Ecological Complexity 7 (2010) 446–457

On competition of predators and prey infection

Ivo Siekmann a,*, Horst Malchow b, Ezio Venturino c

a Auckland Bioengineering Institute, The University of Auckland, Private Bag 92019, Auckland Mail Centre, Auckland 1142, New Zealandb Institute of Environmental Systems Research (USF), University of Osnabruck, Barbarastr. 12, 49069 Osnabruck, Germanyc Dipartimento di Matematica, Universita di Torino, via Carlo Alberto 10, 10123 Torino, Italy

A R T I C L E I N F O

Article history:

Received 11 March 2009

Received in revised form 5 October 2009

Accepted 19 October 2009

Available online 24 November 2009

Keywords:

Competition diagram

Eco-epidemiological model

Mass action transmission

Standard incidence transmission

Fold-Hopf bifurcation

A B S T R A C T

A class of prey–predator models with infected prey is investigated. Predation terms are either of Holling

type II or III, infection is either modelled by mass action or standard incidence. It is shown that the key for

understanding the model behaviour is the competition of predators versus infection. In the presented

models the predator is not susceptible to the infection and the infection of the prey has no influence on

the ability of the predator of catching the prey. However, the prey population can be seen as a resource

which both the predators and the infection depend on. The competition for this resource is strong—the

principle of competitive exclusion holds for biologically meaningful choices of parameters as long as there

is no destabilisation by a Hopf bifurcation. The representation of models in competition diagrams which

are introduced in this article can be used for a wide range of competition models which seems to be a

promising method with many potential applications.

� 2009 Elsevier B.V. All rights reserved.

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1. Introduction

Prey–predator models with infected prey have already beenanalysed extensively, see Bairagi et al. (2007); Ghosh et al. (2007);Bhattacharyya and Bhattacharya (2006); Xiao and Chen (2001);Chattopadhyay and Arino (1999); Venturino (1995, 1994). Veryrecently, also more complex situations like the infection ofpredators through the consumption of prey, see Hsieh and Hsiao(2008) and the references therein, as well as the influence of preyinfection on the chaotic dynamics of a three-trophic food chain, seeDas et al. (2009), were considered.

The work presented in this paper is based upon a thoroughstudy of Siekmann et al. (2008); Sieber et al. (2007); Hilker andMalchow (2006); Hilker et al. (2006); Malchow et al. (2005, 2004)as well as Beltrami and Carroll (1994) where not only differenttransmission terms for the infection were investigated, but alsodifferent functional responses and growth terms of infected prey.

In the following, different types of infection transmission andfunctional response of the predator will be investigated in order toget some general insight in the interdependencies of predators andprey infection. As the prey also plays the role of the host of aninfectious disease, to avoid confusion it shall be noted that the preyis sometimes referred to as ‘‘the host’’ if only the subsystem of theinfection without predator is concerned. The notion ‘‘prey’’,

* Corresponding author. Tel.: +64 9 373 7599; fax: +64 9 367 7157.

E-mail address: ivo.siekmann@auckland.ac.nz (I. Siekmann).

1476-945X/$ – see front matter � 2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.ecocom.2009.10.005

unfortunately, will be used ambiguously, in cases where onlythe disease-free prey–predator subsystem as well as where thewhole prey population, consisting of both susceptible and infectedprey, is regarded. The authors hope that it will be clear from thecontext which population is referred to in any particular case.

It is assumed that the predator feeds with the same catch rateon both susceptible and infected prey. In Bairagi et al. (2007), also aclass of prey–predator models is analysed but the authors assumethat consumption of infected prey is harmful for the predator. Acomparison of their results with our study can be found in Section4. Ghosh et al. (2007) and Bhattacharyya and Bhattacharya (2006)study a model considering viral particles by an explicit equationbut under some simplifying assumptions which do not restrict themodels presented in this paper. Furthermore, they assume that thepredator feeds on infected prey only. The model presented in Xiaoand Chen (2001) is based upon functional differential equations. InChattopadhyay and Arino (1999) the disease is not transmitted tothe offspring of infected which replicate at the same rate assusceptibles. Finally, in earlier work (Venturino, 1995), infected arenot assumed to contribute to intraspecific competition.

Much of the stability analysis of the models developed inSiekmann et al. (2008); Sieber et al. (2007); Hilker and Malchow(2006); Hilker et al. (2006); Malchow et al. (2005, 2004) can besummarised by the approach developed in this article. The modelspresented here comprehend vertical infection transmission, i.e., thepossibility that infected produce infected offspring; also both typesof infection transmission which were considered in the above-mentioned papers are included in this study. However, most of these

Fig. 1. Competition diagram for a prey–predator system with infected prey, Holling

type II functional response. The diagram shows the stability regions depending on

the stationary levels of the prey population in the disease-free, P1, and the predator-

free subsystem, P2. The diagram is valid for arbitrary parameter sets, different

parameter sets are represented by different points in the diagram. Only b is needed

to determine the locus of the Hopf bifurcation which occurs in the disease-free

subsystem. For the diagram shown here, b ¼ 5 was chosen. In the red region below

the diagonal, P1 > P2 holds; for parameter sets located in this region, the predator

goes extinct. In the light blue region above the diagonal the disease-free subsystem

is stable while in the blue region non-stationary coexistence as well as periodic

disease-free solutions exist. For parameter sets located exactly on the diagonal, to

the right of the Hopf bifurcation, stationary coexistence is possible; however, as the

relation P1 ¼ P2 must hold, these stationary solutions are biologically unrealistic. At

the intersection of the diagonal and the vertical line which represents the Hopf

bifurcation, a fold-Hopf bifurcation takes place. This implies quasi-periodic and

chaotic solutions for some choices of parameters which are located to the left of the

Hopf line. (For interpretation of the references to colour in this figure legend, the

I. Siekmann et al. / Ecological Complexity 7 (2010) 446–457 447

articles consider spatial aspects by adding diffusive movement of allspecies and environmental variability by multiplicative noisewhereas in this work we mainly restrict ourselves to thedeterministic local dynamics.

