Uniformly Strong Persistence for a Delayed Predator-Prey Model

Abstract

An asymptotically periodic predator-prey model with time delay is investigated. Some sufficient conditions for the uniformly strong persistence of the system are obtained. Our result is an important complementarity to the earlier results.

1. Introduction

The dynamical behavior including boundedness, stability, permanence, and existence of periodic solutions of predator-prey systems has attracted a great deal of attention and many excellent results have already been derived. For example, Gyllenberg et al. [1] studied limit cycles of a competitor-competitor-mutualist Lotka-Volterra model. Mukherjee [2] made a discussion on the uniform persistence in a generalized prey-predator system with parasitic infection. Aggelis et al. [3] considered the coexistence of both prey and predator populations of a prey-predator model. Agiza et al. [4] investigated the chaotic phenomena of a discrete prey-predator model with Holling type II. Sen et al. [5] analyzed the bifurcation behavior of a ratio-dependent prey-predator model with the Allee effect. Zhang and Luo [6] gave a theoretical study on the existence of multiple positive periodic solutions for a delayed predator-prey system with stage structure for the predator. Nindjin and Aziz-Alaoui [7] focused on the persistence and global stability in a delayed Leslie-Gower-type three species food chain. Ko and Ryu [8] discussed the coexistence states of a nonlinear Lotka-Volterra-type predator-prey model with cross-diffusion. Fazly and Hesaaraki [9] dealt with periodic solutions of a predator-prey system with monotone functional responses. One can see [10–19] and so forth for more related studies. However, the research work on asymptotically periodic predator-prey model is very few at present.

The so-called asymptotically periodic function is that a function 𝑎(𝑡) can be expressed by the form 𝑎(𝑡)=𝑎(𝑡)+𝑎(𝑡), where 𝑎(𝑡) is a periodic function and 𝑎(𝑡) satisfies lim_𝑡→+∞𝑎(𝑡)=0.

In 2006, Kar and Batabyal [20] investigated the stability and bifurcation of the following predator-prey model with time delay 𝑑𝑥𝑟𝑑𝑡=𝑥𝑟−𝐾𝛼𝑥−¹𝑦𝑎₁−𝛼+𝑥²𝑧𝑎₂,+𝑥𝑑𝑦𝑑𝑡=𝑦−𝑑₁+𝛽₁𝛼₁𝑥(𝑡−𝜏)𝑎₁(),+𝑥𝑡−𝜏−𝛾𝑦𝑑𝑧𝑑𝑡=𝑧−𝑑₂+𝛽₂𝛼₂𝑥(𝑡−𝜏)𝑎₂(),+𝑥𝑡−𝜏−𝛿𝑧 (1.1) with initial conditions 𝑥(0)≥0,𝑦(0)≥0,𝑧(0)≥0, where 𝑧(𝑡) denotes the densities of prey; 𝑦(𝑡) and 𝑧(𝑡) denote the densities of two predators, respectively, at time 𝑡; 𝛾 and 𝛿 denote the intraspecific competition coefficients of the predators; 𝛽₁ and 𝛽₂ denote the conversion of biomass constant; 𝑑₁ and 𝑑₂ are the death rate of first and second predator species, respectively; 𝛼₁ is the maximum values of per capita reduction rate of 𝑥 due to 𝑦 and 𝛼₂ is the maximum values of per capita reduction rate of 𝑥 due to 𝑧; 𝑎₁ and 𝑎₂ are half saturation constants. 𝜏 is time delay in the prey species. All the parameters are positive constants. For details, one can see [20].

It will be pointed out that all biological and environment parameters in model (1.1) are constants in time. However, any biological or environmental parameters are naturally subject to fluctuation in time. Thus the effects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Therefore, the assumptions of periodicity of the parameters are a way of incorporating the periodicity the environment (such as seasonal effects of weather, food supplies, and mating habits). Inspired by above considerations and considering the asymptotically periodic function, in this paper, we will modify system (1.1) as follows: 𝑑𝑥𝑟(𝑡)+𝑟̃(𝑡)−𝑟(𝑡)+𝑟̃(𝑡)𝑑𝑡=𝑥𝐾(𝑡)+𝐾(𝑡)𝛼𝑥−¹(𝑡)+𝛼₁(𝑡)𝑦𝑎₁(𝑡)+𝑎₁(𝑡)−𝛼+𝑥²(𝑡)+𝛼₂(𝑡)𝑧𝑎₂(𝑡)+𝑎₂(𝑡),+𝑥𝑑𝑦−𝑑𝑑𝑡=𝑦₁(𝑡)+𝑑₁(𝑡)+𝛽₁(𝑡)+𝛽₁(𝑡)𝛼₁(𝑡)+𝛼₁(𝑡)𝑥(𝑡−𝜏)𝑎₁(𝑡)+𝑎₁(𝑡)()−𝛾(𝑡)+𝛾(𝑡)𝑦,+𝑥𝑡−𝜏𝑑𝑧−𝑑𝑑𝑡=𝑧₂(𝑡)+𝑑₂(𝑡)+𝛽₂(𝑡)+𝛽₂(𝑡)𝛼₂(𝑡)+𝛼₂(𝑡)𝑥(𝑡−𝜏)𝑎₂(𝑡)+𝑎₂(𝑡)()−𝛿(𝑡)+𝛿(𝑡)𝑧,+𝑥𝑡−𝜏 (1.2) with initial conditions 𝑥(0)≥0,𝑦(0)≥0,𝑧(0)≥0.

