A DISCRETE-TIME HOST-PARASITOID MODEL

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1 A DISCRETE-TIME HOST-PARASITOID MODEL SOPHIA R.-J. JANG AND JUI-LING YU We study a discrete-time host-parasitoid model proposed by May et al. In this model, the parasitoid attacks the host first then followed by density dependence, where density dependence depends only on those host populations that escaped from being parasitized. Asymptotic dynamics of the resulting system are derived. There exist thresholds for which both populations can coexist indefinitely. Copyright 2006 S. R.-J. Jang and J.-L. Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction It is well known that the sequence of density dependence and parasitism in the host life cycle can have a significant effect on the population dynamics of the host-parasitoid interaction. Consequently, the effect can have important implications for biological control. In [10], May et al. proposed and numerically simulated three host-parasitoid models based on the timing of parasitism and density dependence. In this work, we will study a model proposed by May et al. [10] in which parasitism occurs first then followed by density dependence. However, density dependence only depends on the remaining host population that escaped being parasitized. 2. The model Let N t be the host population at time t. The parasitoid population at time t is denoted by P t. An individual parasitoid must find a host to deposit its eggs so that the parasitoid can reproduce. It is assumed that parasitism occurs first then followed by density dependence. Let β be the average number of offsprings that a parasitized host can reproduce for a parasitoid individual. It is assumed that the number of encounters between host and parasitoid populations at any time t 0 follows that of simple mass action, bn t P t, where the searching efficiency b is a constant. We assume that the number of encounters is distributed randomly with a Poisson distribution. Consequently, the probability that an individual host will escape from being parasitized when the parasitoid population is Hindawi Publishing Corporation Proceedingsof the Conference on Differential & Difference Equations and Applications, pp

2 452 A discrete-time host-parasitoid model of size P is e bp. For simplicity, the host population in the absence of the parasitoid is modeled by a simple Beverton-Holt equation λn/(1+kn), where parameters λ and k are positive. Since density dependence occurs after parasitism, the interaction between the host and the parasitoid is governed by the following system of difference equations: λn N t t+1 = 1+kN t e bpt e bpt, (2.1) ( ) P t+1 = βn t 1 e bp t, (2.2) N 0,P 0 0. Steady state E 0 = (0,0) always exists. The Jacobian matrix can be given by where ( ) J 11 J 12 J = β ( 1 e bp) βbne bp, (2.3) J 11 = J 12 = λe bp ( 1+kNe bp ) 2, (2.4) ( λbne bp 1+kNe bp ) 2. Note that ( ) λ 0 J(0,0) =. (2.5) 0 0 Thus it can be easily seen that E 0 is the only steady state of system (2.1) ifλ<1 and it is globally asymptotically stable. Indeed, N t+1 = 1+kN t e bpt e bpt = kn t + e bpt 1+kN t <, (2.6) for t 0 implies lim t N t = 0asλ<1. As a result, we can show that lim t P t = 0and hence E 0 = (0,0) is globally asymptotically stable. Suppose now λ>1. Then (2.1) has another boundary steady state E 1 = ((λ 1)/k,0)= ( N,0) and the Jacobian matrix of the system associated with E 1 is 1 ( ) J 12 E1 J( N,0)= λ 0 βb λ 1. (2.7) k

3 S. R.-J. Jang and J.-L. Yu 453 Thus E 1 is locally asymptotically stable if βb(λ 1)/k = βb N <1. We show that (2.1)has no interior steady state if βb N <1. Notice that the P-component of an interior steady state (N,P ) must satisfy λ = e bp + kh(p), (2.8) where h(p) = P/β(1 e bp )forp>0. Since lim P 0 + h(p) = 1/βb, h (P) > 0forP>0and lim t h( ) =,weseethat(2.8) has a positive solution P if and only if βb + k βb <λ iff βb N >1. (2.9) In this case P > 0 is unique and there is a unique interior steady state E 1 = (N,P )if βb N >1. We conclude that if λ>1andβb N <1, then E 1 is locally asymptotically stable and there is no interior steady state. We show that solutions of (2.1) withn 0 > 0all converge to E 1. To this end, λn N t t+1 =, (2.10) e bpt + kn t 1+kN t for t 0 implies limsup t N t (λ 1)/k by a simple comparison argument. Then for any ɛ > 0 there exists t 0 > 0suchthatN t < (λ 1)/k + ɛ for t t 0.Sinceβb N <1, we choose ɛ > 0suchthat But then βb( N + ɛ) < 1. (2.11) P t+1 = βn t ( 1 e bp t ) <β( N + ɛ) ( 1 e bpt) βb( N + ɛ)p t, (2.12) for t t 0 implies lim t P t = 0. Consequently, we can prove that liminf t N t (λ 1)/k if N 0 > 0. Therefore, lim t N t = N and E 1 is globally asymptotically stable. Suppose now λ>1andβb N >1. Notice E 0 and E 1 are unstable and (2.1)hasaunique interior steady state. We prove that the system is uniformly persistent by using a result of Hofbaur and So [6]. Clearly, system (2.1)hasaglobalattractorX.LetY ={(N,P) R 2 + : N = 0orP = 0}, that is, Y is the union of nonnegative coordinate axes, and let M be the maximal invariant set in Y. ThenM ={E 0,E 1 },where{e 0 } and {E 1 } are isolated in X. We claim that the stable set W + (E 0 ) ={(N,P) R 2 + : N t 0,P t 0ast }lies in Y. For suppose there exists a solution (N t,p t )of(2.1) withn 0 > 0, P 0 > 0suchthat lim t (N t,p t ) = E 0, then since λ>1, we can choose ɛ > 0suchthatλ e bɛ > 0. For this ɛ > 0 there exists t 1 > 0suchthatP t < ɛ for t t 1, and consequently N t+1 = λn > t, (2.13) e bpt + kn t e bɛ + kn t for t t 1.Henceliminf t N t > (λ e bɛ )/k > 0 and we obtain a contradiction. Therefore W + (E 0 )liesony. Similarly, if there exists a solution (N t,p t )of(2.1)withn 0,P 0 > 0such