It will be shown that the qualitative behaviour of a certain classof prey–predator models with infected prey is largely determinedby the competition of predators versus infection. In this rationale,the prey population may be considered as a ‘‘resource’’ which boththe predators and the infection depend on. On the one hand, if theinfection kills too much prey the predator goes extinct because itlacks food, on the other hand, if the predator feeds too much prey,the host population is too low for the spreading of the infection.Exactly this relationship is discovered in the mathematicalmodels: let P1 denotes the stationary solution of the prey in theprey–predator subsystem and let P2 denotes the stationarysolution of infected hosts in the infection subsystem. Then thesubsystem with the lower level of Pn, n ¼ 1;2 will survive whereasthe other subsystem goes extinct or, stated in a more compactform:

This means that the principle of competitive exclusion, seeVolterra (1926a,b); Gause (1934) as well as Hardin (1960), holds ina slightly generalised sense. Coexistence of predators and infectionis only possible for the biologically unrealistic condition P1 ¼ P2,see Section 2. This observation not only applies independently ofthe possibility of vertical infection transmission or the type oftransmission (mass action or standard incidence), however,interestingly, it also remains valid if the system is extended byexplicit modelling of viral agents, see Section 3. From the theory ofprey–predator systems it is well known that competitive exclusion

can be overcome if the system is destabilised by a Hopf bifurcation(Armstrong and McGehee, 1976a,b; McGehee and Armstrong,1977). Oscillatory rather than stationary coexistence is possible,provided that a strictly positive limit cycle solution exists. This alsooccurs in the models considered here and as in the classicalArmstrong–McGehee model, see Abrams et al. (2003), morecomplex dynamics like chaotic solutions can be observed, seeFig. 3.

The complicated interdependency of predators and infectionwhich seemingly depends on many parameters boils down tosimple parameter diagrams (see Figs. 1, 2, 4 and 5) depending onlyon the population levels P1 and P2. In the simplest case thediagram can be constructed by using only one additionalparameter which is needed to calculate the Hopf line. As it iseasy to generalise the approach to other models which arecharacterised by competition, the diagrams will often be denotedcompetition diagrams in the following. In Section 4, possibleapplications of competition diagrams in ecosystem managementare presented.

2. A class of prey–predator models with infected prey

A class of models of the following form is investigated:

dS

dt¼ a1S 1� Sþ I

K

� �� l

SI

ðSþ IÞk� Fm

max

SðSþ IÞm�1

Hm þ ðSþ IÞmZ; (1)

dI

dt¼ a2I 1� Sþ I

K

� ��mII þ l

SI

ðSþ IÞk� Fm

max

IðSþ IÞm�1

Hm þ ðSþ IÞmZ; (2)

dZ

dt¼ eFm

max

ðSþ IÞm

Hm þ ðSþ IÞmZ �mZZ: (3)

These equations model the dynamics of a prey–predator systemwith infected prey. The prey population consists of an infectedsubpopulation I and a susceptible subpopulation S. A predator Z

feeds on both subpopulations. Dimensional time units aredenoted t.

The susceptibles S grow logistically at a growth rate a1 until acarrying capacity of K is reached. Also infected individuals I areassumed to be able to replicate—potentially at a smaller growthrate a2, i.e. 0 � a2 � a1—but they suffer from an increasedmortality rate mI which is called virulence. Of course, alsoinfections which cause immediate loss of the ability ofreplication, are included in this model by choosing a2 ¼ 0. Thetransmission of the infection is either modelled by mass action

(k ¼ 0) or standard incidence (k ¼ 1) with transmission coefficientl, see Hethcote (2000). The functional responses of the predator,see Holling (1959), are either Holling type II (m ¼ 1) or Hollingtype III (m ¼ 2), the underlying type II prey–predator model wasconsidered by Rosenzweig and MacArthur (1963). Sigmoidalfunctional responses like Holling type III appeared in themodelling literature in the 1970s (Noy-Meir, 1975; Oaten andMurdoch, 1975) and later in textbooks (Yodzis, 1989). However,it seems that Truscott and Brindley (1994) were the first todescribe excitability which is the most interesting feature of theHolling type III model and the reason why this model is oftenreferred to as the Truscott–Brindley model. It is assumed thatpredators have no preference for either susceptible or infectedprey, thus the maximum catch rate Fmax and also the half-saturation constant H are equal for both subpopulations of theprey. The predator Z grows proportionally to the amount ofcaught prey scaled by an efficiency factor e chosen between 0 and1. It dies at a linear rate with mortality mZ .

reader is referred to the web version of the article.)

I. Siekmann et al. / Ecological Complexity 7 (2010) 446–457448

The dimensionless equations are as follows:

ds

dt¼ sð1� s� iÞ � l

si

ðsþ iÞk� am sðsþ iÞm�1

1þ bmðsþ iÞmz; (4)

di

dt¼ rið1� s� iÞ �mI iþ l

si

ðsþ iÞk� am iðsþ iÞm�1

1þ bmðsþ iÞmz; (5)

dz

dt¼ am ðsþ iÞm

1þ bmðsþ iÞmz�mZz: (6)

by choosing

s ¼ S

K; i ¼ I

K; z ¼ Z

eK; t ¼ a1t

and dimensionless parameters

r ¼ a2

a1; mI ¼

mI

a1; mZ ¼

mZ

a1; l ¼ l

K1�k

a1;

a ¼ Fmax K

H

e

a1

� �ð1=mÞ; b ¼ K

H:

In the following, by a simple transformation of (4)–(6), it will beshown that stationary coexistence is only possible if the condition:

mass action :mI þ 1� rlþ 1� r

standard incidence : 1� l�mI

1� r

9>>=>>; ¼ P2 ¼ P1 ¼

mZ

a� bmZ

; Holling IIffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimZ

a2 � b2mZ

s; Holling III

8>>><>>>:

(7)

holds, where Pn :¼ sþ i, n ¼ 1;2, denotes the stationary solutionsfor the total prey population in the predator–prey subsystem andthe infected subsystem, respectively. The attentive reader willhave noticed that in the above equation, P2 cannot be evaluated forstandard incidence transmission if r ¼ 1, which means thatsusceptible and infected prey produce offspring at the samegrowth rate. For the sake of clarity we will not consider this specialcase here but point the interested readers to Malchow et al. (2004)where it is investigated in detail.

