ARTICLE IN PRESS

0022-5193/$ - se

doi:10.1016/j.jtb

�CorrespondE-mail addr

nisbet@lifesci.u

Journal of Theoretical Biology 247 (2007) 733–742

www.elsevier.com/locate/yjtbi

Semi-discrete host–parasitoid models

Abhyudai Singha,�, Roger M. Nisbetb

aDepartment of Electrical and Computer Engineering, University of California at Santa Barbara, CA 93106-9610, USAbDepartment of Ecology, Evolution and Marine Biology, University of California at Santa Barbara, CA 93106-9610, USA

Received 1 December 2006; received in revised form 15 March 2007; accepted 4 April 2007

Available online 10 April 2007

Abstract

Arthropod host–parasitoid interactions constitute a very important class of consumer resource dynamics. Discrete-time models are a

tradition for such interactions and are characterized by an updating function, which relates the population densities at a fixed date in one

year to those at the same date in the previous year. Previous workers have investigated the effects of functional response and density

dependence on the stability of the host–parasitoid interaction by heuristically incorporating them in the updating function. Such an

approach ignores the effects of population changing continuously within a year due to different processes (for example intraspecific

competition, mortality from parasitism) that may act simultaneously. Their cumulative effect on the updating function is not obvious

and a more systematic methodology is needed. This paper uses a hybrid approach to formulate the updating function. This is done by

modeling the dynamics of various within-year processes in continuous-time, and reproduction as a discrete event. Using this formalism

we derive results connecting the stability of the host–parasitoid interaction with different forms of density dependence and the form of

the functional response. The latter results contradict previous conclusions from heuristically formulated models, and illustrate the need

for such a hybrid approach in discrete-time host–parasitoid theory.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Consumer–resource dynamics; Host–parasitoid models; Semi-discrete models

1. Introduction

One of the central themes in ecology is the interactionbetween populations of consumers (e.g. predators, para-sitoids) and resources (e.g. prey, host). There is a largebody of literature that studies such interactions using twowell-developed approaches: continuous-time models, whichare used to model populations with overlapping genera-tions and all year round reproduction, and discrete-timemodels, which are more suited for populations whichreproduce in a discrete pulse determined by season(Murdoch et al., 2003).

A large body of theory has focused on the dynamics ofarthropod host–parasitoid systems, dating back to theseminal work of Nicholson and Bailey (1935). Reviews canbe found in Hassell (2000) and Murdoch et al. (2003).

e front matter r 2007 Elsevier Ltd. All rights reserved.

i.2007.04.004

ing author. Tel.: +1805 403 2533.

esses: abhi@engineering.ucsb.edu (A. Singh),

csb.edu (R.M. Nisbet).

Discrete-time models are the traditional framework used, achoice that reflects the univoltine life histories of manytemperate-region insects. In such systems the host species isusually vulnerable to a particular parasitoid species at onlyone stage of its life cycle, commonly the larval stage. Onecommon life cycle involves adult hosts that emerge duringspring or summer, lay eggs and then die. The eggs matureinto larvae and for a short interval of time are vulnerable toattack by the parasitoids. Surviving host larvae pupate,overwinter in the pupal stage, and emerge as adult hoststhe following year. Adult parasitoids search for hostsduring the window of time that host larvae are present,then die. Parasitized hosts mature into a juvenile para-sitoid, pupate, overwinter, and emerge as adults thefollowing year. Synchronized life cycles, little overlap oflife stages, and no overlap of generations in both hosts andparasitoids suggest that discrete-time models are appro-priate for these systems.The updating function in a discrete-time, host–parasitoid

model relates the population densities at a fixed date in one

ARTICLE IN PRESS

Fig. 1. Life cycle of the host in year t.

A. Singh, R.M. Nisbet / Journal of Theoretical Biology 247 (2007) 733–742734

year to those at the same date in the previous year, anddescribes the cumulative effect of all the processes thathave occurred within the year. However, life processes arecontinuous, and the form taken by this cumulative effectmay be far from intuitive, especially where multipleprocesses operate simultaneously. A systematic way toformulate the updating function is presented in Box 4.1 ofMurdoch et al. (2003), and involves a hybrid approach,where the continuous processes involved in the within-yeardynamics are described by a continuous-time model, andreproduction is modeled as a discrete event. We followPachepsky et al. (2007), and call such models semi-discrete.

