0.4
0.8
1.2
1.6
2
2.2 x 104
ps
x (Ticks per ha)
(b)
00.2 0.4 0.6 0.8 1
0.6
1
1.4
1.8
2.2
2.6 x 104
x (Ticks per ha)
(d)
pL
360 Theor Ecol (2015) 8:349–368
density transitioning into the adult class are small in ampli-
tude and the type II functional response to nymph density
further dampens any oscillations in the production of adult
ticks resulting in the very low amplitude adult oscillations
which are in phase with the nymphs.
The response of tick density to oscillations in H3(t) gives
a quite different picture to that of the small-host oscilla-
tion case. While adults ticks continue to exhibit very low
amplitude oscillations, the larvae and nymph populations
now oscillate in phase (Fig. 5b) resulting in large amplitude
oscillations in total tick density. Larvae oscillate in phase
with H3(t) because of the large numbers of larvae produced
per adult. Because H1and H2are held constant in this sce-
nario, the larvae oscillations get transmitted directly into
nymph oscillations with only a slight modification in the
phase caused by the time spent in the larval life-stage. The
amplitude of oscillations in nymph density is damped by the
type II functional response, which is damped further by the
time the ticks reach the adult stage.
Lotka-Volterra oscillations may occur at different fre-
quencies depending on the predator type and on seasonal
fluctuations in resource availability, so we also examine
changes in the period of host oscillations (Fig. 6). Increas-
ing the period of host oscillations increased the magnitude
of oscillations in total tick density. Results were most sensi-
tive to oscillations in large hosts (Fig. 6c). When the larger
hosts undergoes oscillations, the average tick density is
lower than is predicted from the scenario on non-oscillating
hosts; however, the fluctuations in tick density are large.
The two types of small host did not have the same effect
on tick density. When H1(t) oscillates, the average tick den-
sity is lower than in the non-oscillating case. When H2(t)
oscillates, the average tick density is the same as in the
non-oscillating case. However, tick oscillations were more
sensitive to the period of oscillations in H2hosts rather
than H1. The sensitivity to H2oscillations is due to the
sensitivity of total tick densities to the rate that nymphs
become adults which is enhanced by increases in H2
hosts.
All of the results of Section 3continued to hold when
we examined predation on H2and H4individually, or with
simultaneous (synchronous) predation on H1and H2or on
H3and H4. Since synchronous oscillations in small or large
host populations occurs only very rarely in natural popula-
tions, we also examined the cases when the small or large
hosts are cycling asynchronously. In accordance to some
works (Holt and Roy 2007), we find that asynchronous pre-
dation on small hosts can increase average tick densities, but
the effect is very limited (data not shown). Similarly, asyn-
chronous predation on both large hosts may decrease tick
densities, but again, the magnitude of the effect is negligible.
We have also examined situations where one of the small
or large hosts have higher predation levels than the other, as
well as cases where both the small and large hosts are being
predated on, but the results do not differ from the previous
cases (data not shown).
Discussion
Ticks can transmit numerous pathogens and so it is impor-
tant to understand the factors that regulate and limit tick
abundance. Ticks are opportunistic feeders and have been
shown to feed on a wide range of hosts, but whether
the abundance of ticks is a function of host abundance
remains controversial (Keesing et al. 2006). Evidence of
altered host behaviour or space use in the presence of het-
erospecifics is likely to affect tick-host encounter rates,
which Keesing et al. (2006)termedencounter augmentation
and this is one example of a mechanism that could gener-
ate complex relationships between tick and host abundance.
More generally, the hosts are part of a complex ecosystem
and studies have suggested that community composition of
hosts are an important factor in determining tick abundance
(Giardina et al. 2000). By constructing a tick stage-
structured model, we were able to determine how changes
in biodiversity that affect the ecological processes of host
competition, behaviour and predation influence tick popu-
lations. In so doing, we determined the conditions where
biodiversity may potentially regulate tick populations. Our
results show that, though increasing biodiversity can in
many cases have a regulating influence on tick densities, it is
in itself too coarse of a measure to predict the magnitude and
direction of the regulating effect. Instead, the effect can be
better understood and predicted by examining the underly-
ing ecological processes affecting ticks, which, in turn, can
be translated into meaningful tick management approaches.