The simple condition (7) already contains all essential featuresof the systems modelled by (4)–(6). First of all, it shows—as hasalready been observed by Hilker and Malchow (2006)—thatstationary coexistence of susceptible and infected prey is onlypossible in the biologically unrealistic setting that the parametervalues are fixed by (7). Thus, the authors are inclined to say that a(generalised) principle of competitive exclusion, see Volterra(1926a,b); Gause (1934); Hardin (1960) holds. Although mathe-matically possible, coexistence would never be observed inpractice due to the strong stochastic fluctuations which arecharacteristic for biological parameters; environmental noiseexcludes any possibility that algebraic relationships like (7) holdbetween the model parameters.

Secondly, if (7) is fulfilled the system is degenerated from amathematical point of view. A line of positive equilibria existswhich can be computed by solving (8) for z. For a graphicalrepresentation of this line, see again Hilker and Malchow (2006).The rank of the Jacobian is not maximal if (7) holds; thus, theJacobian has a zero eigenvalue along the line of equilibria. Theinteresting implications for the complex dynamics of this systemand linkages with recent progress in mathematical research will bereferred to later in this paragraph.

Thirdly, because zero eigenvalues generally indicate changes ofstability, already the simple observation that, for coexistence, P1 ¼ P2

must hold, implies that the stability of the stationary solutions P1 andP2 is reversed as soon as (under parameter variation) P1 < P2 changesto P1 > P2 or vice versa. Thus, survival of the predator or the infection

can be expressed by using only the values of the stationary preypopulations P1 and P2. This is independent of the variation of singleparameter values, in this sense we have found a generic parameter-

free stability condition for the stationary solutions of (4)–(6).For showing (7), the models (4)–(6) are now transformed to a

more convenient form where the processes of infection andpredation are separated. Introducing the total prey population,P :¼ sþ i, and the prevalence of the infection, q :¼ i=ðsþ iÞ and takinginto account the stationary solutions P1 and P2 as in (7), thetransformed model equations read:

dP

dt¼ P ð1� PÞ½1� ð1� rÞq� �mIqf g � am Pm

1þ bmPm z; (8)

dq

dt¼ qð1� qÞrq P � P2½ �; (9)

dz

dt¼ z

1þ bmPm

mZ

Pm1

½Pm � Pm1 �: (10)

with

rq ¼1� rþ l for k ¼ 01� r for k ¼ 1:

�

Note that (8) and (10) contain only terms related to demographicprocesses (although the virulence mI occurring in (8) actually is adisease-induced influence on demography) and predation,whereas (9) is the only equation where the transmissioncoefficient l appears while there are no terms related to predation.This complete separation of both influences on the prey populationcan only be accomplished if the predator has no preference forinfected over susceptible prey, i.e., if both catch rates are equal. Inthis special case the predator is not affected by the disease of itsprey. Consistently, the prevalence, i.e., the ratio of the infected overthe total population is not influenced by predation. The assump-tion that predators catch susceptible and infected prey at the samerate is nevertheless justified in some ecological systems. Forzooplankton grazing on phytoplankton the probability of prey tobe caught is probably independent of individual characteristics asinfection.

It is interesting to investigate the destabilisation of the(degenerate) equilibria along the above-mentioned line by aHopf bifurcation. Lines of equilibria in these and similar modelswere already observed by Farkas (1984) and Hilker and Malchow(2006)—Farkas named this phenomenon zip bifurcation. Acomplete theory of Hopf bifurcations occurring on lines ofequilibria was developed by Fiedler et al. (2000). The authorsdetermined generic phase portraits for two different situations.Our situation corresponds to the so-called elliptic case where theHopf point divides the line of equilibria in two parts. In one partthe equilibria are unstable, in the other part the equilibria are(neutrally) stable; unstable and stable equilibria are joined byheteroclinic orbits, see Fiedler et al. (2000) for a graphicalrepresentation. Thus, which stationary solution on the line isreached in the end depends on the choice of initial conditions. Thisclearly demonstrates that the models (4)–(6) are structurallyunstable in case that P1 ¼ P2 holds.

The Hopf condition for (4)–(6) can be stated explicitly for aHolling type II predator. In case of Holling type III predation werestrict ourselves to show a polynomial equation which can be

I. Siekmann et al. / Ecological Complexity 7 (2010) 446–457 449

solved for given values of b:

Holling type II : PH1 ¼

b� 1

2b(11)

Holling type III : 2b2ðPH1 Þ

3 � b2ðPH1 Þ

2 þ 1 ¼ 0 (12)

From the two conditions (11) and (12) it follows that in both cases,Hopf bifurcations only occur if b is above a certain threshold; forHolling type II it is easy to see that b must exceed unity. Increasingb leads to a monotonic increase of the corresponding Hopfthreshold PH

1 which tends to PH1 ¼ 1=2 for b!1.

For Holling type III one can show by evaluating thediscriminant that the cubic polynomial (12) has a double rootfor b ¼ 3

ffiffiffi3p�5:196 located at P1 ¼ 1=3. For higher values of b the

double root splits into two simple roots; one above and one below1=3. For b!1 the former tends to P1 ¼ 1=2 whereas the lattertends to 0. Thus, for Holling type III there are always two Hopfbifurcations in the interval P1 2 ½0;1=2�provided that b>3

ffiffiffi3p

. If P1

is below the lower of the two Hopf bifurcations the system isexcitable. In this parameter range, disturbing the stationarysolution leads to an ‘‘excitation’’, i.e. rapid replication of prey andpredator before both populations return to the resting state.Truscott and Brindley (1994) gave a possible explanation ofphytoplankton blooms based upon excitability; thus they couldshow that phytoplankton blooms might occur as a result of prey–predator interactions and independent of environmental influ-ences. In Section 4.2.1, we investigate the influence of a viralinfection in an extension of the Truscott–Brindley model in theexcitable parameter range.

Thus, the asymptotic behaviour of the presented prey–predatormodels with infected prey looks surprisingly simple if representedin a competition diagram, i.e., a parameter diagram depending on P1

and P2, see Figs. 1 and 2. The system is characterised by thegeneralised form of the competitive exclusion principle as definedbefore in this section. Above the diagonal, the disease-freesubsystem dominates the infection, below the predator goesextinct. Coexistence is possible only for parameter sets which arelocated on the diagonal but this implies that the biologicallyunrealistic constraint P1 ¼ P2, see (7) has to be met which isextremely unlikely under the usually high stochastic variability ofbiological parameters.

