Permanence, stability, and coexistence of a diffusive predator–prey model with modified Leslie–Gower and B–D functional response

Abstract

This paper investigates a diffusive predator-prey system with modified Leslie-Gower and B-D (Beddington-DeAngelis) schemes. Firstly, we discuss stability analysis of the equilibrium for a corresponding ODE system. Secondly, we prove that the system is permanent by the comparison argument of parabolic equations. Thirdly, sufficient conditions for the global asymptotic stability of the unique positive equilibrium of the system are proved by using the method of Lyapunov function. Finally, by using the maximum principle, Poincare inequality, and Leray-Schauder degree theory, we establish the existence and nonexistence of nonconstant positive steady states of this reaction-diffusion system, which indicates the effect of large diffusivity.

A Criterion for the Existence of the Unique Periodic Solution of One-Dimensional Periodic Differential Equation

“…is an ω-periodic continuous function. It follows from (14) and Lemma 1 that equation ( 11) has a unique ω-periodic continuous solution as follows:…”

“…As we all know, the existence, uniqueness, and stability of periodic solutions of differential equations have always been an important research hotspot in the field of differential equations (see [10][11][12][13][14][15][16][17][18][19][20]). However, the above literatures are basically about the study of periodic solutions of specific equations, rather than the study of general differential equations.…”

Stability and Coexistence of a Diffusive Predator-Prey System with Nonmonotonic Functional Response and Fear Effect

“…However, the results of eorem 10 show that two species can coexist in a biological system if their diffusivity satisfies certain conditions at the same time. In fact, we have used different methods to study the similar dynamic behavior of the solution on another predator-prey model in reference [26], and one can refer to it for more detailed results.…”

Permanence and uniform asymptotical stability of a ratio-dependent Leslie system with feedback controls on time scales

“…where x 1 and x 2 stand for the population (the density) of the preys and of the predators, respectively, p is the so-called predator functional response to predator and prey. In the last decades, the dynamical behaviors for the continuous-time Leslie predator-prey systems such as Hopf bifurcation [2,3], permanence [4], periodic solution [5,6], almost periodic solution [7,8], and stability [4], etc., have been widely investigated.…”