Below, we discuss the results and their management
implications.
Competition type and encounter augmentation
When we considered aspects of biodiversity that affect host
competition, then increasing host biodiversity did not nec-
essarily have a regulating effect on tick densities. In order
for high host biodiversity to lower tick densities, the compe-
tition among the hosts had to be direct, rather than indirect
(Fig. 1). With indirect competition, the presence of another
competitor will not affect the hosts’ ability to forage and
so tick-host encounter rate is independent of the presence
of host heterospecifics; while, with direct competition, the
presence of a competitor affects the host ability to forage
and encounter ticks (e.g. through changes in behaviour).
This difference affects tick densities because, with indirect
competition, the hosts do not modify their encounter rate
with ticks and tick densities are not lowered by increased
Theor Ecol (2015) 8:349–368 361
Fig. 5 The effect of host
oscillations on larval, nymph
and adult tick densities as a
function of time. In a,the
preferred small host density
oscillates according to
H1(t) =50(1+cos(2πt/10)),
all other host densities are held
constant. In b, the preferred
large host density oscillates
according to
H3(t) =0.1(1+cos(2πt/10)),
all other host densities are held
constant. In each plot, the solid
lines with the circles indicates
total tick density (x(t)) while
the dashed line indicates larval
density(x1(t)), the solid line is
nymph density (x2(t))andthe
dotted line is the adult density
(x3(t)). Unless otherwise stated
Hs=100, HL=0.2,
ps=pL=0.5andσn(y) =1
450 455 460 465 470 475 480 485 490 495 500
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
Time, years
Ticks per ha
(b)
450 455 460 465 470 475 480 485 490 495 500
0
0.5
1
1.5
2x 104
Time, years
Ticks per ha
(a)
biodiversity. In fact, an ecosystem, with a single host species
supporting low tick loads leads to the lowest tick densities.
In contrast, with direct competition, the hosts’ encounter
rate with ticks is modified in a non-linear manner, and in
a diverse host environment, results in decreased total tick
densities. These results suggest that the changes in tick-
host encounter rates, which can often be associated to direct
competition, are one of the fundamental ecological pro-
cesses that can determine whether increasing biodiversity
will regulate tick populations. Indeed, there is ample evi-
dence that the presence of other small hosts greatly affect
the behaviour of deer mice and other rodents, so that their
encounter rate with ticks would necessarily be also modi-
fied (Mitchell et al. 1990; Yunger et al. 2002); similarly, the
behaviour of deer also change when other large hosts are
present (Hobbs et al. 1996;Latham1999). As well, recent
work has even shown that rodents may also change their
behaviour in the presence of large hosts (Munoz and Bonal
2007). The situation of indirect competition would most
likely occur when the resources available are plentiful, and
there is no need to compete directly (Wooton 1994). High
resource availability may occur around spatially heteroge-
neous agricultural areas where there is often the presence of
food subsidies, and also habitat regions that support small
hosts (e.g. orchards or corn fields).
Host biodiversity may also play a role in answering the
question posed by Ostfeld (Ostfeld RS 2011): “why is the
relationship between deer and tick abundance so variable?”
Luo and Wu (2014) proposed one answer based on tick
seeking assumptions. They found frequency-dependent con-
tact between ticks and hosts, which is expected at high
host densities, resulted in no relationship between tick
and deer densities, while density-dependent contact, which
is expected to occur when host densities are lower, led
to a positive relationship between tick and deer densi-
ties. We offer a different answer based on the biodiversity
of the ecosystem in which the deer reside. If deer mod-
ify their behaviour in response to other animals present
in the ecosystem, then the composition of the ecosys-
tem can alter the tick-host encounter rate. Looking at Fig.
