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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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