The Hopf bifurcation which destabilises the interior equili-brium of the disease-free subsystem, however, ensures that at leastoscillatory coexistence is possible. At the intersection of the Hopf

Fig. 2. Competition diagram for a prey–predator system with infected prey, Holling

type III functional response. The parameter diagram shows the stability regions

depending on the stationary levels of the prey population in the disease-free, P1,

and the predator-free subsystem, P2. The loci of the Hopf bifurcations which occur

in the disease-free subsystem were computed using (12) with a choice of b ¼ 7. For

further details, see the caption of Fig. 1.

condition, (11) or (12), respectively, with the diagonal, a fold-Hopf

bifurcation occurs, which means that the Jacobian has one zero aswell as two purely imaginary eigenvalues. Normal form analysis ofthis bifurcation revealed that for some parameter sets, invarianttori might emerge in phase space which can lead to quasi-periodicor chaotic solutions (Kuznetsov, 1995; Wiggins, 2003). Later in thissection, we will demonstrate this numerically.

The competition diagram for a Holling type III predator issimilar to the corresponding diagram for Holling type II functionalresponse, the only difference is the occurrence of a second Hopfbifurcation in the disease-free prey–predator system, see Fig. 2.

The considerations from this section show that the asymptoticbehaviour of the presented prey–predator models with infectionof the prey population is to a great extent independent of thecharacteristic features of the infection like the type of transmis-sion (mass action vs. standard incidence) or the possibility ofvertical transmission. The stationary population level of the preyin the disease-free subsystem, P1, and in the predator-freesubsystem, P2, are biologically meaningful indicators for thestrength of predation versus infection. The lower P1 the higher isthe strength of the predators, the lower P2 the more powerfulis the infection. The only region in the competition diagram whichis not determined by just P1 and P2 is the area where the system isdestabilised by Hopf bifurcation(s) occurring in the disease-freesubsystem.

Actually, the solutions for parameter sets lying in the regionslabelled ‘‘coexistence or disease-free (non-stationary)’’ may bequite complex. To demonstrate this, some numerical solutionsfor a system with Holling type II functional response and massaction type of infection transmission are presented, see Fig. 3.The numerical solutions were produced for values of P1, thestationary prey population in the disease-free subsystem, whichare quite apart from the Hopf threshold which lies at PH

1 �0:43 forb ¼ 7. For values of P2 which are close to P1�0:322, see Fig. 3(a),the solutions first remind of torus oscillations but for high valuesof t they relax to periodic oscillations. This phenomenon—already observed in Hilker and Malchow (2006) for the samemodel with standard incidence infection transmission—occurswhen a torus has originated in the phase space because of thefold-Hopf bifurcation. This can either lead to torus oscillations orperiodic oscillations which appear if there exists a stable limitcycle on the torus. For details on these phenomena related to thefold-Hopf bifurcation which is still a topic of ongoing mathe-matical research, see Kuznetsov (1995), Wiggins (2003), as wellas the references therein. Increasing mI which also implies anincrease of P2, the solutions first become periodic withoutpreceding torus-like oscillations, see Fig. 3(b). Then period-doubling starts, see Fig. 3(c). For values of P2�0:467, it is alreadyhard to determine the multiplicity of the limit cycles, seeFig. 3(d). At further increased values of P2, the cycles again havelower multiplicity, finally reaching simple periodic solutionsagain. The population of infected prey gets lower and lower, forvalues of P2 above 0.5 the infection goes extinct and the limitcycle of the disease-free prey–predator system is reached. Theauthors have been informed that a more detailed study of thisespecially interesting parameter region is forthcoming (Sieberand Hilker, in preparation).

3. Models explicitly considering viral particles

It is interesting that the extension of the infection modelspresented above by an explicit equation for viral particles stillshows the same characteristics of a competition model. Themodelling of viral particles is based upon a model proposed byBeretta and Kuang (1998). The extension of this model by a Hollingtype II predator was first presented in Siekmann et al. (2008). We

Fig. 3. Numerical solutions of the model (4)–(6) for Holling type II functional response and mass action type of infection transmission (m ¼ 1; k ¼ 0)—susceptible

prey sðtÞ is plotted in green, infected prey iðtÞ is plotted red and the predator zðtÞ is shown in blue. The dynamics is explored by increasing the virulence mI

(which leads to an increase of P2, see (7)). At (a), i.e., close to the diagonal, the solutions look like torus oscillations but they become ‘‘phase-locked’’ and

finally approach a limit cycle for t!1. For increasing values of P2 regular oscillations appear—without preceding torus oscillations (see (b)). Just a bit above, at (c),

a cascade of period-doublings starts, at (d) it is already hard to count the multiplicity of the cycle. If P2 is increased above 0.5 the infection goes extinct.

Parameters: r ¼ 0, l ¼ 5, mZ ¼ 0:495, a ¼ 5 and b ¼ 7. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of

the article.)

I. Siekmann et al. / Ecological Complexity 7 (2010) 446–457450

restrict ourselves to infected which are incapable of replication butslightly generalise (Siekmann et al., 2008) by including also Hollingtype III functional response.

dV

dt¼ �lSV �mV V þ BmII; (13)

dS

dt¼ a1S 1� Sþ I

K

� �� lSV � Fm

max

SðSþ IÞm�1

Hm þ ðSþ IÞmZ; (14)

dI

dt¼ �mII þ lSV � Fm

max

IðSþ IÞm�1

Hm þ ðSþ IÞmZ; (15)

dZ

dt¼ eFm

max

ðSþ IÞm

Hm þ ðSþ IÞmZ �mZZ: (16)

Eqs. (13)–(16) are based upon the same assumptions as (1)–(3)extended by a more detailed model of infection transmissionwhich takes into account that infectious diseases are in facttransmitted by pathogens and not simply by direct contacts ofsusceptibles and infected. Therefore, a population V is intro-

duced which stands for pathogens which move freely in theenvironment. The infection dynamics follows a mass action lawwith transmission coefficient l, i.e., pathogens which infectsusceptible hosts decrease at a rate of lSV , whereas susceptiblespass over to the infected state at the same rate. Note that here, l

describes disease transmission by contacts between pathogens

and susceptibles whereas in the models (1)–(3) the sameparameter was used for disease transmission from infected tosusceptibles. Infected hosts lose the capability of reproductionand die at an increased disease-induced mortality rate mI like inthe models (1)–(3). When infected hosts die, reproducedpathogens are set free at a rate BmII; the replication factor B

simply stands for the number of pathogens which are set free onaverage per deceased host. The model is simplified byintroducing scaled variables

v ¼ V

K; s ¼ S

K; i ¼ I

K; z ¼ Z

eK; t ¼ lKt;

and scaled parameters

rS ¼a1

lK; mV ¼

mV

lK; mI ¼

mI

lK; mZ ¼

mZ

lK; b ¼ K

H;

