Dynamic behaviors of a nonautonomous predator–prey system with Holling type II schemes and a prey refuge

In this paper, we consider a nonautonomous predator–prey model with Holling type II schemes and a prey refuge. By applying the comparison theorem of differential equations and constructing a suitable Lyapunov function, sufficient conditions that guarantee the permanence and global stability of the system are obtained. By applying the oscillation theory and the comparison theorem of differential equations, a set of sufficient conditions that guarantee the extinction of the predator of the system is obtained.


Introduction
The dynamic relationship between predator and prey 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]. Furthermore, the study of dynamic behaviors of predatorprey system incorporating a prey refuge become one of the most important research topic; see . In [2], Amant proposed a Lotka-Volterra predator-prey model with a constant prey refuge: x(t) = rxβ(xm)y, y(t) = e(xm)ydy, (1.1) incorporating Holling type II and a constant prey refuge: where K is the prey environmental carrying capacity, a is the amount of prey needed to achieve one-half of β. They show that the effect of prey refuges would have a stabilizing influence on the dynamical consequences of system (1.2). Some scholars argued that the nonautonomous case is more realistic, because many biological or environmental parameters do subject to fluctuate with time, thus more complex equations should be introduced. Many scholars studied the dynamic behaviors of nonautonomous predator-prey system incorporating prey refuge. In [4], Zhu proposed and studied the nonautonomous predator-prey system incorporating prey refuge:

t) a(t)b(t)x(t)c(t) x(t)m(t) y(t), y(t) = y(t) -r 1 (t)k(t)y(t) + d(t) x(t)m(t)
, (1.3) where x(t) and y(t) denote the density of prey and predator populations at time t, respectively. a(t), b(t), c(t), m(t), r 1 (t), k(t), d(t) are nonnegative continuous function that have the upper and lower bounds. a(t) is the intrinsic per capita growth rate of prey, b(t) is the per capita death rate of prey, c(t) is the maximal per capita consumption rate of predators, m(t) is the maximum capacity of refuge, r 1 (t) is the per capita death rate of predator, k(t) is the density constraints on predator populations, and d(t)/c(t) is the conversion coefficient. In [4], the sufficient conditions to guarantee the global asymptotic stability of the system (1.3) are obtained. On the basis of system (1.3), Wu [5] further studied the extinction of predator populations. In this paper, we study the nonautonomous predator-prey system incorporating prey refuge and Holling type II schemes: where a 1 (t) is the amount of prey needed to achieve one-half of c(t).
Based on the biological significance of systems, we consider system (1.4) together with the following initial conditions: Furthermore, for a bounded continuous function g(t) defined on R, The following work is organized as follows. Sufficient conditions which guarantee the positive and permanence of system (1.4) are given in Sect. 2. In Sect. 3, we obtained a set of sufficient conditions for global stability of the system (1.4). In Sect. 4, the extinction of predator are studied and a set of sufficient conditions that guarantee the extinction of predator are obtained. In Sect. 5, three examples together with their numerical simulations show the feasibility of the main results. This paper ends by a brief conclusion.

Lemma 2.2
For every positive solution (x(t), y(t)) T that satisfies the initial condition (1.5), if the system satisfies (2.1) and then, for every positive solution (x(t), y(t)) T that satisfies the initial condition (1.5), system Here, .
Proof For every positive solution (x(t), y(t)) T that satisfies the initial condition (1.5), from the first equation of system (1.4), condition (2.1) and the third condition of (2.2), it follows thatẋ Hence, according to Lemma 2.3 in [32], it is directly found that For any small positive constant ε > 0, there exists T 1 > 0 such that, for all t ≥ T 1 , It follows from (2.5) and the second equation of the system (1.4) thaṫ (2.6) According to Lemma 2.3 in [32], it follows that Because of the arbitrariness of ε, let ε → 0, it follows from (2.7) that Conditions (2.8) implies that, for any small ε > 0, there exists a T 2 > T 1 , such that, for all t > T 2 , Then, for t > T 2 , from the first equation of system (1.4), it follows thaṫ (2.10) According to Lemma 2.3 in [32], it follows that :=m 1 . (2.12) Let 0 < ε < 1 2 m 1 be any positive constant small enough. Then it follows from (2.12) that there exists a T 3 > T 2 , such that, for all t > T 3 , From the second equation of the system (1.4) together with (2.5) and (2.13), it followṡ (2.14) According to Lemma 2.3 in [32], it follows that This completes the proof of Lemma 2.2.

Global stability
We introduce some notations before we state the main result of this section. Set That is, the system (1.4) shows global stability.
Proof From (3.1), it follows that (3.5) Then, for t > T, we have and Set (3.7) Let ε → 0, then (3.8) Here A and B are defined in Eq. (3.1). Setting α = min{A, B} > 0. Then (x(t), y(t)) is stable under the meaning of Lyapunov. Integrating Eq. (3.8) from T to t, then, for t > T, we get hence,

The extinction of predator
Consider the following equation: Then, for any arbitrary ε > 0, there exists a T 1 (> T), such that From the first equation of system (1.4), it follows thaṫ By the applying comparison theorem of differential equations, we get Thus, combined with (4.5), we have Therefore, from the second equation of system (1.4), it follows that, for t ≥ T 1 , Let ε → 0, we have Therefore, from the condition (4.3), it follows that lim t→∞ y(t) = 0.

Numeric simulations
Now let us consider the following three examples.
Example 5.1 (5.1) In system (5.1), corresponding to system (1.4), we assume that a(t) = 11 + cos t, b(t) = 3, c(t) = 5, m(t) = 0.5, a 1 (t) = 2. It follows that, when the maximum capacity of the refuge is larger than the maximum of the prey population, the predator population will go extinct. Numerical simulation (see Fig. 3) also supports this conclusion.
y(t) = y(t) -1.7 -4y(t) + (4 + 0.5 sin t)(x(t) -1.5) x(t) + 1 . It follows that, when the maximum capacity of refuge is lower than the maximum of the prey population, the predator population will go extinct if the per capita death rate of predator is high enough. Numerical simulation (see Fig. 4) also supports this conclusion.

Conclusion
In this paper, we consider a nonautonomous predator-prey model with Holling type II schemes and a prey refuge. Firstly, by applying the comparison theorem of differential equations, a set of conditions that ensure the permanence of the system is obtained. Secondly, by constructing a suitable Lyapunov function, sufficient conditions that guarantee the permanence and global stability of the system are investigated. Lastly, by applying the oscillation theory and the comparison theorem of differential equations, a set of sufficient conditions that guarantee the extinction of the predator of the system is obtained. Condition (4.3) implies two situations: (1) When the maximum capacity of refuge is larger than the maximum of the prey population, the predator population will go extinct.
(2) When the maximum capacity of refuge is lower than the maximum of the prey population, as long as per capita death rate of predator is large enough, then the predator population will go extinct.