Stability Property for the Predator-Free Equilibrium Point of Predator-Prey Systems with a Class of Functional Response and Prey Refuges

Abstract

We investigate the stability property for the predator-free equilibrium point of predator-prey systems with a class of functional response and prey refuges by using the analytical approach. Under some very weakly assumption, we show that conditions that ensure the locally asymptotically stable of the predator-free equilibrium point are consistent with that of the globally asymptotically stable ones. Our result supplements the corresponding result of Ma et al., 2009.

1. Introduction

Stability analysis of a predator-prey system incorporating a given functional response becomes a popular issue in mathematical ecology during the last decade [1–10]. Ma et al. [10] considered the effects of prey refuges on a predator-prey model with a class of functional response, for example, 𝑋̇=𝑟𝑋1−𝑋𝐾̇−𝑝𝜑𝑋−𝛽𝑋𝑌,𝑌=𝑞𝜑𝑋−𝛽𝑋−𝑑𝑌, (1.1) where 𝑋(𝑡) and 𝑌(𝑡) denote the density of prey and predator populations at time 𝑡, respectively. The term 𝜑(𝑋) represents the functional response of the predator population and satisfies the following assumption:𝜑(0)=0,𝜑^′(𝑋)>0(𝑋>0). (1.2) By using the change of variables:𝜙∶𝑅⁺₀²⟶𝑅⁺₀²()=𝑥,𝜙𝑋,𝑌,𝑦1−𝛽1−𝛽, (1.3) system (1.1) is equivalent to the following model:𝑥𝑥̇=𝑟𝑥1−𝐾(𝑥)(𝑥)1−𝛽−𝑝𝜑𝑦,𝑦̇=𝑞𝜑−𝑑𝑦. (1.4)

Concerned with the stability property of the predator-free equilibrium, by analysing the Jacobian matrix, the authors obtained the following conclusions.

Conclusion 1. 𝐸₁((1−𝛽)𝐾,0) is locally asymptotically stable if and only if 𝑞𝜑((1−𝛽)𝐾)−𝑑<0.
Based on this conclusion, without any other deduction, they declared (see [10, Theorems  4.1(3) and 4.2(3)]).

Conclusion 2. If 1−𝜑⁻¹(𝑑/𝑞)/𝐾<𝛽<1, then predator goes extinct while prey population reaches its maximum environment carrying capacity.

Note that Conclusion 1 is local one while Conclusion 2 reflects the globally property of the system (1.4). Obviously there is a gap between these two conclusions. To show the Conclusion 1 implies Conclusion 2, some more detail analysis is needed. The aim of this paper is try to show that under some very weakly assumption on 𝜑, the local asymptotical stability of the predator-free equilibrium point do implies the global ones. More precisely, we obtain the following result.

Theorem 1.1. Assume that 𝜑(𝑥)=𝜑₁(𝑥)𝑥 and there exists a positive constant 𝐿 such that 𝜑₁(𝑥)≤𝐿 for all 𝑥>0 holds, assume further that 1−𝜑⁻¹(𝑑/𝑞)/𝐾<𝛽<1 holds. Then predator species will be extinct while prey population reaches its maximum environment carrying capacity.

Remark 1.2. Many explicit forms for the predator functional response that have been used are satisfy above assumption. For example, 𝑏𝑥,𝑎+𝑥HollingtypeII𝑏𝑥²𝑎+𝑥²,𝑎{}[],HollingtypeIII1−exp−𝑐𝑥Ivlev𝑏𝑥^𝛾,0<𝛾<1Rosenzweig. (1.5)