Semi-discrete models have previously been used toinvestigate interactions between adults and juveniles withina season with discrete between-season dynamics (Eskolaand Geritz, 2007), to model systems where a host orresource has discrete generations and is attacked by aconsumer with non-seasonal dynamics (e.g. Dugaw et al.,2004; Bonsall and Hassell, 1999; Briggs and Godfray,1996), and systems with discrete consumer generations andvery fast resource dynamics (Geritz and Kisdi, 2004). Semi-discrete models have also been used to address specificquestions relating to host–parasitoid dynamics; for exam-ple the stabilizing effects of within-season movement ofparasitoids (e.g. Rohani et al., 1994), and the consequencesof phenological asynchrony between parasitoids and theirhosts (Munster-Swendsen and Nachman, 1978; Godfrayet al., 1994). However, the main body of discrete-time,host–parasitoid theory does not take advantage of thisformalism.

In this paper we revisit discrete-time host–parasitoidtheory using semi-discrete models. We derive new resultsrelating stability to the form of the functional response,and to different forms of density dependence. Someprevious conclusions from heuristically formulated modelssurvive, but others are contradicted, notably those relatingthe form of the functional response to stability. Ourapproach can easily be extended to include more complexbiological interactions in contexts where a number ofprocesses can occur concurrently.

2. Model structure

A general model describing host–parasitoid dynamics indiscrete-time is given by

Htþ1 ¼ F ðHt;PtÞ, (1a)

Ptþ1 ¼ GðHt;PtÞ, (1b)

where Ht and Pt are the adult host and adult parasitoiddensities, respectively, at a fixed date near the start of year t

(where t is an integer). The host life cycle is illustrated inFig. 1 where Ht adult hosts give rise to eggs which matureinto RHt host larvae at the start of the vulnerable stage.Here R41 denotes the number of viable eggs produced byeach adult host. The time within the vulnerable larvae stageis denoted by t which varies from 0 to T corresponding to

the start and end of the vulnerable stage, respectively. Theupdate functions F, G are obtained by first writing acontinuous-time model that describes the various contin-uous-mortality sources that occur during the vulnerablestage of the host and is given by the following system ofordinary differential equations:

dLðt; tÞdt

¼ � g1½Lðt; tÞ;Pðt; tÞ; Iðt; tÞ�Pðt; tÞLðt; tÞ

� g2½Lðt; tÞ;Pðt; tÞ; Iðt; tÞ�Lðt; tÞ, ð2aÞ

dPðt; tÞdt

¼ �g3½Lðt; tÞ;Pðt; tÞ; Iðt; tÞ�Pðt; tÞ, (2b)

dIðt; tÞdt

¼ g1½Lðt; tÞ;Pðt; tÞ; Iðt; tÞ�Pðt; tÞLðt; tÞ

� g4½Lðt; tÞ;Pðt; tÞ; Iðt; tÞ�Iðt; tÞ, ð2cÞ

where Pðt; tÞ, Lðt; tÞ and Iðt; tÞ are the density ofparasitoids, un-parasitized and parasitized host larvae,respectively, at a time t within the vulnerable stage in yeart. For convenience of presentation, in the rest of the paperwe suppress the dependence on t and t and represent theabove differential equations by

dL

dt¼ �g1LP� g2L;

dP

dt¼ �g3P,

dI

dt¼ g1LP� g4I . (3)

The functions g2, g3 and g4 (units: time�1) represent(potentially density-dependent) host mortality due tocauses other than parasitism, parasitoid mortality, andmortality of the parasitized larvae, respectively. Thefunction g1 (units: time�1 parasitoid�1) is the (potentiallydensity-dependent) attack rate of the parasitoids. Theproduct g1L is frequently referred to as the functional

response and represents the instantaneous rate at whichhosts are attacked per parasitoid in the population. Theabove set of ordinary differential equations are thenintegrated starting from t ¼ 0 with initial conditions

Lð0; tÞ ¼ RHt; Pð0; tÞ ¼ Pt; Ið0; tÞ ¼ 0, (4)

up to t ¼ T . Assuming that each parasitized host larvaegives rise to k adult parasitoids in the next generation, theupdate functions can be obtained as

F ðHt;PtÞ ¼ LðT ; tÞ, (5a)

GðHt;PtÞ ¼ kIðT ; tÞ. (5b)

ARTICLE IN PRESSA. Singh, R.M. Nisbet / Journal of Theoretical Biology 247 (2007) 733–742 735

The stability analysis for the resulting discrete-time modelcan then be investigated using the standard Jury conditions(Elaydi, 1996).

