Dynamics of a Discrete Host-Parasitoid System with Stocking

Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2015, Article ID 905957, 6 pages http://dx.doi.org/10.1155/2015/905957 Research Article Dynamics of a Discrete Host-Parasitoid System with Stocking Sophia R.-J. Jang Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, USA Correspondence should be addressed to Sophia R.-J. Jang; Received 25 March 2015; Accepted 3 June 2015 Academic Editor: Zhengqiu Zhang Copyright © 2015 Sophia R.-J. Jang. 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. Motivated by the biological control of pests, we present discrete-time models of host-parasitoid interactions to study the effects of external stocking upon the systems. It is assumed that density dependence of the hosts occurs first followed by parasitism. We prove that the constant stocking can eliminate the pest population if the stocking is sufficiently large. Furthermore, stocking can simplify the dynamics of the interaction by stabilizing the coexisting steady state. 1. Introduction There are many natural host populations that are also pests. Biological control is the reduction of pest populations by natural enemies, also known as the biological control agents. Many species of wasps and some flies are parasitoids and most of the parasitoids have a narrow host range which can be used as biological control agents. Biological controls often involve supplemental release of natural enemies. Relatively few natural enemies may be released at a critical time of the season (inoculative release) or literally millions may be released in a single time (inundative release) [1]. Several discrete-time mathematical models have been proposed to study the dynamical effects of external stocking or the inoculative release of the control agents. See, for example, AlSharawi and Rhouma [2], Chow and Jang [3], Elaydi and Yakubu [4], Jang and Yu [5], Kulenović and Nurkanović [6], and references cited therein. The classical Leslie-Gower competition model with stocking in one of the two populations is analyzed in [6] and the Ricker type competition system with stocking occurring in one of the two competing population is studied in [4]. The work in [2] investigates a multispecies population model with constant harvesting/stocking and a model of three interacting populations with stocking in one of the two competing populations is analyzed in [3]. The theory of optimal control is applied to study a host-parasitoid model in [5] where the constant stocking of parasitoids is used as a control strategy for the hosts. Insect populations frequently suffer from some density dependent effect in addition to mortality from insect parasitoids [7]. The ordering of density dependence and parasitism in the host life cycle can have significant impacts on the dynamics of the interactions [7]. In this investigation, we propose a discrete-time host-parasitoid model to study the effect of external stocking upon the host-parasitoid interaction. Unlike the work in [5], where parasitism occurs first followed by density dependence, it is assumed in the present study that density dependence of the hosts acts first followed by parasitism. Furthermore, the hosts are also pests and the parasitoids are used as biological control agents to control the pests. There is a constant level of the external parasitoids released into the interaction at each generation. It is shown that external stocking of the parasitoids can eliminate the hosts population and can also simplify the dynamics of the interaction. In the following section, the host-parasitoid model with no stocking is proposed and analyzed first. The corresponding model with stocking is then presented and studied. We derive sufficient conditions for the existence of interior steady state and discuss its magnitude relative to the model parameters for both systems. Numerical examples using MatLab are provided to illustrate our findings. The final section provides a brief summary. 2. The Models and Stability Analysis In this section, we present two general host-parasitoid models, one with stocking and the other with no stocking. 2 Discrete Dynamics in Nature and Society 2.1. The Model of No Stocking. Let 𝑥(𝑡) and 𝑦(𝑡) denote the host and parasitoid populations in generation 𝑡 = 0, 1, . . ., respectively. The parasitoid has a very narrow range of hosts and is specialized to this particular host population. It is assumed that density dependence of the hosts occurs first followed by parasitism. Moreover, the number of encounter between hosts and parasitoids is distributed randomly and the probability of an individual host escaping from being parasitized is modeled by the zero term of a Poisson distribution (cf. [7]). The parameter 𝑎 > 0 denotes the average number of encounters per unit time per parasitoid and is also referred to as the searching efficiency of the parasitoid. The host-parasitoid interaction without external stocking of the parasitoids is described by the following system: −𝑎𝑦(𝑡) 𝑥 (𝑡 + 1) = 𝜆𝑥 (𝑡) 𝑔 (𝑥 (𝑡)) 𝑒 , 𝑦 (𝑡 + 1) = 𝛽𝑥 (𝑡) 𝑔 (𝑥 (𝑡)) (1 − 𝑒−𝑎𝑦(𝑡) ) , (1) where 𝜆 > 0 is the intrinsic growth rate of the hosts and 𝛽 > 0 is the average number of parasitoids produced by each parasitized host. The per capita growth rate 𝑔 of the host population satisfies the following assumptions; namely, (H1) 𝑔 ∈ 𝐶2 [0, ∞), 𝑔(0) = 1, 𝑔(𝑥) > 0, 𝑔󸀠 (𝑥) < 0 for 𝑥 ≥ 0, lim𝑥 → ∞ 𝑔(𝑥) = 0, and sup{𝑥𝑔(𝑥) : 𝑥 ≥ 0} = 𝑙 < ∞. It is clear that the classical Beverton-Holt and Ricker type growth rates [8] satisfy (H1). Furthermore, the growth rate of the Beverton-Holt model has the following monotone properties: 󸀠 (𝑥𝑔 (𝑥)) > 0 󸀠󸀠 (𝑥𝑔 (𝑥)) < 0 ∀𝑥 ≥ 0, (2) ∀𝑥 ≥ 0. (3) By defining new state variable 𝑦̂ = 𝑎𝑦 and new parameter 𝛽̂ = 𝛽𝑎 and by ignoring the hats, system (1) is converted into 𝑥 (𝑡 + 1) = 𝜆𝑥 (𝑡) 𝑔 (𝑥 (𝑡)) 𝑒−𝑦(𝑡) , (a) If 𝜆 < 1, then 𝐸00 = (0, 0) is globally asymptotically stable in R2+ . (b) If 𝜆 > 1, then 𝐸00 is unstable and 𝐸10 = (𝑥, 0) exists. In addition, if (2) is satisfied, then 𝐸10 is globally asymptotically stable in {(𝑥, 𝑦) ∈ R2+ : 𝑥 > 0} if 𝛽𝑥𝑔(𝑥) < 1. If the monotone property (2) is not assumed, then dynamics of the host population in the absence of the parasitoids may be complicated. For example, the Ricker model undergoes a cascade of period-doubling bifurcations to chaos as the intrinsic growth rate of the population increases [8]. In such a case, dynamics of the host-parasitoid model (4) may not equilibrate even when 𝛽𝑥𝑔(𝑥) < 1. Let 𝜆 > 1. Note that the 𝑥 component of an interior steady state satisfies 𝐻0 (𝑥) := ln (𝜆𝑔 (𝑥)) − 𝛽𝑥𝑔 (𝑥) + 𝛽𝑥 = 0. 𝜆 (6) Let 𝐿(𝑥) = ln(𝜆𝑔(𝑥)) and 𝑅0 (𝑥) = 𝛽𝑥𝑔(𝑥) − 𝛽𝑥/𝜆. Then 𝐿 (0) = ln 𝜆 > 0, (4) 𝐿󸀠 (𝑥) = with nonnegative initial conditions. We first study system (4). Notice that system (4) always h (...truncated)