The principle object of this paper is to explore the uniformly strong persistence of system (1.2). There are very few papers which deal with this topic, see [10, 21].

In order to obtain our results, we always assume that system (1.2) satisfies (H1)𝛼_𝑖(𝑡), 𝛽_𝑖(𝑡), 𝑎_𝑖(𝑡), 𝑑_𝑖(𝑡)(𝑖=1,2),𝑟(𝑡), 𝛾(𝑡), 𝛿(𝑡), 𝐾(𝑡) are continuous, nonnegative periodic functions; 𝛼_𝑖(𝑡),  𝛽_𝑖(𝑡), 𝑎_𝑖(𝑡), 𝑑_𝑖(𝑡)(𝑖=1,2), 𝑅(𝑡), 𝛾(𝑡), 𝛿(𝑡), 𝐾(𝑡) are continuous, nonnegative asymptotically items of asymptotically periodic functions.

2. Uniformly Strong Persistence

In this section, we will present some result about the uniformly strong persistence of system (1.2). For convenience and simplicity in the following discussion, we introduce the notations, definition, and Lemmas. Let 0<𝑓^𝑙=lim_𝑡→+∞inf𝑓(𝑡)≤lim_𝑡→+∞sup𝑓(𝑡)=𝑓^𝑢<+∞. (2.1) In view of the definitions of lower limit and upper limit, it follows that for any 𝜀>0, there exists 𝑇>0 such that 𝑓^𝑙−𝜀≤𝑓(𝑡)≤𝑓^𝑢+𝜀,for𝑡≥𝑇. (2.2)

Definition 2.1. The system (1.2) is said to be strong persistence, if every solution 𝑥(𝑡) of system (1.2) satisfied 0<lim_𝑡→+∞inf𝑥(𝑡)≤lim_𝑡→+∞sup𝑥(𝑡)≤𝛿<+∞. (2.3)

Lemma 2.2. Both the positive and nonnegative cones of 𝑅² are invariant with respect to system (1.2).

It follows from Lemma 2.2 that any solution of system (1.2) with a nonnegative initial condition remains nonnegative.

Lemma 2.3 (see [10]). If 𝑎>0,𝑏>0, and 𝑥̇(𝑡)≥(≤)𝑥(𝑡)(𝑏−𝑎𝑥^𝛼(𝑡)), where 𝛼 is a positive constant, when 𝑡≥0 and 𝑥(0)>0, we have 𝑥(𝑡)≥(≤)𝑏𝑎^1/𝛼1+𝑏𝑥^−𝛼(0)𝑎−1𝑒^{−𝑏𝛼𝑡−1/𝛼}. (2.4)

In the following, we will be ready to state our result.

Theorem 2.4. Let 𝑃₁, 𝑃₂, 𝑃₃, and 𝑄₁ be defined by (2.7), (2.10), (2.13), and (2.16), respectively. Assume that conditions (H1) and (H2)𝛼^𝑢_𝑖𝛽^𝑢_𝑖>𝑑^𝑙_𝑖(𝑖=1,2),𝑟^𝑙𝑎^𝑙₁𝑎^𝑙₂>𝑎^𝑢₁𝑎^𝑙₂𝑃₂+𝑎^𝑙₁𝑎^𝑢₂𝑃₃, (H3)𝛼^𝑙₁𝛽^𝑙₁𝑄₁>𝑑^𝑢₁(𝑎^𝑢₁+𝑃₁),𝛼^𝑙₂𝛽^𝑙₂𝑄₁>𝑑^𝑢₂(𝑎^𝑢₂+𝑃₁)hold, then system (1.2) is uniformly strong persistence.