Fig. 4. Competition diagram for a prey–predator system with infected prey, Holling

type II functional response, explicit modelling of viruses. By comparison with Fig. 1

it can be seen that due to the explicit consideration of a virus population an

additional Hopf bifurcation is inherited from the underlying infection model by

Beretta and Kuang (1998). In the extended model there is no line of equilibria for

parameter sets which are located exactly on the diagonal, instead a region of

bistability, where depending on the chosen initial conditions either the predator or

the infection goes extinct, has appeared. A stationary coexistence solution exists in

this region but is always unstable. Parameter values: b ¼ 5, rS ¼ 8, mV ¼ 1, and

mI ¼ 5.

Fig. 5. Competition diagram for a prey–predator system with infected prey, Holling

type III functional response, explicit modelling of viruses. Differences to the system

without explicit modelling of viruses (Fig. 2) are the same as Fig. 4 differs from its

counterpart Fig. 1. See the caption of Fig. 4 for details. Parameter values: b ¼ 8,

rS ¼ 1, mV ¼ 0:5 and mI ¼ 1.

l Complexity 7 (2010) 446–457 451

a ¼

Fmax

H

e

l; m ¼ 1

Fmax

H

ffiffiffiffiffiffieK

l

r; m ¼ 2

8>><>>: ;

which leads to dimensionless equations

dvdt¼ �sv�mVvþ BmIi; (17)

ds

dt¼ rSs 1� s� ið Þ � sv� am sðsþ iÞm�1

1þ bmðsþ iÞmz; (18)

di

dt¼ �mIiþ sv� am iðsþ iÞm�1

1þ bmðsþ iÞmz; (19)

dz

dt¼ am ðsþ iÞm

1þ bmðsþ iÞmz�mZz: (20)

Generalising the stability analysis of the predator-free stationarysolution, apart from the Hopf condition for the predator-freesubsystem which was already analysed in Beretta and Kuang(1998), the additional condition:

mZ >am ðs� þ i�Þm

1þ bmðs� þ i�Þm

is found by expanding the last row of the Jacobian. Bysubstituting P2 ¼ s� þ i� and mZ ¼ amPm

1 =ð1þ bmPm

1 Þ this condi-tion simplifies to P2 < P1 as in the models above. However, by theintroduction of viral particles the predator ‘‘gains’’ someadditional area in the parameter plane. From the stabilityconstraint for the extinction of the infection one finds a region ofbistability for both Holling type II and III functional response. TheJacobian in this case has a symmetric structure which leads to adecomposition of the characteristic polynomial in two quadraticpolynomials. One of the polynomials coincides with thecharacteristic polynomial of the underlying prey–predatormodel. The other polynomial is

pðxÞ ¼ x2 þ ðP1 þ g þmV Þxþ gðP1 þmV Þ � BmIP1

with

g ¼ rSð1� P1Þ þmI:

As the linear coefficient is always positive, only the sign of gðP1 þmV Þ � BmIP1 has to be determined. This condition can betransformed to

B>gP1 þmV

mIP1: (21)

By taking into account Siekmann et al. (2008) we can use

P2 ¼ mV

mI þ rS

ðB� 1ÞmI þ rSmV

¼ 1þ A

C þ A;

with A :¼ rS

mI

; C :¼ B� 1

mV

¼ s�1

which by solving for B and substituting into (21) leads to:

P2 >P1

1þ rSmV ðmIþrSÞ

P1ð1� P1Þ:

or

P2 >P1

1þ Að1þAÞmV

P1ð1� P1Þ:

It has to be mentioned that there exists a stationary coexistencesolution which can be computed explicitly, see Siekmann et al.(2008). However, it can be shown that this solution is unstable inits whole feasibility range. After transforming (17)–(20) to theprevalence form again, the Jacobian can be simplified by using the

I. Siekmann et al. / Ecologica

following substitutions which can be deduced by requiring that allequations vanish for positive values of v; P; q and z:

mZ ¼ am Pm

1þ bmPm ; v ¼ gq ¼ g 1þmV

P

� �� BmI

where g is defined as above. With these substitutions, a manage-able representation of the Jacobian determinant of this stationarysolution is obtained:

det J ¼ mmZP2ð1� qÞð1þ bm

PmÞ2g 1þmV

P

� �� BmI

h i� 2v

n o

¼ mmZP2ð1� qÞð1þ bm

PmÞ2ð�vÞ: (22)

From (22) it is seen that for the determinant to be positive, v wouldhave to be negative which is impossible. Instability follows fromthe Routh-Hurwitz criterion, see Murray (2002).

Thus, the competition diagrams for the extended models notonly contain an additional Hopf bifurcation of the infectionsubsystem which is inherited from the underlying Beretta/Kuanginfection model, see Beretta and Kuang (1998), but also a region of

I. Siekmann et al. / Ecological Complexity 7 (2010) 446–457452

bistability of the stationary subsystems where only the predator orthe infection survives, see Fig. 4.

The competition diagram for a Holling type III predator issimilar to the corresponding diagram for Holling type II functionalresponse, the only difference is the occurrence of a second Hopfbifurcation in the disease-free prey–predator system, see Fig. 5.

4. Conclusions

A review of prey–predator models with infected prey was given.It was shown that competition of predators versus infection is thekey for understanding prey–predator models with infected prey.

Competition diagrams which were proposed in this studyprovide a useful way of representing the qualitative modelbehaviour depending on two aggregated parameters, however,averting the danger of over-simplification. One advantage of thisapproach is that different but equivalent parameter sets of thesame model are easily identified—only those areas in the P1 � P2

plane where the system is destabilised by a Hopf bifurcation haveto be analysed more thoroughly to decide if multiple-periodic,quasi-periodic or chaotic oscillations occur. Furthermore, thepopulation levels P1 and P2 of the prey population in the prey–predator or the infected subsystem, respectively, are biologicallyplausible indicators for the strength of the predator versus thestrength of the infection. This is supported by the main result. Theone of the two influences on the prey population, predator orinfection, which drives it to the lowest level, makes the other goextinct.