If g2 ¼ g4 ¼ 0, i.e. host larvae mortality is solely throughparasitism and there is no parasitized host mortality, thenfrom (2) we have that Lðt; tÞ þ Iðt; tÞ is constant through-out the vulnerable period and is equal to RHt. In that casethe model (1) takes the more familiar form

Htþ1 ¼ RHtf ðHt;PtÞ, (6a)

Ptþ1 ¼ kRHt½1� f ðHt;PtÞ�, (6b)

where

f ðHt;PtÞ :¼F ðHt;PtÞ

RHt

(7)

denotes the fraction of host larvae that escape parasitismand is calculated from the solution of the system (2) ofdifferential equations.

3. Effects of functional responses

If the attack rate g1 is assumed to be a constant c,implying a linear functional response, and ifg2 ¼ g3 ¼ g4 ¼ 0, then integrating the corresponding con-tinuous-time model leads to the classic Nicholson–Baileymodel with

f ðHt;PtÞ ¼ expð�cTPtÞ, (8)

in the discrete-time model (6) (Box 4.1, Murdoch et al.,2003). A typical population time series of this model is asfollows: at low densities of hosts and parasitoids both thepopulations grow, when the population of hosts becomeslarge enough the parasitoid begins to overexploit the hostleading to a crash of the host population, followed by acrash of the parasitoids. Such cycles of over exploitationand crashes make this interaction unstable and both hostsand parasitoids show diverging oscillations. Variousauthors have investigated Type II (Rogers, 1972) and TypeIII functional responses (Hassell and Comins, 1978) withthe conclusion that they do not stabilize such discretegeneration host–parasitoid populations. They did this byassuming that the attack rate is a function of the initial hostlarvae density. For example, for a Type II functionalresponse, it was taken as

g1 ¼c

1þLð0; tÞ

LH

¼c

1þRHt

LH

(9)

for some constants c and LH . By contrast, our semi-discretemodel recognizes that in a real system, the number of hostslarvae declines throughout the vulnerable period due toattack by the parasitoids. Thus it is assumed that the attackrate is a function of Lðt; tÞ, the current host larvae densityinstead of Lð0; tÞ. We illustrate next, using a simple class offunctional responses which have the form cLmþ1, that thewithin season variation in host density can change thequalitative dynamics from instability to stability. These

accelerating functional responses are mathematically easyto analyze and provide insight on host–parasitoid interac-tions for the more complicated Type III functionalresponse which we investigate in Section 3.3. We beginby considering quadratic functional responses.

3.1. Quadratic functional responses

We consider the dynamical interaction

dL

dt¼ �cL2P;

dI

dt¼ cL2P;

dP

dt¼ 0, (10)

i.e. g1 ¼ cL, g2 ¼ 0, g3 ¼ 0 and g4 ¼ 0 which incorporatesa quadratic functional response. The solution to the abovecontinuous-time model is

Lðt; tÞ ¼RHt

1þ cRHtPtt; Pðt; tÞ ¼ Pt,

Iðt; tÞ ¼ RHt � Lðt; tÞ ð11Þ

which yields the following function f ðHt;PtÞ in thediscrete-time model (6)

f ðHt;RtÞ ¼LðT ; tÞ

RHt

¼1

1þ cTRHtPt

. (12)

Stability analysis of the model (see Appendix A) revealsthat the equilibrium point

H� ¼1ffiffiffiffiffiffiffiffiffiffiffiffi

kTRcp ; P� ¼ kðR� 1ÞH� (13)

is neutrally stable with a period of 2p= arctanðffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � 1p

Þ forsmall amplitude oscillations. Thus, the cycle period is verylong when R is only slightly greater than one (the minimumvalue for viable populations), drops to six for R ¼ 2, andasymptotically approaches a four year period of oscillationwith increasing R.The neutral stability persists if we add a background

density-independent parasitoid mortality, i.e, g3 ¼ c1. Weobtain the same form for the function f as (12) with cT

being replaced by c½1� expð�c1TÞ�=c1. This does not affectthe above neutral stability result which holds independentof c.

3.2. Higher order functional responses

We now consider a generalization of the above model bytaking the functional response to be cLmþ1 with m40, i.e.g1 ¼ cLm. Using similar analysis to that above, we obtain

f ðHt;RtÞ ¼1

½1þ cTmðRHtÞmPt�

1=m, (14)

and the following equilibrium densities

H� ¼Rm � 1

cTkmRmðR� 1Þ

� �1=ðmþ1Þ

; P� ¼ kðR� 1ÞH�.

