Persistence in Non-Autonomous Lotka-Volterra System with Predator-Prey Ratio-Dependence and Density Dependence

The main purpose of this article is considering the persistence non-autonomous Lotka-Volterra system with predator-prey ratio-dependence and density dependence. We get the sufficient conditions of persistence of system, further have the necessary conditions, also the uniform persistence condition, which can be easily checked for the model is obtained.


Introduction
Predator-prey behavior is a form of very common biological interaction in nature.There are many mathematical models to model predator-prey behavior such as Lotka-Volterra system [1][2][3][4][5][6], Rosenzweig-MacArthur system, Kolmogorov system, etc.Recently, models with such a prey-dependent-only response function have been facing challenges from the biology and physiology communities.Some biologists [7][8][9] have argued that in many situations, especially when predators have to search for food (and therefore, have to share or compete for food), the functional response in a predator-prey model should be predator-dependent.
The certain environment confines for the predator to be density dependent.The theories on the model of the predator-prey in which the predator has density dependence are not perfect [10][11][12].Kartina [13] shows that predator dependence is important at not only very high predator densities on per captia predation rate but also at low predator densities.In ecology, we should consider both prey and predator density dependence, and need to take into account realistic levels of predator dependence.The qualitative analysis for the model will be difficult compared to the model with only density dependent prey [10][11][12].
In this paper, we will consider the permanence of nonautonomous density dependent and ratio-dependent predator-prey system ( ) ( ) ( ) , ( )

c t y x x a t b t x m t y x f t x y y d t e t y m t y x
where

Preparation
In this paper, we will always assume that the parameters in system (1) are periodic continuous on R and with common period 0 , where ( ) f t is a continuous and periodic function with period  .
Motivated by the biological background of system (1), this paper only considers positive solutions of system (1).We can directly integrate the two equations of system (1) to obtain 0 Hence, it is obvious that the solutions ( ), ( ) x t y t is positive if and only if the initial value (0) 0, (0) 0 x y   .
In order to describe in the following results, we need first to discuss system (1) in the absence of the predator, namely, the Riccatti equation with initial value , the solution is given by Clearly, the null solution exists in equation (2).By the uniqueness of solutions, we can see that solutions with positive initial values remain positive.For all solution  ( ), ( ) x t y t of (1) with positive initial values.

Proof. If is solution of the following equation
and assume (1), we can obtain that by comparison theorem and Lemma 2.1, we have there is a constant such that And if is solution of the following equation similarly, we can obtain that there is a constant such that We complete the proof of Theorem for all solution   ( ), ( ) x t y t of (1) with positive initial values.
Proof.If the conclusion ( 6) is not true, then there is a sequence , , By Theorem 2.1, for a given positive integer , there is a such that n ( ) Thus, for any , ( ) For , there is a positive integer . By integrating (7) from for any , we obtain There are constants , such that for , 0 0, Thus, we can get as and choosing sufficiently small positive numbers 1 x and by the following Equation ( 8), we can obtain which is a contradiction.This completes the proof of Theorem 2.2.By Lemma 2.1 and 0 ( ) H , the following equation, ( ) ( ) ( ) ( ) has a unique positive where ** ( ) x t is the unique periodic solution of equation ( 9), then exist positive constants  and lim sup ( ) , for all solution   ( ), ( ) x t y t of ( 1 q By Theorem 2.1, for a given integer , there is a , such that Because of q , there is a positive integer . Integrating the above inequality from ( ) , .
By (12), we obtain that there are constants , such that for , From ** ( ) x t of ( Integrating the above inequality from ( ) 12).This is a contradiction.We complete proof of Theorem 2.3.

The Main Conclusions
From Definition 2.1 and Theorems From (1), easily get Together with (15), we have .