It was shown that the asymptotic behaviour of the analysedmodels is independent of the transmission type of the infection asthey lead to the same competition diagrams. A major differencebetween mass action and standard incidence transmission is thatin standard incidence models disease-induced extinction of theinfection is possible. Note that in (7), P2 can have negative valuesfor positive choices of the parameters r, mI and l for standardincidence transmission whereas this is impossible for mass actiontransmission. However, the case of negative P2 is exactly the caseof disease-induced extinction. In the context of competition

between predators and infection this is not especially interestingbecause for negative P2, the condition P1 > P2 holds trivially, i.e.disease-induced extinction is always stable independent from thepredator. However, if, for example, the predator feeds selectivelyon infected prey, it can prevent disease-induced extinction byreducing the infected individuals.

4.1. Comparing models with competition diagrams

But also the comparison with other models becomes possible.For this purpose we first choose (Bairagi et al., 2007) because theauthors also undertake a comparative study whose results theyrepresent in the traditional way in a table of parameter relation-ships indicating changes of the system behaviour. By representingtheir results in competition diagrams, we are going to demonstrateon the one hand that by using this new visual representation a

Table 1Stability ranges for (23)–(25), see Bairagi et al. (2007), in terms of the stationary populat

Lotka–Volterra

Extinction ð0;0;0ÞSusceptible prey only ð1;0;0ÞExtinction of infection ðs1;0; z1Þ

Extinction of predator ðs2; i2;0Þ s1 > s2 � A1ð1� s2Þ

Coexistence ðs�; i�; z�Þ

For the stability range of the predator-free solution ðs2; i2; 0Þ we define f mðs2Þ :¼Amð1

much more intuitive understanding of the system can be achievedand on the other hand how our approach can be easily adapted to acompletely different setting. In 4.1.2 we summarise what is gainedby extending the models (1)–(3) by an explicit modelling ofpathogens as in (13)–(16) by comparing Figs. 1 and 2 with Figs. 4and 5.

4.1.1. Harmful consumption of infected prey

Bairagi et al. (2007) investigate a predator feeding onsusceptible and infected prey but opposed to the assumptions ofour models they are interested in the case that the predator cancatch infected prey more easily than susceptibles. The dimension-less equations of their models are restated here with the scalingused by the authors but with parameter names adapted to ournotation:

ds

dt¼ rsð1� s� iÞ � si� am

S

sm

1þ bmsm

z; (23)

di

dt¼ si� aI iz�mI i; (24)

dz

dt¼ �eaI iz�mZzþ eam

S

sm

1þ bmsm

z: (25)

These equations describe logistic growth of the susceptible preywith growth rate r. Bairagi et al. (2007) account for theirassumption that the predator z catches infected prey i more easilythan susceptibles by choosing the classical mass action rate ofpredation as in the classical model by Lotka (1925); Volterra(1926a,b). They justify this by interpreting the parameter b of thenowadays more common Holling type II functional response as anexpression of the handling time, i.e. the amount of time which apredator needs for searching, catching and digesting prey. Fromthis point of view, a Lotka–Volterra type of predator can beunderstood as a ‘‘quick’’ predator with a handling time of b ¼ 0.The infection transmission is of mass action type inducing anincreased mortality mI in infected prey which also lose the abilityof replication. Susceptible prey is consumed by the predator with acatch rate of Holling type II or type III; Bairagi et al. (2007) consideralso Lotka–Volterra predation which is included in (23)–(25) bychoosing m ¼ 1 and b ¼ 0. The predator z dies at a mortality ratemz and is further decreased by the consumption of infected prey.

Bairagi et al. (2007) formulate their results in Theorems 4.1–4.5,5.1 and 6.1 which we are going to rewrite here in terms ofstationary levels of susceptible prey s (not total prey P ¼ sþ i asabove!). The results can be found in Table 1.

As an example we give a diagram for the case of Holling type IIIpredation, see Fig. 6, the corresponding diagrams for Holling type IIand Lotka–Volterra predation have only one or no Hopf line and thebistability region has a different shape. We further mention thatthe original article (Bairagi et al., 2007) seems to contain someinconsistencies in the interpretation of the stability results(although the stability ranges seem to be correct). The authorshave seemingly not observed that there is a bistable region where

ion of susceptible prey s1 in the disease-free and s2 in the predator-free subsystem.

Holling II Holling III

Unstable

s1 >1 and s2 >1

s1 < s2

s1 >s2 � f 1ðs2Þ1þ b f 1ðs2Þ

s1 >s2

2 � f 2ðs2Þ1þ b2

f 2ðs2Þ

Always unstable

� s2Þð1þ bmsm

2 Þ with Am :¼ðr=r þ 1ÞðaI=amS Þ.

Fig. 6. In (a), the competition diagram for (23)–(25) with Holling type III predation is shown. Different colours correspond to different types of asymptotic behaviour, cf. Fig. 2.

In the region labelled ‘bistability’ (bounded by the curve above the diagonal) depending on the initial conditions either the predator or the infection prevails. In the dark blue

region, susceptible prey and predator oscillate if the infection goes extinct. Bistability—the location of the parameter set in (a) b ¼ 7, A1 ¼ 0:1 is indicated by a white ‘*’—is

demonstrated numerically in (b) i0 ¼ 0:2 and (c) i0 ¼ 0:23 for the parameters r ¼ 1, aS ¼ 5, aI ¼ 5, b ¼ 7, mI ¼ 0:6, mZ ¼ 0:2, e ¼ 0:6, s0 ¼ z0 ¼ 0:1 and i0 as in the captions.

Susceptible prey sðtÞ is plotted in green, infected prey iðtÞ is plotted red and the predator zðtÞ is shown in blue. (For interpretation of the references to colour in this figure

legend, the reader is referred to the web version of the article.)

I. Siekmann et al. / Ecological Complexity 7 (2010) 446–457 453

either the disease-free or the predator-free equilibrium is attaineddepending on the initial conditions. Furthermore, they claim thatthere is no limit cycle for Holling type III predation although bylooking at the competition diagram it clearly shows that thereshould be one and it can be demonstrated to exist by numericalsolutions, see Fig. 6(b) and (c).