4 454 A discrete-time host-parasitoid model that lim t (N t,p t ) = E 1 = ((λ 1)/k,0), then for any ɛ > 0 there exists t 2 > 0suchthat N t > (λ 1)/k ɛ if t t 2.Sinceβb N >1, we choose ɛ > 0suchthatβb((λ 1)/k ɛ) > 1. But then ( λ 1 (1 ) P t+1 >β ɛ) e bp t, (2.14) k for t t 2 implies liminf t P t > 0 and we obtain a contradiction. Therefore W + (E 1 )lies on Y and system (2.1) is uniformly persistent by Hofbauer and So [6, Theorem 4.1]. We summarize the above discussion in the following theorem. Theorem 2.1. Dynamics of system (2.1) can be summarized below. (a) If λ<1,thensolutionsof(2.1)allconvergetoe 0 = (0,0). (b) If λ>1, thensystem(2.1) has another boundary steady state E 1 = ( N,0). In addition if βb N <1, then solutions of (2.1) withn 0 > 0 all converge to E 1.Ifβb N >1, then system (2.1) has a unique interior steady state E 2 = (N,P ) and (2.1) is uniformly persistent, that is, there exists M>0 such that liminf t N t M and liminf t P t M for all solutions (N t,p t ) of (2.1)withN 0 > 0 and P 0 > Discussion In this short chapter we investigated a model proposed by May et al. [10], where parasitism occurs before density dependence and density dependence depends only on the remaining population that escaped from being parasitized. The model exhibits simple asymptotic dynamics. Both populations go to extinction if the intrinsic growth rate λ of the host is less than 1. When the host intrinsic growth rate is greater than 1, then the host can stabilize in a positive steady state N in the absence of the parasitoid. Therefore the parasitoid population becomes extinct if βb N <1, where βb N can be interpreted as the growth rate of the parasitoid when the host is stabilized at the level N. Both populations can coexist indefinitely if λ>1andβb N >1. Notice the per capita population growth rate of the host in the absence of the parasitoid population is a decreasing function of the host population. Allee effects occur when the per capita growth rate of a species is initially an increasing function of the population size [1]. Allee effects may due to a variety of causes ranging from mating limitation, predator saturation, and antipredator defense and so forth. Among these is the uncertainty of finding mates to reproduce or lack of cooperative individuals to exploit resources efficiently in spars populations. We refer the reader to [1, 2, 4, 5]formorebiological discussion about Allee effects.see also [3, 7 9, 11 14] and references cited therein for models of Alee effects. We will next incorporate Allee effects into the host population and examine the Allee effects upon the dynamics of the host-parasitoid interaction studied in this manuscript. References [1] W. C. Allee,The Social Life of Animals, William Heinemann, London, [2] M. Begon, J. Harper, and C. Townsend, Ecology: Individuals, Populations and Communities, Blackwell Science, New York, 1996.

5 S. R.-J. Jang and J.-L. Yu 455 [3] J.M.Cushing,The Allee effect in age-structured population dynamics, Mathematical Ecology (Trieste, 1986) (T. Hallam, L. Gross, and S. Levin, eds.), World Scientific, New Jersey, 1988, pp [4] B. Dennis, Allee effects: population growth, critical density, and the chance of extinction, Natural Resource Modeling 3 (1989), no. 4, [5], Allee effectsin stochasticpopulations, Oikos96 (2002), [6] J. Hofbauer and J. W.-H. So, Uniform persistence and repellors for maps, Proceedings of the American Mathematical Society 107 (1989), no. 4, [7] S. R.-J. Jang, Allee effectsin a discrete-timehost-parasitoid model,journalofdifference Equations and Applications 12 (2006), no. 2, [8] S. R.-J. Jang and S. L. Diamond, A host-parasitoid interaction with Allee effects on the host, submitted to Computers and Mathematics with Applications. [9] M. R. S. Kulenović and A.-A. Yakubu, Compensatory versus overcompensatory dynamics in density-dependent Leslie models, Journal of Difference Equations and Applications 10 (2004), no , [10] R. M. May, M. P. Hassell, R. M. Anderson, and D. W. Tonkyn, Density dependence in hostparasitoid models, Journal of Animal Ecology 50 (1981), no. 3, [11] A. Morozov, S. Petrovskii, and B.-L. Li, Bifurcations and chaos in a predator-prey system with the Allee effect,proceedings of the Royal Society. Series B 271 (2004), [12] S. Schreiber, Allee effects, extinctions, and chaotic transientsin simple population models, Theoretical Population Biology 64 (2003), [13] A.-A. Yakubu, Multiple attractors in juvenile-adult single species models, JournalofDifference Equations and Applications 9 (2003), no. 12, [14] S. Zhou, Y. Liu, and G. Wang, The stability of predator-prey systems subject to the Allee effects, Theoretical Population Biology 67 (2005), Sophia R.-J. Jang: Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA , USA address: jang@louisiana.edu Jui-Ling Yu: Department of Applied Mathematics, Providence University, Taichung 43301, Taiwan address: jlyu@pu.edu.tw