Proof. It follows from (2.2) that for any 𝜀>0, there exists 𝑇₁>0 such that for 𝑡≥𝑇₁, 𝑟^𝑙−𝜀≤𝑟(𝑡)≤𝑟^𝑢𝑟̃(𝑡)𝐾+𝜀,−𝜀<<𝜀,_𝑙−𝜀≤𝐾(𝑡)≤𝐾^𝑢𝐾(𝑡)𝑎+𝜀,−𝜀<<𝜀,_𝑙1−𝜀≤𝑎₁(𝑡)≤𝑎^𝑢₁+𝜀,−𝜀<𝑎₁(𝑡)𝑎<𝜀,_𝑙2−𝜀≤𝑎₂(𝑡)≤𝑎^𝑢₂+𝜀,−𝜀<𝑎₂(𝑡)𝛼<𝜀,_𝑙1−𝜀≤𝛼₁(𝑡)≤𝛼^𝑢₁+𝜀,−𝜀<𝛼₁(𝑡)𝛼<𝜀,_𝑙2−𝜀≤𝛼₂(𝑡)≤𝛼^𝑢₂+𝜀,−𝜀<𝛼₂(𝑡)<𝜀. (2.5) Substitute (2.5) into the first equation of system (1.2), then we have 𝑑𝑥𝑟(𝑡)+𝑟̃(𝑡)−𝑟(𝑡)+𝑟̃(𝑡)𝑑𝑡=𝑥𝐾(𝑡)+𝐾(𝑡)𝛼𝑥−¹(𝑡)+𝛼₁(𝑡)𝑦𝑎₁(𝑡)+𝑎₁(𝑡)−𝛼+𝑥²(𝑡)+𝛼₂(𝑡)𝑧𝑎₂(𝑡)+𝑎₂(𝑡)𝑟(𝑡)+𝑟̃(𝑡)−𝑟(𝑡)+𝑟̃(𝑡)+𝑥≤𝑥𝐾(𝑡)+𝐾(𝑡)𝑥≤𝑥(𝑡)𝑟^𝑢−𝑟+2𝜀^𝑙−2𝜀𝐾^𝑢𝑥(𝑡.+2𝜀) (2.6) By Lemma 2.3, we get lim_𝑡→+∞(𝑡)≤𝑟sup𝑥^𝑢𝐾^𝑢𝑟^𝑙∶=𝑃₁. (2.7) Then for any 𝜀>0, there exists 𝑇₂>𝑇₁>0 such that 𝑥(𝑡)≤𝑃₁+𝜀,𝑡≥𝑇₂. (2.8) Similarly, from (2.2) and the second equation of system (1.2), we obtain that for any 𝜀>0, there exists 𝑇₃>𝑇₂>0 such that 𝑦̇(𝑡)=𝑦−𝑑₁(𝑡)+𝑑₁(𝑡)+𝛽₁(𝑡)+𝛽₁(𝑡)𝛼₁(𝑡)+𝛼₁(𝑡)𝑥(𝑡−𝜏)𝑎₁(𝑡)+𝑎₁(𝑡)()−𝛾(𝑡)+𝛾(𝑡)𝑦(𝑡−𝑑+𝑥𝑡−𝜏≤𝑦)^𝑙₁−2𝜀−𝛾^𝑙−2𝜀𝑦(𝑡)+𝛽^𝑢₁+2𝜀𝛼^𝑢₁+2𝜀. (2.9) In view of Lemma 2.3, we derive lim_𝑡→+∞(𝑡)≤𝛼sup𝑦^𝑢1𝛽^𝑢₁−𝑑^𝑙₁𝛾^𝑙∶=𝑃₂. (2.10) Then for any 𝜀>0, there exists 𝑇₄>𝑇₃>0 such that 𝑦(𝑡)≤𝑃₂+𝜀,𝑡≥𝑇₄. (2.11) From (2.2) and the third equation of system (1.2), we obtain that for any 𝜀>0, there exists 𝑇₅>𝑇₄>0 such that 𝑧̇(𝑡)=𝑧−𝑑₂(𝑡)+𝑑₂(𝑡)+𝛽₂(𝑡)+𝛽₂(𝑡)𝛼₂(𝑡)+𝛼₂(𝑡)𝑥(𝑡−𝜏)𝑎₂(𝑡)+𝑎₂(𝑡)()−𝛿(𝑡)+𝛿(𝑡)𝑧(𝑡−𝑑+𝑥𝑡−𝜏≤𝑧)^𝑙₂−2𝜀−𝛿^𝑙−2𝜀𝑧(𝑡)+𝛽^𝑢₂+2𝜀𝛼^𝑢₂+2𝜀. (2.12) In view of Lemma 2.3, we derive lim_𝑡→+∞(𝑡)≤𝛼sup𝑧^𝑢2𝛽^𝑢₂−𝑑^𝑙₂𝛿^𝑙∶=𝑃₃. (2.13) Then for any 𝜀>0, there exists 𝑇₆>𝑇₅>0 such that 𝑧(𝑡)≤𝑃₃+𝜀,𝑡≥𝑇₆. (2.14) According (2.8), (2.11), (2.14) and the first equation of system (1.2), we obtain that for any 𝜀>0, there exists 𝑇₇>𝑇₆>0 such that 𝑑𝑥𝑟(𝑡)+𝑟̃(𝑡)−𝑟(𝑡)+𝑟̃(𝑡)𝑑𝑡=𝑥𝐾(𝑡)+𝐾(𝑡)𝛼𝑥−¹(𝑡)+𝛼₁(𝑡)𝑦𝑎₁(𝑡)+𝑎₁(𝑡)−𝛼+𝑥²(𝑡)+𝛼₂(𝑡)𝑧𝑎₂(𝑡)+𝑎₂(𝑡)(𝑡𝑟+𝑥≥𝑥)^𝑙−𝑟−2𝜀^𝑢+2𝜀𝐾^𝑙𝑥(𝑡)−𝛼−2𝜀^𝑢1+2𝜀𝑃₂+𝜀𝑎^𝑙₁−𝛼−2𝜀^𝑢2+2𝜀𝑃₃+𝜀𝑎^𝑙₂.