1c, with direct competition between large hosts, we see
that increasing the proportion of deer, pLcould lead to
an increase or decrease in tick numbers depending on
the relative abundance of the other large hosts in the
system and on how those other large hosts affect deer
behaviour.
Our analytical results suggest that the importance of any
particular host or tick life-stage in regulating tick densities
depends on whether they cause a rate limiting step in the
tick population cycle. If the host densities are low, or if the
362 Theor Ecol (2015) 8:349–368
0 10 20 30 40 50
0.8
1
1.2
1.4
1.6
1.8
2x 104
Period
(c)
0 10 20 30 40 50
0.8
1
1.2
1.4
1.6
1.8
2x 104
Period
(b)
0 10 20 30 40 50
0.8
1
1.2
1.4
1.6
1.8
2x 104
Period
x (Ticks per ha)
(a)
Fig. 6 The effect of host oscillation period. aVarying the period of
oscillations in H1,whereH1(t) =50(1+cos(2πt/period)).Thesolid
line indicates the case where is constant H1=50. When H1(t) oscil-
lates, so does the equilibrium total tick density, and the bold solid line
is the average tick density (averaged over the period of the attractor),
the dashed lines indicate the maximum and minimum tick densities
over the period of the attractor. bVarying the period of oscillations
in H2,whereH2(t ) =50(1+cos(2πt/period)),thelines are as
described in (a). cVarying the period of oscillations in H3,where
H3(t) =0.1(1+cos(2πt/period)),thelines are as described in a.
Unless otherwise stated Hnis a constant, Hs=100, HL=0.2,
ps=pL=0.5andσn(y) =1
transition rate to one life stage is slower than the transi-
tion rates for other stages, then that host or tick life-stage
will cause a rate limiting step that effectively regulates the
rate of total tick production. Notably, the particular rate
limiting influence of nymphs that we found here is based
on the parameterisation derived from the North-Eastern
Ixodes scapularis tick; in other tick species, the rate limit-
ing step may be in another life-stage—highlighting the need
to consider the ecological properties of the tick-host system.
Our results are consistent with the finding that moused-
based interventions had only weak effects on tick abundance
(Brisson et al. 2008). Reducing mouse densities is equiva-
lent to reducing ps, whereby the competitors of the mice
replace those mice that are removed (Keesing et al. 2009).
Reducing psincreased tick densities because, in our model,
the competitor of the mice (chipmunks or birds) supported
higher loads of nymph ticks and thus allowed higher num-
bers of ticks to transition to adults (see Fig. 1a).
A common tick-management practice targets the adult
ticks feeding on large hosts; the practice essentially attracts
deer to a device that applies acaricides (i.e. tick-specific
pesticides) on the deer as it feeds (Schulze et al. 2009).
Applying acaricides is equivalent to reducing λ3,3in the
model. Figure 3a shows that reducing tick loads on deer had
one of the largest impacts on total tick densities. Lowering
nymph loads on alternative small hosts such as chipmunks,
was the only strategy that would lower tick densities more.
Targeting large hosts have been shown to be effective, cou-
pling the practice to a similar approach that targets nymphs
and small hosts could increase the reduction of tick den-
sities. One approach may be to cull small hosts, but this
is impractical given their ubiquity and high density (Myers
et al. 1998). Another promising method would be similar
to that employed with deer, where small hosts could be
given access to nest bedding that is imbibed with acaricides
(Jaenson et al. 1991); in so doing, they would kill the
ticks that have attached to the rodents when they return to
their nest. Other innovative approaches may involve orally
vaccinating small hosts against the tick bites, which has
been shown to have some preliminary success (Gomes-
Solecki et al. 2006). A side from management practices
that modify λi,n, climate can also modify λi,n .High
humidity can cause ticks to quest higher on vegetation
which increases their encounter rate with larger hosts and
Theor Ecol (2015) 8:349–368 363
reduces their encounter rate with smaller hosts such as mice
(LoGiudice et al. 2008).
