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

and Applied Analysis 3 Proposition 2.2. Let x1 t , x2 t be any positive solution of system 1.1 , then lim n→ ∞ supxi n ≤ Mi, i 1, 2, 2.3


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 ẋ1 t with initial conditions where x 1 t and x 2 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.r 1 t , r 2 t stand for the intrinsic growth rates of the prey and the predator, respectively.a 11 t is the intraspecific competition rate of the prey.a 12 t is the capturing rate of the predator.The predator grows with the carrying capacity x t /a 21 t proportional to the population size of the prey or prey abundance.a 21 t is a measure of the food quality that the prey provided for conversion into predator birth.Assumed that r i t , a ij 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-ratiodependent 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 r i t , a ij 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 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 .

Permanence
In this section, we establish a permanence result for system 1.1 .Lemma 2.1 see 9 .If a > 0, b > 0 and ẋ ≥ x b − ax , when t ≥ 0 and x 0 > 0, one has: If a > 0, b > 0 and ẋ ≤ x b − ax , when t ≥ 0 and x 0 > 0, one has: Proposition 2.2.Let x 1 t , x 2 t be any positive solution of system 1.1 , then where

2.4
Proof.Let x 1 t , x 2 t be any positive solution of system 1.1 , from the first equation of system 1.1 one has ẋ1 t ≤ r 1 t x 1 t .

2.5
By integrating both sides of the above inequality from t − τ t to t with respect to t, we obtain By substituting 2.6 into the first equation of system 1.1 , one has By Lemma 2.1, according to 2.7 , it immediately follows that lim

2.8
It follows that for any small positive constant ε > 0, there exists a T 1 > 0 such that 2.9 By substituting 2.9 into the second equation of system 1.