Abstract
In this paper, a nonautonomous reaction-diffusion predator-prey model with modified Leslie–Gower Holling-II schemes and a prey refuge is proposed. Applying the comparison theory of differential equation, sufficient average criteria on the permanence of solutions and the existence of the positive periodic solutions are established. Moreover, the existence region of the positive periodic solutions is an invariant region dependent on t. Then, constructing a suitable Lyapunov function, we obtain sufficient conditions to guarantee the global asymptotic stability of the positive periodic solutions. Finally, we do some numerical simulations to verify our main results and investigate the effect of prey refuge on the dynamics of the system.
Highlights
In the past two decades, population biology has been greatly developed [3, 6, 10, 11, 20, 26]
Based on the above consideration, many nonautonomous predator-prey models with modified Leslie–Gower Holling-II schemes have been proposed to investigate the effect of the periodic environment on the dynamics of population [13, 22, 28]
The existence region of the positive periodic solutions is an invariant region dependent on t, which is different from the previous results in Xie et al [22]
Summary
In the past two decades, population biology has been greatly developed [3, 6, 10, 11, 20, 26]. Based on the above consideration, many nonautonomous predator-prey models with modified Leslie–Gower Holling-II schemes have been proposed to investigate the effect of the periodic environment on the dynamics of population [13, 22, 28]. [28] proposed the periodic predator-prey model with modified Leslie–Gower Holling-II schemes and obtained the existence and global attractivity of the positive periodic solutions.
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