Predation and host population cycles
Changes in biodiversity can also lead of host oscillations,
as associated with increasing predation pressure or fluctua-
tions in host resources (Ostfeld and Keesing 2000). Under
these scenarios, the average tick densities did not deviate
significantly from the cases with no oscillations in host den-
sity. Moreover, oscillations in small hosts were damped out
in the tick population. However, oscillations in large hosts
were transmitted to the ticks to give rise to oscillations in
tick densities (see Fig. 5). Therefore, one may expect that
sudden changes in large host densities to have a more dra-
matic impact on tick densities than corresponding changes
to the density of small hosts. Lengthening the period of the
host oscillations (see Fig. 6) further increased the amplitude
of oscillations in tick numbers. These changes in the period
of the host oscillations may occur as a result of changes in
the type or behaviour of the predators; but, the change in
period may also result from changes in the host dynamics
themselves, which may be sensitive to resource availabilities
(e.g. production of acorns during masting events) or climate
variations (Ostfeld et al. 2006). While these fluctuations in
resource availability will certainly occur in wildlife situa-
tions, they are likely not observed in areas close to human
land use, as there may be sufficient food subsidies to support
stable populations of hosts. Our results remained insensi-
tive to various combinations of asynchronous predation and
predation on small and large hosts.
Implications of Lyme disease
In Appendix B, we provide a simple extension of our model
following (Lou and Wu 2014) which allows us to calcu-
late the basic reproduction number R0for lyme disease and
hence allows us to relate out findings to disease transmis-
sion, the focus of many tick studies. We assume that the H2
hosts are not competent reservoirs for the disease, but the
H1hosts are. The basic reproduction number is given by
R0=βHγ
2
μ2(a2+x∗
2)+γ
2
βLα
2
H1α2μH1μ2+γ2
a2+x∗
2x∗
2,
(12)
(βHand βLare the transmission coefficients of the infec-
tion to H1hosts and larval ticks, respectively. γ
i(α
i)is
the contribution to γi(αi) that comes from feeding on
H1hosts only. Lastly, x∗
1and x∗
2are the equilibrium tick
densities).
Our formulae in Appendix Bare very similar to those
of Lou and Wu (2014), only our stage-structured model
of tick dynamics, and consequently the epidemiological
model, differs in two key ways:
(a) Since each host can carry a maximum number of ticks
the production terms in Eqs. 1–3saturate with a type II
functional response in tick density compared to a type I
response in Lou and Wu (2014).
(b) The inclusion of additional host types and the modifi-
cation of encounter rate in response to host biodiversity
is omitted in Lou and Wu (2014)
These two differences both modify the expression for
R0calculated in Lou and Wu (2014). In particular, (a)
results in R0no longer being a simple increasing func-
tion of nymph density, instead (12) can increase and then
decrease as we increase nymph density. The decrease only
happens if γ
2is small meaning that most of the hosts
that nymphs feed on are type H2rather than H1.The
decrease in R0at high densities of H2is a ’dilution
effect’, whereby ticks feed on hosts that are not a dis-
ease reservoir (H2hosts), so do not transmit the disease,
resulting in the pathogen being diluted and maintained
in the environment at a much lower level (e.g. Schmidt
and Ostfeld 2001). However, (b) allows γ
2to also be
low when H2hosts are less abundant. If the presence of
other hosts modify the tick-host encounter rate in such
a way that very few ticks feed on H1hosts, in other
words φ10.5 then we still have a dilution effect,
except the pathogen is diluted because of the effects of
direct competition between the hosts, importantly this result
means that the existence and strength of a dilution effect is
likely to depend on the biodiversity of the ecosystem in a
complex way.