(15)

ARTICLE IN PRESS

Fig. 2. Plots of the fraction of host larvae escaping parasitism f as a

function of host density Ht, as given by (14) (dashed line, functional

response is a function of the current host larvae density) and (16) (solid

line, functional response is a function of the initial host larvae density) for

R ¼ 2;T ¼ 1; c ¼ :01;m ¼ 1 and Pt ¼ 1.

A. Singh, R.M. Nisbet / Journal of Theoretical Biology 247 (2007) 733–742736

In Appendix B we show that functional response of theform cLmþ1 can stabilize the host–parasitoid interaction form41.

It is instructive to contrast this result with previous work(Rogers, 1972; Hassell and Comins, 1978) where functionalresponses were incorporated by assuming that the attackrate g1 is a constant and equal to cLð0; tÞm ¼ cðRHtÞ

m, i.e. afunction of only the initial number of host larvae available.Such an assumption does not prevent the parasitoids fromoverexploiting the hosts as a large initial host density leadsto a high attack rate throughout the vulnerable stage inspite of the fact that the host population has been reducedto low levels towards the end of the stage. Not surprisingly,the equilibrium of the corresponding discrete-time modelgiven by

f ¼ exp½�cTðRHtÞmPt� (16)

is unstable for all mX0. The difference between thisfunction and the one obtained from a semi-discrete model(Eq. (14)) is illustrated in Fig. 2. Note the larger values of f

obtained from the semi-discrete approach, i.e. more hostlarvae escaping parasitism when Ht is large as compared to(16). This is simply because the semi-discrete approachtakes into account that the attack rate is not a constant butdecreases with the host larval densities throughout thevulnerable period. This leads to a larger host populationescaping parasitism and prevents overexploitation, whichcontributes to the stability of the semi-discrete model.

3.3. Type II and III functional responses

Changing the attack rate to

g1 ¼

cL

LH

� �m

1þL

LH

� �mþ1(17)

with mX1 incorporates a sigmoidal functional responsewith maximal value cLH . Analysis in Appendix C showsthat this yields the implicitly defined discrete-time model

Htþ1 �Lmþ1

H

mHmtþ1

¼ RHt �Lmþ1

H

mðRHtÞm � cTLHPt, (18a)

Ptþ1 ¼ kðRHt �Htþ1Þ (18b)

with the following equilibrium densities.

H� ¼ LH

Rm � 1

mRmðR� 1ÞðkcTLH � 1Þ

� �1=ðmþ1Þ

,

P� ¼ kðR� 1ÞH�; kcTLH41. ð19Þ

The product cTLH represents the number of hostsparasitized per parasitoid during the entire larval stagewhen the host larvae densities are large. Hence, the aboveequilibrium is only feasible if the population of parasitoidscan grow, i.e. their reproductive rate ðkcTLH Þ is larger thanone, when a large number of hosts are available.

The equilibrium in the above discrete-time model isstable for m�om where constant m� is the solution to

Rm�m�

Rm� � 1¼

kcTLHR� 1

ðkcTLH � 1ÞðR� 1Þ(20)

and has a value greater than one. If kcTLhb1 then m� � 1and if Rb1 then m� � kcTLH=ðkcTLH � 1Þ. From theabove analysis we conclude that a Type III functionalresponse can stabilize a host–parasitoid interaction as longas the initial portion of the functional response acceleratessufficiently strongly. Also, as m� is always greater than one,a Type II functional response, which would correspond tom ¼ 0, will lead to an unstable equilibrium.

4. Effects of density dependence

It is well known that density-dependent effects, ex-pressed through dependence of functions g1; g2; g3 and g4

on initial population density can stabilize the equilibria inthe Nicholson–Bailey model. For example, May et al.(1981) showed the stabilizing effect of density-dependentself-limitation in the host by introducing the discrete formof the logistic equation to the host equation and Hasselland Varley (1969) provided an analysis of the stabilizingeffects of density-dependent mutual interference betweensearching parasitoid adults. In this section we willinvestigate some density-dependent effects by incorporat-ing them in the continuous-time portion of our hybridmodel. A key advantage of using the hybrid approach here

ARTICLE IN PRESSA. Singh, R.M. Nisbet / Journal of Theoretical Biology 247 (2007) 733–742 737

is that these density-dependent effects can act concurrentlywith host mortality from parasitism. This contrasts withprevious studies where for ease of discrete-time modelformulation, it was explicitly or implicitly assumed that thedensity-dependent mortality acts at a stage before or afterthe stage where hosts are attacked by parasitoids.

