Permanence of a Semi-Ratio-Dependent Predator-Prey System with Nonmonotonic Functional Response and Time Delay

Abstract

Sufficient conditions for permanence of a semi-ratio-dependent predator-prey system with nonmonotonic functional response and time delay _{x 1} (t)=_x1 (t)[_r1 (t)-_a11 (t)_x1 (t-τ(t))-_a12 (t)_x2 (t)/(^m2 +^x12 (t))], _{x 2} (t)=_x2 (t)[_r2 (t)-_a21 (t)_x2 (t)/_x1 (t)], are obtained, where _x1 (t) and _x2 (t) stand for the density of the prey and the predator, respectively, and m≠0 is a constant. τ(t)≥0 stands for the time delays due to negative feedback of the prey population.

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Recommended by Paul Eloe

School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, Fujian 350007, China

Received 14 May 2009; Accepted 28 July 2009

1. Introduction

The dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance [1]. These problems may appear to be simple mathematically at first sight, but they are, in fact, often very challenging and complicated [2, 3].

Recently, Ding et al. [4] studied dynamics of a semi-ratio-dependent predator-prey system with the nonmonotonic functional response and delay [figure omitted; refer to PDF] with initial conditions [figure omitted; refer to PDF] where _x1 (t) and _x2 (t) stand for the density of the prey and the predator, respectively, and m≠0 is a constant. τ(t)≥0 stands for the time delays due to negative feedback of the prey population. _r1 (t), _r2 (t) stand for the intrinsic growth rates of the prey and the predator, respectively. _a11 (t) is the intraspecific competition rate of the prey. _a12 (t) is the capturing rate of the predator. The predator grows with the carrying capacity x(t)/_a21 (t) proportional to the population size of the prey or prey abundance. _a21 (t) is a measure of the food quality that the prey provided for conversion into predator birth. Assumed that _ri (t), _aij (t), i,j=1,2 , are continuously positive periodic functions with period ω , by using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for the semi-ratio-dependent predator-prey system with nonmonotonic functional responses and time delay is established. For the ecological sense of the system (1.1) we refer to [5-8] and the references cited therein.

As we know, permanence is one of the most important topics on the study of population dynamics. One of the most interesting questions in mathematical biology concerns the survival of species in ecological models. Biologically, when a system of interacting species is persistent in a suitable sense, it means that all the species survive in the long term. It is reasonable to ask for conditions under which the system is permanent. However, Ding et al. [4] did not investigate this property of the system (1.1).

Motivated by the above question, we will consider the permanence of the system (1.1). Unlike the assumptions of Ding et al. [4], we argue that a general nonautonomous nonperiodic system is more appropriate, and thus, we assume that the coefficients of system (1.1) satisfy: (A) _ri (t), _aij (t), τ(t), i,j=1,2, are nonnegative functions bounded above and below by positive constants.

Throughout this paper, for a continuous function g(t) , we set [figure omitted; refer to PDF]

It is easy to verify that solutions of system (1.1) corresponding to initial conditions (1.2) are defined on [0,+∞) and remain positive for all t≥0 . In this paper, the solution of system (1.1) satisfying initial conditions (1.2) is said to be positive.

The aim of this paper is, by using the differential inequality theory, to obtain a set of sufficient conditions to ensure the permanence of the system (1.1).

2. Permanence

In this section, we establish a permanence result for system (1.1).

Lemma 2.1 (see [9]).

If a>0, b>0 and x...≥x(b-ax) , when t≥0 and x(0)>0 , one has: [figure omitted; refer to PDF] If a>0, b>0 and x...≤x(b-ax) , when t≥0 and x(0)>0 , one has: [figure omitted; refer to PDF]

Proposition 2.2.

Let (_x1 (t),_x2 (t)) be any positive solution of system (1.1), then [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

Let (_x1 (t),_x2 (t)) be any positive solution of system (1.1), from the first equation of system (1.1) one has [figure omitted; refer to PDF] By integrating both sides of the above inequality from t-τ(t) to t with respect to t , we obtain [figure omitted; refer to PDF] By substituting (2.6) into the first equation of system (1.1), one has [figure omitted; refer to PDF] By Lemma 2.1, according to (2.7), it immediately follows that [figure omitted; refer to PDF] It follows that for any small positive constant [straight epsilon]>0 , there exists a _T1 >0 such that [figure omitted; refer to PDF]

By substituting (2.9) into the second equation of system (1.1), one has

[figure omitted; refer to PDF] By Lemma 2.1, according to (2.10), we get [figure omitted; refer to PDF] Setting [straight epsilon][arrow right]0 yields that [figure omitted; refer to PDF]

This completes the proof of Proposition 2.2.

Now we are in the position of stating the permanence of the system (1.1).

Theorem 2.3.

Assume that ^r1l -^a12u_M2 /^m2 >0 hold, then system (1.1) is permanent, that is, there exist positive constants _mi , _Mi , i=1,2, which are independent of the solutions of system (1.1), such that for any positive solution (_x1 (t),_x2 (t)) of system (1.1) with initial condition (1.2), one has [figure omitted; refer to PDF]

Proof.

By applying Proposition 2.2, we see that to end the proof of Theorem 2.3, it is enough to show that under the conditions of Theorem 2.3, [figure omitted; refer to PDF] From Proposition 2.2, for all [straight epsilon]>0 , there exists a _T2 >0, for all t>_T2 , [figure omitted; refer to PDF] By substituting (2.15) into the first equation of system (1.1), it follows that [figure omitted; refer to PDF] By integrating both sides of the above inequality from t-τ(t) to t with respect to t , we obtain [figure omitted; refer to PDF] By substituting the above inequality into the first equation of system (1.1), one has [figure omitted; refer to PDF] By Lemma 2.1, under the conditions of Theorem 2.3, it immediately follows that [figure omitted; refer to PDF] Setting [straight epsilon][arrow right]0 , yields that [figure omitted; refer to PDF] It follows that for the above positive constant [straight epsilon]>0 , there exists a _T2 >0 such that [figure omitted; refer to PDF] By substituting (2.21) into the second equation of system (1.1), one has [figure omitted; refer to PDF] By Lemma 2.1, according to (2.22), it immediately follows that [figure omitted; refer to PDF] Setting [straight epsilon][arrow right]0 yields that [figure omitted; refer to PDF] This completes the proof of Theorem 2.3.

References

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[4] X. Ding, C. Lu, M. Liu, "Periodic solutions for a semi-ratio-dependent predator-prey system with nonmonotonic functional response and time delay," Nonlinear Analysis: Real World Applications , vol. 9, no. 3, pp. 762-775, 2008.

[5] Y.-H. Fan, W.-T. Li, L.-L. Wang, "Periodic solutions of delayed ratio-dependent predator-prey models with monotonic or nonmonotonic functional responses," Nonlinear Analysis: Real World Applications , vol. 5, no. 2, pp. 247-263, 2004.

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[7] Q. Wang, M. Fan, K. Wang, "Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey systems with functional responses," Journal of Mathematical Analysis and Applications , vol. 278, no. 2, pp. 443-471, 2003.

[8] D. Xiao, S. Ruan, "Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response," Journal of Differential Equations , vol. 176, no. 2, pp. 494-510, 2001.

[9] F. Chen, M. You, "Permanence for an integrodifferential model of mutualism," Applied Mathematics and Computation , vol. 186, no. 1, pp. 30-34, 2007.