Limitations of the results
For our analysis, we made a number of simplifying assump-
tions. First, in terms of the ecology of ticks, tick-host
dynamics are more complex than modelled. While research
has suggested that tick-host dynamics can be at equilibrium
in wild populations (Wikel 1996; Bull and Burzacott 1993;
Lack 1954; Irvine 2006), other studies have found that ticks
can affect the behaviour as well as the fitness of domestic
hosts (e.g. White et al. 2003;Bocketal.2004) and wild
hosts (McKilligan 1996). Similarly, while the tick stages do
usually feed on small or large hosts as we described, it is
possible for any stage of a tick to feed on any host, since
they are opportunist feeders. The inclusion of these consid-
erations would modify the dynamics of the tick-host system,
364 Theor Ecol (2015) 8:349–368
Table 3 Analytical criteria for tick eradication (A, B) and the absence of tick cycles (C, D) (see Appendix Afor details of the derivations). Note
that (A) and (B) are alternative criteria, only one of these needs to be satisfied and similarly for (C) and (D)
Result Criteria
(A) (0,0,0) is globally asymptotically stable. μ3>α1−γ3
a3.
(B) (0,0,0) is globally asymptotically stable. μ1>α1
a3,μ2>α2
a1and μ3>α3
a2.
(C) There exist no invariant closed curves and the omega μ2+μ3>α1−γ3
a3.
limit set of any orbit is a single equilibrium.
(D) There exist no invariant closed curves and the μ1+μ2>α1
a3,μ2+μ3>α2
a1
omega limit set of any orbit is a single equilibrium. and μ3+μ1>α3
a2.
as the ticks would then be causing population fluctuations
in the hosts, since they can influence host fitness, and likely
also the potential for changes in tick-stage transition rates,
since they may feed on small and large hosts.
However, changes in tick loads or transition rates would
only affect the location of the rate limiting step and which
tick life stage is acting as a bottleneck. The trends of our
results are thus robust to changes in parameters and the
dynamics we identified should be robust for a wide range of
parameter values.
Conclusion
Our paper examined how two ecological processes, com-
petition and predation, that underlie biodiversity can poten-
tially regulate tick populations. While biodiversity can in
many cases regulate tick populations, this will depend more
on the ecological processes determining the relationship
between the ticks and their hosts. Significantly, the ecolog-
ical processes may often trump the predictions of biodiver-
sity, such that increased biodiversity may have no or the
opposite effect than intended. Hence, though biodiversity
may be a good initial measure of whether disease emer-
gence may occur, the ecological processes that govern the
vector-host dynamics must be examined more closely. The
added benefit of examining the ecological processes is that
it would lead to specific management implications that can
be implemented.
Acknowledgments This research is the direct result of the Pacific
Institute for the Mathematical Sciences (PIMS) 11th Industrial Prob-
lem Solving Workshop held at the University of Alberta. The authors
are grateful for the support given by PIMS and are particularly appre-
ciative of the hard work put in by the local organizers. JT acknowl-
edges the support of the PHARE training grant. CAC acknowledges
the support of Royal Society grant TG090850 which funded a visit to
work with JT. We also acknowledge the contributions during the ini-
tial development of the model from David Laferriere, Babak Pourziaei,
Juan Ramirez, Marc D. Ryser, Wing Hung Sze, Hannah Dodd, Herb
Freedman, and Ognjen Stancevic.