These inaccuracies shall not be overestimated, though. Theauthors of this article are well aware from their own experiencethat the interpretation of eco-epidemiological models with theircharacteristic multistability can be confusing. However, we wouldlike this to be taken as another supportive argument for choosing amore intuitive representation of complex models.

4.1.2. Direct infection transmission versus explicit modelling of

viruses

Going back to the models (1)–(3) which we started from, onemay ask what is gained by explicitly considering pathogens as in(13)–(16). This question can be addressed by comparing thecompetition diagrams from Section 2 and 3, see Figs. 1 and 4 orFigs. 2 and 5, respectively. The diagonal which in the s� i� z

models indicates special choices of parameters where stationarycoexistence of predators and infection is possible, is replaced by aregion of bistability in the extended model where one of thesubsystems, either the disease-free or the predator-free is reached,depending on the initial conditions. The principle of competitive

exclusion is thus fulfilled strictly in the extended models as long asno destabilisation due to a Hopf bifurcation occurs because anypossibility for stationary coexistence has vanished—predators andinfection can only coexist oscillatory in these models. One mayargue that the extended models are more realistic because therather artificial condition P1 ¼ P2 has disappeared. The additionalHopf bifurcation which is inherited from the underlying Beretta/Kuang infection model may have interesting consequences for thedynamics by interaction with the Hopf bifurcation from thedisease-free subsystem.

While the article so far mainly demonstrated how competitiondiagrams can be computed, now, some examples shall be givenhow competition diagrams can be used in modelling applications.

4.2. Applications to biological control problems

Although all models which were presented here are verysimple, interesting ecological conclusions can nevertheless be

drawn. For this purpose we are going to present two applications ofcompetition diagrams to biological control problems.

At first glance the idea of bio-control by infecting harmfulspecies with a virus seems attractive. By choosing a specialisedvirus, the danger of undesired side-effects seems to be ruled out.The models which were presented in this article, however, remindof the sensitivity of ecological systems. They show that predatorsmay lose their life resource when the population level of the preywhich they depend on is reduced by infection. In the context ofplankton systems this means that biological control of ‘‘harmfulalgal blooms’’ by viral infections shall be applied with extremecaution as it might lead to the eradication of zooplankton whichdepend on phytoplankton. Also among biologists, the risks ofbiological control are controversially debated, see, for example,Simberloff and Stiling (1996) (which appeared in the special issueof Ecology, vol. 77(7) on biological control) and the comments(Frank, 1998; Simberloff and Stiling, 1998).

4.2.1. Example: excitability and the biological control of

phytoplankton blooms

There is evidence that viral infections might be responsible forthe termination of phytoplankton blooms (Gastrich et al., 2004;Jacquet et al., 2002; Brussard et al., 1996; Bratbak et al., 1995).Rhodes et al. (2008) got the same result with two models ofplankton dynamics which they extended by viral infection. Asone of their models can be seen as a simplified version of ourmodel (13)–(16) with m ¼ 2 we show that also our modelsupports termination of blooms by infections using the sameparameter set as Rhodes et al. (2008), see Fig. 7, and Table 2 forparameter values.

By computing P1�0:035 and P2�0:217 with the parametervalues of Table 2 we can easily see that our choice of parameters isin the ‘‘right’’ parameter range. By evaluating (12) with P1 andb�18:95 we find that the result is positive. It follows that P1 isbelow the lower of the two Hopf thresholds, i.e. the prey–predatorsystem is in the excitable range, see Fig. 7(a). Also, P2 is clearlyabove P1, i.e. the parameter set lies above the diagonal of thecorresponding competition diagram, which means that theinfection is only transient. By simulating the parameter set fromTable 2 with different initial values of viruses v0, it is found that foran intermediate level of viruses, see Fig. 7(c), the bloom isterminated by the infection whereas for lower (Fig. 7(b)) or highervalues (Fig. 7(d)) a weaker bloom occurs between t�150 and

Fig. 7. The termination of a phytoplankton bloom depending on the density of viral particles v0 which is added at the onset of the bloom: (a) shows the bloom without viral

infection (v0 ¼ 0), in (b)–(d) the effect of adding different amounts of viruses is demonstrated. Adding v0 �1:25 dimensionless units of viruses shows the best results, see (c).

Adding less (b) or more viral particles (d) leads to a peak before the infection goes extinct.

I. Siekmann et al. / Ecological Complexity 7 (2010) 446–457454

t�220 which, however, is much weaker than the considerablystronger bloom with no control by virus.

4.2.2. Example: an ecosystem management application

For giving an example of the power of competition diagrams inecosystem management applications, we consider a population offood fish (e.g. trouts or carps) in a pond. We would like to catchmore fish, so we decide to control a predator, like, for example,pike, which feeds on the fish we are interested in. Reduction of thepredator can be done either by selectively killing or removing it

Table 2Parameter values for the simulation shown in Fig. 7.

Parameter Unit Value Dimen

K [mg N=l] 108

r [d�1] 0.3 rS

Fmmax [d�1] 0.7 a

H [mg N=l] 5.7

bmZ [d�1] 0.012 mZ

e 0.05

l [l=ðmg NÞd�1] 0.01

mI [d�1] 0.16 mI

mV [d�1] 1.23 mV

B 15 B

from the pond. This is modelled by a linear increase of the predatormortality mZðtÞ over time, all parameters for (4)–(6) are given inthe caption of Fig. 8.

mZðtÞ ¼mmin

Z ; for t � t0

mminZ þ ðmmax

Z �mminZ Þ t � t0

t1 � t0; for t0 < t< t1

mmaxZ ; for t� t1

8>><>>: (26)

In Fig. 8(a) we see the desired result. By raising the mortality of thepredator over time, the stock of fish is increased. Fig. 8(b) shows

sionless Value

Carrying capacity

0.278 Maximum growth rate

3.41 Maximum grazing rate

Half-saturation constant

18.95 Dimensionless handling time

0.01 Zooplankton mortality

Assimilation efficiency

Transmission rate

0.148 Infected phytoplankton mortality

1.139 Viral mortality

15 Burst rate

Fig. 8. Controlling the predator increases the infection risk. Predator control has been modelled by a linear increase of the mortality mZðtÞ of the predator z, see (26) with

mminZ ¼ 0:68254, mmax

Z ¼ 0:75, t0 ¼ 50 and t1 ¼ 400. Eqs. (4)–(6) are simulated assuming mass action transmission and Holling type II predation (m ¼ 1, k ¼ 0), the other

parameters are r ¼ 0, a ¼ 5, b ¼ 5, l ¼ 3 and mI ¼ 1. In (a), the desired result is shown. The stock of susceptible prey s, which we would like to protect, increases. In (b), we see

that an undesired scenario is also possible, though. After a certain threshold for the predator mortality mZ has been exceeded, a disease breaks out which drives the predator z

to extinction. Also, if only the non-infected fraction can be used, for example, because the quality of the infected subpopulation i is compromised, the utilisable stock s is even

below the level where the predator control was started.