4.1. Density dependent host mortality

We first consider the dynamical interaction

dL

dt¼ �cLP� c1L

2;dI

dt¼ cLP;

dP

dt¼ 0, (21)

i.e. g1 ¼ c; g2 ¼ c1L and g3 ¼ g4 ¼ 0. Here not only doboth the host mortalities occur concurrently but also thedensity-dependent mortality rate is a function of thecurrent density of hosts available. We assume thatparasitized larvae immediately becomes juvenile parasi-toids which do not face competition for resources with thehost larvae, and hence, have no density-dependencemortality acting on them, i.e. g4 ¼ 0. In the absence ofparasitoids, it is known that this system would lead toBeverton–Holt dynamics in the host (Gurney and Nisbet,1998, p. 125).

The solution to the above continuous-time modelis

Lðt; tÞ ¼RHt

expðcPttÞ þ c1RHt

expðcPttÞ � 1

cPt

, (22a)

Iðt; tÞ ¼cPt

c1ln 1þ c1RHt

1� expð�cPttÞcPt

� �, (22b)

Pðt; tÞ ¼ Pt. (22c)

which gives the discrete-time host–parasitoid model as

Htþ1 ¼ LðT ; tÞ ¼RHt expð�cTPtÞ

1þ c1RHt

1� expð�cTPtÞ

cPt

, (23a)

Ptþ1 ¼ IðT ; tÞ ¼kcPt

c1ln 1þ c1RHt

1� expð�cTPtÞ

cPt

� �.

(23b)

The above system has two non-trivial equilibrium points.The first is the no-parasitoid equilibrium that is set by thestrength of the host-density dependence

H�NC ¼R� 1

c1TR; P� ¼ 0. (24)

The Beverton–Holt-like host dynamics imply that the hostpopulation alone can never have overcompensation orexhibit cycles. Stability analysis using the Jury conditionsshow that the above equilibrium is stable in the full systemfor

lnRoc1

kc, (25)

a condition that precludes growth of a small parasitoidpopulation when the hosts are at carrying capacity. Thus,sufficiently large values of c1=ck, which can be interpretedas the strength of density-dependent mortality vs. parasit-ism, stabilize the no-parasitoid equilibrium. Similar stabi-lity analysis on Mathematica reveals that the secondequilibrium, where both host and parasitoid are presentand given by

H� ¼exp

c1

kc

� �� 1

1�exp

c1

kc

� �R

0BBBB@

1CCCCA

cP�

c1R; P� ¼

lnR�c1

kccT

, (26)

is stable for

z�oc1

cko lnR, (27)

where z� is the solution to

z� þ 1 ¼RðlnR� z�Þ

R� expðz�Þ. (28)

The constant z� is an increasing function of R, howeverincreasing not as fast as lnR. Thus the size of the stabilityregion specified in (27), increases with R as is illustrated inFig. 3. For c1=kc4 lnR the no-parasitoid equilibrium (24) isstable and as we will see, for c1=kcoz� one typically getspersistence of both host and parasitoid populations in thesense of bounded oscillations. Fig. 4 plots the aboveequilibrium densities and illustrates the different transitionsbetween stability and instability. Note that for each value ofthe parameters there is at most one stable equilibrium of thediscrete-time model (23). This is in contrast to the modelstudied by Neubert and Kot (1992) which reduces tothe Ricker model for the host in the absence of parasitoids.Due to the overcompensation in the Ricker model, theNeubert–Kot model may exhibit two locally stable attrac-tors, one where the consumer persists and other where it isextinct, for intermediate values of c1. Fig. 5 illustrates asimulation of the discrete-time model (23), with attack rate c

chosen such that c1=kc takes values below, in, and above thestable region (27). In summary, for a fixed parasitism rate,stability arises at intermediate levels of density-dependentmortality. At low levels of density-dependent mortality thehost population becomes large enough for the parasitoids tooverexploit it, which leads to familiar cycles of increase andcrashes. On the other hand, strong density-dependentmortality causes a very fast decrease of the hosts throughoutthe vulnerable stage and prevents the parasitoids fromexploiting the resource at all, which leads to their extinction.Stability occurs in the intermediate situation where para-sitoids can get enough recruits for the next generation butalso do not overexploit the hosts. The quantity

q ¼R exp

c1

kc� 1

� �lnR�

c1

kc

� �ðR� 1Þ R� exp

c1

kc

� � (29)

ARTICLE IN PRESS

Fig. 4. Plot of host and parasitoid equilibrium densities as a function of the strength of density-dependent mortality (c1). Other parameter were taken as

R ¼ 2;T ¼ 1; k ¼ 1 and c ¼ 1. H� and P� refer to the host–parasitoid equilibrium (26) and H�NC refers to the no-parasitoid equilibrium given by (24). The

solid and dashed lines correspond to stable and unstable equilibrium densities, respectively.