2. Proof of the Main Result

Proof. We first show that under the assumption of Conclusion 2, predator species will be driven to extinction.
It follows from 1−𝜑⁻¹(𝑑/𝑞)/𝐾<𝛽<1 and the continuity of 𝜑, for enough small positive constant 𝜀, the following inequality holds: 𝜑𝛽>1−⁻¹(𝑑−𝜀)/𝑞,𝐾+𝜀 (2.1) that is, 𝜑⁻¹𝑑−𝜀𝑞>1−𝛽(𝐾+𝜀). (2.2) Since 𝜑^′(𝑋)>0, 𝑋>0, inequality (2.2) is equal to the following inequality: 𝑑−𝜀>𝑞𝜑1−𝛽(𝐾+𝜀). (2.3) From the first equation of system (1.4), we have 𝑥𝑥̇≤𝑟𝑥1−1−𝛽𝐾. (2.4) Therefore, limsup_𝑡→+∞𝑥(𝑡)≤1−𝛽𝐾. (2.5) For 𝜀 defined by (2.1), inequality (2.5) shows that there exists an enough large 𝑇 such that 𝑥(𝑡)<1−𝛽(𝐾+𝜀),∀𝑡≥𝑇. (2.6) And so, for 𝑡≥𝑇, from the second equation of system (1.4) and (2.3), one has 𝑦̇≤𝑞𝜑1−𝛽(𝐾+𝜀)−𝑑𝑦<−𝜀𝑦. (2.7) That is, 𝑦(𝑡)≤𝑦(𝑇)exp{−𝜀(𝑡−𝑇)}⟶0,as𝑡⟶+∞. (2.8) For any small positive constant 𝜀₁>0 which satisfies 𝜀₁≤𝑟/2𝑝L, there exists a 𝑇₁>𝑇 such that 𝑦(𝑡)≤𝜀₁,∀𝑡>𝑇₁. (2.9)
On the other hand, since 𝜑(𝑥)=𝜑₁(𝑥)𝑥 and 𝜑₁(𝑥)≤𝐿 for all 𝑥>0. Equation (2.9) together with the first equation of (1.4) leads to 𝑥𝑥̇=𝑟𝑥1−1−𝛽𝐾−𝑝𝜑₁(𝑥)𝑥𝑥𝑦≥𝑟𝑥1−1−𝛽𝐾𝑥−𝑝𝐿𝜀₁𝑥1≥𝑟𝑥2−𝑥𝐾,1−𝛽 (2.10) for all 𝑡≥𝑇₁. From this different inequality, one could easily obtain that, liminf_𝑡→+∞𝑥(𝑡)≥1−𝛽𝐾2. (2.11) Now we introducing a transformation 𝑧=𝑥−(1−𝛽)𝑘, then the first equation of system (1.4) is equivalent to 𝑧̇=−𝑟𝑧𝑘𝑘𝑘1−𝛽𝑧+1−𝛽−𝑝𝜑𝑧+1−𝛽𝑦. (2.12) From (2.5), (2.11), and (2.12) we know that 𝑧(𝑡) is bounded differentiable on (0,∞), Let 𝑧=limsup_𝑡→+∞𝑧(𝑡),𝑧=liminf_𝑡→+∞𝑧(𝑡), According to Fluctuation lemma [11], there exists sequences 𝜏_𝑛→∞, 𝜎_𝑛→∞ such that 𝑧(̇𝜏_𝑛)→0, 𝑧(̇𝜎_𝑛)→0, 𝑧(𝜉_𝑛)→𝑧 and 𝑧(𝜎_𝑛)→𝑧. Also, it follows from (2.12) that 𝑧̇𝜎_𝑛𝜎=−𝑟𝑧_𝑛𝑘𝑧𝜎1−𝛽^𝑛+1−𝛽𝑘−𝑝𝜑𝑧𝜎_𝑛+1−𝛽𝑘𝑦𝜎_𝑛.𝑧̇𝜉_𝑛𝜉=−𝑟𝑧_𝑛𝑘𝑧𝜉1−𝛽^𝑛+1−𝛽𝑘−𝑝𝜑𝑧𝜉_𝑛+1−𝛽𝑘𝑦𝜉_𝑛. (2.13) Since (2.11) implies that lim_𝑛→+∞𝑧𝜉_𝑛+𝑘≥𝐾1−𝛽1−𝛽2. (2.14) Taking limit in (2.13) and (2.14), it follows from (2.8) and (2.16) that 0=𝑧=𝑧, (2.15) that is lim_𝑡→+∞𝑧(𝑡)=0, (2.16) which is equivalent to say that lim_𝑡→+∞𝑥(𝑡)=1−𝛽𝐾. (2.17) This ends the proof of Theorem 1.1.

Acknowledgments

Research was supported by the Natural Science Foundation of Fujian Province (2011J01007) and the Technology Innovation Platform Project of Fujian Province (2009J1007).