I. Siekmann et al. / Ecological Complexity 7 (2010) 446–457 455

that something else might happen, though. After the mortality ofthe predator has exceeded a certain threshold, suddenly, aninfectious disease spreads in our food fish population; at the sametime the predator goes extinct. Where does this infection comefrom? Actually, a certain amount of the trouts or carps probably isinfected with some kind of disease. Also, pathogens might beintroduced into the system from outside; for modelling this westochastically added a small amount of pathogens in each timestep. Somehow, it seems, that the predator has had somecontrolling function; in any case, after removing too many

Fig. 9. Predator control eradicates infection. A spatial simulation of (4)–(6) is shown, t

diffusion coefficient D ¼ 0:05 for all species. In the first row, population levels of susceptib

plotted green, infected prey is red and the predator is shown in blue. Because iðtÞ is at low

values; dark grey values correspond to high population levels, low population levels are p

predator coexist. As in Fig. 8, the predator mortality is increased linearly, see (26) with mmZ

prey is at a low level and has nearly gone extinct at t ¼ 1850. The patterns change from i

prey–predator system. (For interpretation of the references to colour in this figure lege

predators the disease is suddenly able to spread. It can besummarised that the result of our idea to increase our stock of foodfish is catastrophic. Apart from the ecological disaster which hasbeen caused—it was probably not intended to completely removethe predator from the ecosystem!—we have not even reached ourgoal. If the disease makes the infected fish inedible we have evenless fish than before the predator control.

Predator control does not necessarily have to increase theinfection risk, though. If we simulate another choice of parametersin a spatially extended version of the same model (where spatial

he parameter values are r ¼ 0, a ¼ 5, b ¼ 7, l ¼ 5, mI ¼ 1:8 and a dimensionless

le, sðtÞ, and infected prey iðtÞ are represented by different colours. Susceptible prey is

levels at all times, it is also plotted in the second row, where it is represented by grey

lotted in bright grey values. At t ¼ 1100, both susceptible and infected prey and thein ¼ 0:495, mmax

Z ¼ 0:54, t0 ¼ 1000 and t1 ¼ 1500. Already at t ¼ 1350, the infected

rregular patterns to spiral waves which are typical for a non-infected Holling type II

nd, the reader is referred to the web version of the article.)

I. Siekmann et al. / Ecological Complexity 7 (2010) 446–457456

spread is modelled by diffusive movement of all populations(Okubo and Levin, 2001; Malchow et al., 2008)), we get a differentpicture, see Fig. 9. We start from a situation where susceptible andinfected prey and the predator coexist, the solutions arecharacterised by spatio-temporal chaos (Fig. 9(a)). After increasingthe predator mortality, the infected subpopulation of the prey isconsiderably reduced (Fig. 9(b)) and nearly gone extinct at t ¼1850 (Fig. 9(c)). Further control of the predator completelyeradicates the infection.

The first part of the example shows that a predator mightprevent the spread of an infection in its prey—in general, infectionsneed a minimal size of the host population for being able tobecome endemic. Thus, if the predator is reduced so that it cannotkeep the prey below this critical size, the infection breaks out.However, from the second part we see that there are exceptions forthis rule; in a different situation we were able to eradicate theinfection by reducing the predator.

In the competition diagram, see Fig. 10, we can easily see whypredator control may increase the infection risk. If we start abovethe diagonal where the predator is stable, at some point we willhave to cross the diagonal. This explains the sudden outbreak of thedisease. Random immigration of infected individuals will not leadto an outbreak if the parameter set is located above the diagonal.However, after the diagonal has been crossed, even a smallperturbation might lead to infection and extinction of the predator.

However, as we have seen, predator control does notnecessarily risk an infection. If we start in the left part of thecompetition diagram, predator and both susceptible and infectedprey coexist; usually a chaotic attractor is located in this parameterregion. By moving to the right, at the latest at crossing the Hopfthreshold, the infection goes extinct. The simulation shows,though, that already before, the infection is at very low levels—the disease is eradicated although its competitor has beenweakened! Three things are interesting about this example. First,it is impressing that the transition to disease-free solutions showsnot only in the decreasing population level of infected prey but alsoin a drastic change of the spatial pattern. While the predatormortality is increased, spatio-temporal chaos passes over to spiralwaves which are typical for the underlying Holling type IIsubsystem, see Fig. 9. Secondly, while the conclusion that predator

Fig. 10. Predator control may either increase infection risk or drive the infection to

extinction. As predator control increases the stationary prey level P1 without

affecting P2, it can be represented by arrows going from left to right. The arrow on

the right shows a situation where this leads to increased risk of infection and

extinction of the predator, see Fig. 8. The arrow on the left starts from a point where

predator and susceptible and infected prey coexist. Fig. 9 shows that before

reaching the Hopf line the infection goes extinct. That predator control will

eradicate the infection for all parameter sets located above the yellow triangle was

to be expected because at crossing the Hopf line, the disease-free subsystem

becomes stable. (For interpretation of the references to colour in this figure legend,

the reader is referred to the web version of the article.)

control enhances infection risk is hardly surprising, it seemscounter-intuitive that now the same measure eradicates theinfection. Clarifying the ecological mechanisms behind this effectrequires further investigation. Thirdly, the idea to drive aninfection to extinction by controlling its competitor was foundby analysing the competition diagram. This seems to be the mostuseful aspect of the new method. Measures for ecosystemsmanagement can be found, planned, assessed and easily presentedto decision makers with the help of competition diagrams.

Acknowledgements

The very careful report of one anonymous referee is gratefullyacknowledged. IS thanks Helle Frank Skall, Mansour El-Matbouliand Helmut Winkler for answering questions on fish diseases.

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