Fig. 3. The stability region specified in (27), for the discrete-time model (23) as a function of strength of density-dependent mortality vs. parasitism (c1=kc)

and R (the number of viable eggs produced by each adult host). The no-parasitoid equilibrium refers to (24) while the host–parasitoid equilibrium refers to

(26).

A. Singh, R.M. Nisbet / Journal of Theoretical Biology 247 (2007) 733–742738

is of special interest to biological control, for it represent theratio of the host equilibrium with parasitoids and withoutparasitoids. As one varies c1=kc in the stability region, q

increases monotonically between q� and one, where

q� ¼Rðexpðz�Þ � 1ÞðlnR� z�Þ

ðR� 1ÞðR� expðz�ÞÞ. (30)

The quantity q� represent the maximum level of hostdepression that is consistent with a stable dynamics. Wecalculated q� ¼ :34 and :29 for R ¼ 2 and 10, respectively.As 10 appears to be an upper limit on observed values of R

(Murdoch et al., 2003), this shows that strong host

depression is accompanied by instability for biologicallyappropriate parameters.

4.2. Density dependent parasitoid mortality

We first consider the scenario of density-independent

parasitoid mortality which is incorporated in the followingdynamical interaction:

dL

dt¼ �cLP;

dI

dt¼ cLP;

dP

dt¼ �c2P. (31)

ARTICLE IN PRESS

Fig. 5. Population densities of hosts and parasitoids in year t as given by the discrete-time model (23). Parameters taken as R ¼ 2; c1 ¼ :01; k ¼ 1;T ¼ 1

and (a) c ¼ :05, (b) c ¼ :02 and (c) c ¼ :0125 which correspond to values below, in and above the stable region (27), respectively. Initial densities are 20 and

30 for the host and parasitoid, respectively.

A. Singh, R.M. Nisbet / Journal of Theoretical Biology 247 (2007) 733–742 739

This leads to a discrete-time model identical to theNicholson–Bailey model given by (8) with cT replaced byc½1� expð�c2TÞ�=c2. Thus density-independent parasitoidmortality has no effect on stability. We now investigate ifdensity-dependent parasitoid mortality could stabilizethe host–parasitoid interaction. Such density-dependentmortality could arise due to limitation of adultparasitoids by a resource other than the hosts orby a specialist predator that responds to density ofparasitoids. Towards this end, we change the aboveinteraction to

dL

dt¼ �cLP;

dI

dt¼ cLP;

dP

dt¼ �c2P

2, (32)

i.e. g1 ¼ c, g2 ¼ g4 ¼ 0 and g3 ¼ c2P. Analysis of thissystem leads us to the discrete-time model (6) with thefunction f given by

f ðHt;PtÞ ¼1

ð1þ c2TPtÞc=c2

(33)

which has equilibrium densities

H� ¼Rc2=c � 1

kc2TðR� 1Þ; P� ¼

Rc2=c � 1

c2T. (34)

Note that this form of function f is identical to the popularversion of f given by the zero term of the negative binomialdistribution (May, 1978). This similarity is interesting giventhat May’s model was motivated by a different situation—density dependence in the parasitoid attack rate that is inducedby assuming that hosts vary in their susceptibility to attack.Stability arises in this model if c24c (May, 1978). Here, wehave shown that strong density-dependent parasitoid mortality

may similarly stabilize the host–parasitoid interaction. As withMay’s model, stability is associated with an increased hostequilibrium, i.e. the parasitoids have reduced capacity tooverexploit their host.

5. Discussions

The preceding analyses revisit two classic questions inhost–parasitoid theory: how is the stability of interacting

ARTICLE IN PRESSA. Singh, R.M. Nisbet / Journal of Theoretical Biology 247 (2007) 733–742740

populations affected by the form of the functional responseand by density dependence. Our primary findings regardingdensity dependence are unsurprising, but the results on theeffects of the functional response contrast starkly withprevious work that used heuristically formulated models.