Appendix A: Global stability of the extinction
equilibrium and nonexistence of periodic orbits
Consider a system of differential equations dx/dt =f(x),
where x=(x1,x
2,x
3)∈R3and x(t,x0)is a solution
of the equations which satisfies x(0,x
0)=x0.Weusea
generalisation, to higher dimensions, of a criteria of Bendix-
son for the non-existence of invariant closed curves such
as periodic or homoclinic orbits. The theory was devel-
oped by Li and Muldowney (1993,1996) and shows that
oriented infinitesimal line segments, y(t,y0),evolveas
solutions of
dy
dt =∂f
∂x(x(t, x0))y (13)
and oriented infinitesimal areas, z(t, z0)evolve as solutions
of
dz
dt =∂f
∂x
[2]
(x(t , x0))z (14)
where ∂f
∂x
[2]is the second additive compound matrix. For
a general matrix A, the corresponding second additive
compound matrix is given by A[2]as follows,
A=⎡
⎣
a11 a12 a13
a21 a22 a23
a31 a32 a33 ⎤
⎦,
A[2]=⎡
⎣
a11 +a22 a23 −a13
a32 a11 +a33 a12
−a31 a21 a22 +a33 ⎤
⎦.(15)
Theor Ecol (2015) 8:349–368 365
Thus, for Eqs. (5)–(7), the second additive compound
matrix is given by Eq. (16).
∂f
∂x[2]
=⎛
⎜
⎜
⎜
⎜
⎜
⎝
−μ1−μ2−γ1a1
(a1+x1)2−γ2a2
(a2+x2)20−α1a3
(a3+x3)2
−α3a2
(a2+x2)2−μ1−μ3−γ1a1
(a1+x1)2−γ3a3
(a3+x3)20
0α2a1
(a1+x1)2−μ2−μ3−γ2a2
(a2+x2)2−γ3a3
(a3+x3)2
⎞
⎟
⎟
⎟
⎟
⎟
⎠
.(16)
By Theorem 3.3 of Li and Muldowney (1993)iffor
each x0∈R3
+(13)and(14) are uniformly asymptoti-
cally stable then all line segments collapse to the origin
and we have global stability of (0,0,0) and there exists
no invariant closed curves (periodic orbits, homoclinic or
heteroclinic cycles) and the orbits converge to a single
equilibrium.
Asymptotic stability of (13)and(14) is shown by con-
structing Lyapunov functions. Using the Lyapunov function
V(x
1,x
2,x
3)=|x1|+|x2|+|x3|and together with (13),
we have
˙
V(y) =(1,1,1)·∂f
∂x =−μ1+a1(α2−γ1)
(a1+x1)2
−μ2+a2(α3−γ2)
(a2+x2)2−μ3+a3(α1−γ3)
(a3+x3)2
If ˙
V(y) < 0, we have global stability of the zero solu-
tion of (13). Since γ1≥α2and γ2≥α3, then a sufficient
condition for ˙
V(y) < 0isμ3>(α
1−γ3)/a3, condition
(A) in Table 3. Showing that ˙
V(y) =(1,1,1)·∂f
∂x [2]
<
0 guarantees asymptotic stability of (14) and gives
condition (C).
Alternatively, using the Lyapunov function
V(x
1,x
2,x
3)=sup{|x1|,|x2|,|x3|} gives stronger results
(conditions B and D in Table 3).
Appendix B: R0and tick-borne disease dynamics
While ticks can feed on a variety of hosts, it is com-
monly believed that pathogens are associated with a par-
ticular host that acts as a disease reservoir that maintains
the pathogen in the environment (Randolph 2004). For
instance, the spirochete Borrelia burgdorferi s.l. is main-
tained mainly in deer mice: the spirochete is transferred
to the tick when it feeds on an infected deer mouse;
after which, the infected tick can transfer the disease to a
human, causing Lyme disease, or to another deer mouse—
thus maintaining the disease in the environment. If the
tick feeds on an alternate small or large host that is not
a disease reservoir (e.g., pocket mice, rabbits, humans),
the pathogen will either be eliminated by the immune
system, or lead to the death of the host, or not be
transferred to another host, in all cases effectively act-
ing as a dead end that removes the pathogen from the
environment.
Larval ticks typically hatch free from infection and can
acquire infection through a blood meal with an infected
small host, at which point they molt to become infected
nymphs. So larval ticks cannot transmit the disease. Infected
nymphs can transmit the infection to the hosts they feed
upon and the infection remains in the ticks when they molt
to the adult stage. Adopting the approach of Lou and Wu
(2014) we can extend our model in a simple way to cap-
ture the disease dynamics of Lyme disease by describing
the disease status of the individuals in our model. The
rate of change of infected small H1hosts HI
1(t), infected
nymphs xI
2(t) and infected adult ticks xI
3(t) are given by
Eqs. (17)–(19).
