Dynamic Behavior of Positive Solutions for a Leslie Predator-Prey System with Mutual Interference and Feedback Controls

We consider a Leslie predator-prey system with mutual interference and feedback controls. For general nonautonomous case, by using differential inequality theory and constructing a suitable Lyapunov functional, we obtain some sufficient conditions which guarantee the permanence and the global attractivity of the system. For the periodic case, we obtain some sufficient conditions which guarantee the existence, uniqueness, and stability of a positive periodic solution.


Introduction
Leslie [1] introduced the famous Leslie predator-prey systeṁ where ( ) and ( ) stand for the population (the density) of the prey and the predator at time , respectively, and ( ) is the so-called predator functional response to prey.
In system (1), it has been assumed that the prey grows logistically with growth rate and carries capacity / in the absence of predation. The predator consumes the prey according to the functional response ( ) and grows logistically with growth rate and carrying capacity / proportional to the population size of the prey (or prey abundance). The parameter is a measure of the food quality that the prey provides and converted to predator birth. Leslie introduced a predator-prey model, where the carrying capacity of the predator's environment is proportional to the number of prey, and still stressed the fact that there are upper limits to the rates of increasing of both prey and predator , which are not recognized in the Lotka-Volterra model. These upper limits can be approached under favorable conditions: for the predators, when the number of prey per predator is large; for the prey, when the number of predators (and perhaps the number of prey also) is small [2].
In population dynamics, the functional response refers to the numbers eaten per predator per unit time as a function of prey density. Kooij and Zegeling [3] and Sugie et al. [4] considered the following functional response: ( ) = + ( > 0, , are positive constants) . ( In fact, the functional response with ≥ 1 has been suggested in [4]. It is easy to see that system (1) can be written as the following system: When = 1 or = 2 in system (3), the Leslie system (3) can be written as the following system: The Scientific World Journal oṙ( which were considered in [5]. The mutual interference between predators and prey was introduced by Hassell in 1971 [6]. During his research of the capturing behavior between hosts and parasites, he found that the hosts or parasites had the tendency to leave from each other when they met, which interfered with the hosts capturing effects. It is obvious that the mutual interference will be stronger while the size of the parasite becomes larger. Thus, Pan [7] studied a Leslie predator-prey system with mutual interference constant : where ( ), ( ), ( ), ( ) ∈ ( , +) are bounded and periodic functions and ∈ (0, 1] is a constant. Some sufficient conditions are obtained for the permanence, attractivity, and existence of the positive periodic solution of the system (6).
On the other hand, ecosystems in the real world are continuously distributed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Of practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time.
By using a comparison theorem and constructing a suitable Lyapunov function, they obtained some sufficient conditions for the existence of a unique almost periodic solution and the global attractivity of the solutions of the system (7). By using the comparison and continuation theorems, and based on the coincidence degree and by constructing a suitable Lyapunov function, Wang et al. [30] further obtained some sufficient and necessary conditions for the existence and global attractivity of periodic solutions of the system (7) with the periodic coefficients.
However, to the best of our knowledge, up to now, still no scholar has considered the general nonautonomous Leslie predator-prey system with mutual interference and feedback controls. In this paper, we will consider following the Leslie predator-prey system: For the general nonautonomous case, by using differential inequality theory and constructing a suitable Lyapunov functional, we obtain some sufficient conditions which guarantee the permanence and the global attractivity of the system (8). For the periodic case, we obtain some sufficient conditions which guarantee the existence, uniqueness, and stability of a positive periodic solution of the system (8). We would like to mention that the conditions are related to the interference constant .

Preliminaries
In this section, we state several definitions and lemmas which will be useful in proving the main results of this paper.
Let and , respectively, denote the set of all real numbers and the -dimensional real Euclidean space, and + denote the nonnegative cone of . Let be a continuous bounded function on and we set = sup ∈ ( ) and = inf ∈ ( ).
The Scientific World Journal 3 Throughout this paper, we assume that the coefficients of the system (8) then we say that ( ( ), ( ), ( ), V( )) is globally attractive.
For any given the initial conditions of the system (8)
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Periodic Case
To this end, in order to get the existence and uniqueness of a positive periodic solution of the system (8), we further assume that system (8) satisfies the following condition.
It is easy to see that the existence of periodic solution in (8) with assumptions ( 8 ) is equal to prove that the mapping has at least one fixed point.
Proof. According to discussion above, we obtain ( * ) ⊂ * . And we can get that the mapping is continuous by the theory of ordinary differential equation. It is obvious that * is a closed and convex set. Therefore, has at least one fixed point by Brouwer fixed point theorem. That is to say, if system (8) satisfies ( 1 ), ( 2 ), and ( 8 ), system (8) has at least one positive -periodic solution. This completes the proof.
Finally, some sufficient conditions are obtained for the uniqueness of the positive -periodic solution for the system (8).