The key previous result on the effects of the functionalresponse was that of Hassell and Comins (1978) whoargued that accelerating functional response do notcompensate for the instability in the Nicholson–Baileymodel. This implies very different dynamics in discrete-time models than in their continuous-time counterpartswhere accelerating functional responses are stabilizing(Murdoch and Oaten, 1975). Fig. 2 shows the reason forthe different outcomes; essentially, our model recognizesthat parasitism slows down when the host populationbecomes small, thereby precluding the smallest hostpopulations. This leads to an increase in the equilibriumhost density and to stability. The ecological importance ofour findings on the stabilizing effects of acceleratingfunctional responses will depend on the mechanismresponsible for the acceleration. The model in this paperassumes that the parasitoid attack rate depends on theinstantaneous host density. This is obviously an idealiza-tion, as some time delay is inevitably involved. For type IIfunctional responses, there are mechanisms (e.g. thoseinvolving handling time) for which this delay is likely to beshort, but it is harder to defend this assumption withaccelerating functional responses caused by changes inparasitoid behavior. Murdoch and Bence (1987) arguedthat for aquatic predators, non-delayed, type III functionalresponses are seldom observed, and we suspect that asimilar situation holds for parasitoids. Yet the delaysinvolved in parasitoid response may plausibly be muchshorter than the duration of the vulnerable stage of thehost, and in such circumstances our model remains a morereasonable caricature of reality than its discrete-timecounterpart. The take-home message for applications isthe need for careful consideration of underlying mechan-isms and any associated time delays when constructingmodels of any particular system that includes acceleratingfunctional responses. Our conclusion that density depen-dence in either host or parasitoid is stabilizing is, by itself,unsurprising. But the dynamics of our model differs inimportant ways from previous work, not least in theabsence of multiple attractors (Neubert and Kot, 1992;Kon, 2006).

The work reported in this paper illustrates the limita-tions of models in the form of equations (6), namely

Htþ1 ¼ RHtf ðHt;PtÞ, (35a)

A ¼

1þ RH�qf ðHt;Pt

qHt

k R� 1� RH�qf ðHt;

qH

"266664

Ptþ1 ¼ kRHt½1� f ðHt;PtÞ�. (35b)

Much theory on the periods of population cycles inconsumer–resource systems assumes this form or general-izations that allow adult individuals to survive from oneyear to the next (Murdoch et al., 2001, 2003), with oneimportant conclusion being that in a broad range ofsituations, consumer–resource interactions lead to a lowerbound on cycle periods close to 6 time units. Shorterperiods typically require very large values of the geometricrate of increase, R, which is expected to be less than 10 innatural systems (Murdoch et al., 2003). By contrast, themodel in this paper with the quadratic functional responsehas a cycle period of 5 when R ¼ 3:23, well within theecologically feasible range. Yet the data analyzed byMurdoch et al. was broadly consistent with expectationsfrom the simpler model. Further work on cycle periodsusing the semi-discrete formalism is clearly called for.Finally, we note the generality of our approach. Here we

focused on two aspects of consumer–resource interactionsthat have for decades been considered important forstability of consumer–resource interactions. Many otherprocesses can be considered with semi-discrete models. Forexample, we have work in progress on the consequences ofhost-feeding where parasitoids face a choice between eatingor ovipositing in a host. Previous work (e.g. Yamamuraand Yano, 1988; Briggs et al., 1995; Kidd and Jervis, 1989)using continuous-time models has identified situationswhere host feeding has no effect on stability, and situationswhere it is stabilizing. With partially synchronized lifecycles, we lack intuition and systematic investigation isrequired.

Acknowledgments

We thank Cherie Briggs, Bill Murdoch and ElizavetaPachepsky for discussions. A. Singh would like to acknowl-edge support from J.P. Hespanha throughout this work.

