Abstract
This paper is devoted to investigating the dynamics of a diffusive Leslie–Gower predator–prey system with ratio-dependent Holling III functional response. We first establish the stability of positive constant equilibrium, and show the condition under which system undergoes a Hopf bifurcation with the explicit computational formulas for determining the bifurcating properties. Especially, when the positive constant equilibrium loses its stability, a supercritical Hopf bifurcation with spatial homogeneous and stable bifurcating periodic solution occurs. Finally, we discuss the existence and nonexistence of nonconstant positive solutions with the help of Leray–Schauder degree theory and the implicit function theorem.
Highlights
1 Introduction As one of the most common mutual relationships between two populations in nature, predator–prey relationship plays a significant role in ecology and mathematical biology
One way to model this relationship is by using differential equations with various types of predator’s functional responses, which is a key component of a predator– prey relationship
The purpose of the present paper is to investigate dynamics of system (1.2), including the stability of the unique positive equilibrium, the existence and the bifurcating properties of Hopf bifurcation, the existence and nonexistence of nonconstant positive solutions
Summary
As one of the most common mutual relationships between two populations in nature, predator–prey relationship plays a significant role in ecology and mathematical biology. The purpose of the present paper is to investigate dynamics of system (1.2), including the stability of the unique positive equilibrium, the existence and the bifurcating properties of Hopf bifurcation, the existence and nonexistence of nonconstant positive solutions. A larger parameter range of local stability of the unique positive constant equilibrium is established; 2. We investigate the existence and nonexistence of nonconstant positive solutions by using a priori estimates, Leray–Schauder degree theory, and the implicit function theorem. The stability of the equilibria can be obtained by analyzing the distribution of the eigenvalues of the characteristic equation corresponding to the linearized system of system (1.2). Proof We prove the local asymptotic stability of E∗ by analyzing the distribution of eigenvalues at E∗. We give a new local stability region of E∗ (see the blue region) corresponding to case (C) or case (D) of Theorem 2.2
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