We do not track infected large hosts as they can
only transmit the infection to adult ticks which can-
not pass the infection onto their offspring, so the
large hosts are not acting as a reservoir for the dis-
ease the way that the small hosts are. We assume
only the H1small hosts (e.g. deer mice) are a com-
petent reservoir for the disease and that the H2small
hosts are not (Ostfeld and Keesing 2000). βH,βLand
βNare the transmission coefficients of the infection
to H1hosts, larval ticks and nymphal ticks, respec-
tively. γ
iis the contribution to γithat comes from
feeding on H1hosts only similarly for α
i.Forexam-
ple, γ
2=σ1(ps)H1λ2,1. Assuming the tick popula-
tion are at equilibrium then, we can study the dis-
ease dynamics in isolation replacing x1(t) and x2(t)
by their equilibrium values x∗
1and x∗
2and noting that
the equation for infected adult ticks decouples. Hence,
two equations form the epidemiological model, Eqs.
(20)–(21).
366 Theor Ecol (2015) 8:349–368
˙
HI
1=−
death
μH1HI
1+
infected nymphs transmitting
disease to healthy hosts
βH
H1−HI
1
H1
γ
2xI
2
a2+x2
,(17)
˙xI
2=−
death
μ2xI
2+
larvae feeding on infected hosts
molting to become infected nymph
βL
HI
1
H1
α
2x1
a1+x1
−
infected nymphs molting
to become infected adults
γ
2xI
2
a2+x2
,(18)
˙
xI
3=−
death
μ3xI
3+
infected nymphs molting
to become infected adults
α
3(x2−xI
2)
a2+x2
+
uninfected nymph feeding on infected hosts
and molting to become infected adults
βN
HI
1
H1
α
3(x2−xI
2)
a2+x2
−
adults taking final blood meal
γ3xI
3
a3+x3
.(19)
˙
HI
1=−μH1HI
1+βH
H1−HI
1
H1
γ
2xI
2
a2+x∗
2
,(20)
˙xI
2=−μ2xI
2+βL
HI
1
H1
α
2x∗
1
a1+x∗
1
−γ
2xI
2
a2+x∗
2
.(21)
We can calculate the basic reproduction number for the dis-
ease using the next generation matrix method (see Van den
Driessche and Watmough 2002). The transmission matrix
and transition matrix are given by
F=⎛
⎝
0βHγ
2
a2+x∗
2
βLα
2x∗
1
H1(a1+x∗
1)0⎞
⎠and V=μH10
0μ2+γ
2
a2+x∗
2
(22)
respectively. Together these yield the next generation matrix
FV−1=⎛
⎝
0βHγ
2
μ2(a2+x∗
2)+γ
2
βLα
2x∗
1
H1(a1+x∗
1)μH1
0⎞
⎠,(23)
the dominant eigenvalue of which gives the basic reproduc-
tion number R0for the disease.
R0=βHγ
2
μ2(a2+x∗
2)+γ
2
βLα
2x∗
1
H1(a1+x∗
1)μH1
(24)
=βHγ
2
μ2(a2+x∗
2)+γ
2
βLα
2
H1α2μH1μ2+γ2
a2+x∗
2x∗
2
The unique endemic equilibrium is
HI∗
1=H11−1
R2
0(25)
xI∗
2=βL
α
2
α2
x∗
21+γ2−γ
2
γ
2+(a2+x∗
2)μ21−1
R2
0(26)
Applying Theorem 2.1 from Lou and Jianhong (2014)
shows that R0determines the global stability of the endemic
equilibrium. Specifically, if R0>1, the endemic equilib-
rium is globally asymptotically stable.
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