Appendix A. Stability analysis for quadratic functional

responses

Denoting small fluctuation around the equilibrium (13)by htþ1 and ptþ1, one obtains after linearization thefollowing linear discrete system:

htþ1

ptþ1

" #¼ A

ht

pt

" #, (A.1)

with

ÞHt¼H�;Pt¼P�

RH�qf ðHt;PtÞ

qPt

Ht¼H� ;Pt¼P�

PtÞ

t

Ht¼H� ;Pt¼P�

#�kRH�

qf ðHt;PtÞ

qPt

Ht¼H�;Pt¼P�

377775. (A.2)

ARTICLE IN PRESSA. Singh, R.M. Nisbet / Journal of Theoretical Biology 247 (2007) 733–742 741

For stability of the above system we require the followingthree Jury conditions (Elaydi, 1996) to hold

1� trðAÞ þ detðAÞ40, (A.3a)

1þ trðAÞ þ detðAÞ40, (A.3b)

1� detðAÞ40. (A.3c)

Substituting the function

f ðHt;RtÞ ¼1

1þ cTRHtPt

(A.4)

and the equilibrium point

H� ¼1ffiffiffiffiffiffiffiffiffiffiffiffi

RcTkp ; P� ¼ kðR� 1ÞH� (A.5)

in (A.2) we have

trðAÞ ¼2

R; DetðAÞ ¼ 1 (A.6)

which corresponds to a neutrally stable equilibrium. Theroots of the characteristic equation, DetðA� lIÞ ¼ 0, ofthe matrix A are given by

l ¼ expð�yiÞ; y ¼ arctanðffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � 1

pÞ, (A.7)

and hence, a period of oscillation of 2p= arctanðffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � 1p

Þ.

Appendix B. Stability analysis for higher order functional

responses

Substituting the function

f ðHt;RtÞ ¼1

½1þ cTmðRHtÞmPt�

1=m(B.1)

and the equilibrium

H� ¼Rm � 1

cTkmRmðR� 1Þ

� �1=ðmþ1Þ

; P� ¼ kðR� 1ÞH�

(B.2)

in (A.2) we have

trðAÞ ¼1� R�m þ ðR� 1ÞmR�m

ðR� 1Þm,

DetðAÞ ¼Rð1� R�mÞ

ðR� 1Þm. ðB:3Þ

Using the fact that for all m41, R41 we have trðAÞ40,14DetðAÞ40 and

1� trðAÞ þ detðAÞ ¼ðmþ 1Þð1� R�mÞ

m40, (B.4)

one can see that the Jury conditions (A.3) hold, and hencethere is a stable equilibrium, for m41.

Appendix C. Stability analysis for Type II and III functional

responses

The solution for the corresponding continuous-timemodel

dLðt; tÞdt

¼ �

cLðt; tÞ

LH

� �m

1þLðt; tÞ

LH

� �mþ1

26664

37775Lðt; tÞPðt; tÞ, (C.1a)

Pðt; tÞ ¼ Pt, (C.1b)

RHt ¼ Iðt; tÞ þ Lðt; tÞ, (C.1c)

is

Lðt; tÞ �Lmþ1

H

mLmðt; tÞ¼ RHt �

Lmþ1H

mðRHtÞm � cLHPtt (C.2)

which gives us the implicitly defined discrete-time model

Htþ1 �Lmþ1

H

mHmtþ1

¼ RHt �Lmþ1

H

mðRHtÞm � cTLHPt, (C.3a)

Ptþ1 ¼ kðRHt �Htþ1Þ (C.3b)

and the following fixed point

H� ¼ LH

Rm � 1

mRmðR� 1ÞðkcTLH � 1Þ

� �1=ðmþ1Þ

,

P� ¼ kðR� 1ÞH�; kcTLH41. ðC:4Þ

Linearizing the above discrete-time model about thisequilibrium, we obtain the linear discrete system

htþ1

ptþ1

" #¼ A

ht

pt

" #, (C.5)

with

A ¼

qHtþ1

qHt

Ht¼H� ;Pt¼P�

qHtþ1

qPt

Ht¼H� ;Pt¼P�

qPtþ1

qHt

Ht¼H� ;Pt¼P�

qPtþ1

qPt

Ht¼H� ;Pt¼P�

266664

377775: ðC:6Þ

Using (C.3) and (C.4) we find that

trðAÞ ¼Rþ

gRm þ kcTLH

1þ g; DetðAÞ ¼

RkcTLH

1þ g,

g ¼ðkcTLH � 1ÞðR� 1ÞmRm

Rm � 1. ðC:7Þ

Substituting these in the Jury conditions (A.3), one findsthat inequalities (A.3a)–(A.3b) hold for all m40;kcTLH41;R41. The stability condition comes from

ARTICLE IN PRESSA. Singh, R.M. Nisbet / Journal of Theoretical Biology 247 (2007) 733–742742

inequality (A.3c) which holds for all m�om where constantm�41 is the solution to

m�

1�1

Rm�

� � ¼ kcTLHR� 1

ðkcTLH � 1ÞðR� 1Þ. (C.8)

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