Abstract
In this paper, a single-species population model with distributed delay and Michaelis-Menten type harvesting is established. Through an appropriate transformation, the mathematical model is converted into a two-dimensional system. Applying qualitative theory of ordinary differential equations, we obtain sufficient conditions for the stability of the equilibria of this system under three cases. The equilibrium A1 of system is globally asymptotically stable when br−c>0 and η<0. Using Poincare-Bendixson theorem, we determine the existence and stability of limit cycle when br−c>0 and η>0. By computing Lyapunov number, we obtain that a supercritical Hopf bifurcation occurs when η passes through 0. High order singularity of the system, such as saddle node, degenerate critical point, unstable node, saddle point, etc, is studied by the theory of ordinary differential equations. Numerical simulations are provided to verify our main results in this paper.
Highlights
Most biological populations are multi-species, there is no single population in the strict sense, but because of artificial breeding, there are a lot of single population resources in many human-created environments, which provide indispensable roles for human production and life
This paper is mainly aimed at qualitative analysis of a single-species population model with distributed delay and nonlinear harvest
If O(0, 0) is locally stable, and A2 is always a saddle point, and A3 is unstable, there is a stable limit cycle generated by Hopf bifurcation in the neighborhood of A3 when 0 < β 1
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Lian et al [14] considered a predator-prey system with Holling type IV functional response and time delay They studied the effects of time delay on this system and obtained the conditions of local asymptotic stability of the positive equilibrium and the existence of local Hopf bifurcations by applying the delay as a bifurcation parameter. Many single-species population models with time delay were studied and rich results were obtained (see [16,17,18,19,20,21,22]). This paper is mainly aimed at qualitative analysis of a single-species population model with distributed delay and nonlinear harvest.
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