−2𝜀 (2.15) Using Lemma 2.3 again, we have lim_𝑡→+∞(𝑡)≥𝐾inf𝑥^𝑙𝑟^𝑙𝑎^𝑙₁𝑎^𝑙₂−𝑎^𝑢₁𝑎^𝑙₂𝑃₂−𝑎^𝑙₁𝑎^𝑢₂𝑃₃𝑎^𝑙₁𝑎^𝑙₂𝑟^𝑢∶=𝑄₁. (2.16) Thus for any 𝜀>0, there exists 𝑇₈>𝑇₇>0 such that 𝑥(𝑡)≥𝑄₁−𝜀. (2.17) According (2.8), (2.11), (2.14) and the second equation of system (1.2), we obtain that for any 𝜀>0, there exists 𝑇₉>𝑇₈>0 such that 𝑦̇(𝑡)=𝑦−𝑑₁(𝑡)+𝑑₁(𝑡)+𝛽₁(𝑡)+𝛽₁(𝑡)𝛼₁(𝑡)+𝛼₁(𝑡)𝑥(𝑡−𝜏)𝑎₁(𝑡)+𝑎₁(𝑡)()−𝛾(𝑡)+𝛾(𝑡)𝑦(𝑡)−𝑑+𝑥𝑡−𝜏≥𝑦^𝑢₁+2𝜀−𝛾^𝑢𝑦(𝑡)+𝛽+2𝜀^𝑙1−2𝜀𝛼^𝑙₁−2𝜀𝑄₁−𝜀𝑎^𝑢₁+2𝜀+𝑃₁.+𝜀 (2.18) Using Lemma 2.3 again, we have lim_𝑡→+∞(𝑡)≥𝛽inf𝑦^𝑙1𝛼^𝑙₁𝑄₁−𝑑^𝑢₁𝑎^𝑢₁+𝑃₁𝛾^𝑢𝑎^𝑢₁+𝑃₁∶=𝑄₂. (2.19) Thus for any 𝜀>0, there exists 𝑇₁₀>𝑇₉>0 such that 𝑦(𝑡)≥𝑄₂−𝜀. (2.20) According (2.8), (2.11), (2.14) and the third equation of system (1.2), we obtain that for any 𝜀>0, there exists 𝑇₁₁>𝑇₁₀>0 such that 𝑧̇(𝑡)=𝑧−𝑑₂(𝑡)+𝑑₂(𝑡)+𝛽₂(𝑡)+𝛽₂(𝑡)𝛼₂(𝑡)+𝛼₂(𝑡)𝑥(𝑡−𝜏)𝑎₂(𝑡)+𝑎₂(𝑡)()−𝛿(𝑡)+𝛿(𝑡)𝑧(𝑡)−𝑑+𝑥𝑡−𝜏≥𝑦^𝑢₂+2𝜀−𝛿^𝑢𝑧(𝑡)+𝛽+2𝜀^𝑙2−2𝜀𝛼^𝑙₂−2𝜀𝑄₁−𝜀𝑎^𝑢₂+2𝜀+𝑃₁.+𝜀 (2.21) Using Lemma 2.3 again, we have lim_𝑡→+∞(𝑡)≥𝛽inf𝑧^𝑙2𝛼^𝑙₂𝑄₁−𝑑^𝑢₂𝑎^𝑢₂+𝑃₁𝛿^𝑢𝑎^𝑢₂+𝑃₁∶=𝑄₃. (2.22) Thus the proof of Theorem 2.4 is complete.

Acknowledgments

This work is supported by National Natural Science Foundation of China (no. 11261010 and no. 11161015), Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), Natural Science and Technology Foundation of Guizhou Province (J[2012]2100), Governor Foundation of Guizhou Province ([2012]53), and Doctoral Foundation of Guizhou University of Finance and Economics (2010).