Abstract
This paper is concerned with a predator–prey model with hyperbolic mortality and prey harvesting. The parameter regions for the stability and instability of the unique positive constant solution of ODE and PDE are derived, respectively. Especially, the global asymptotical stability of positive constant equilibrium of the diffusive model is obtained by iterative technique. The stability and direction of periodic solutions of ODE and PDE are investigated by center manifold theorem and normal form theory, respectively. Numerical simulations are carried out to depict our theoretical analysis.
Highlights
In the study of ordinary differential equations, the analysis of periodic solutions is an important goal
Various techniques have been proposed to construct Dulac functions, which range from algebraic methods for special systems, methods for the construction of Lyapunov functions to techniques involving the solutions of certain partial differential equations
We extended the techniques for the construction of Dulac functions
Summary
In the study of ordinary differential equations, the analysis of periodic solutions is an important goal. Deciding whether an arbitrary differential equation has periodic orbits or not is a difficult question that remains open. For the two-dimensional case, the Bendixson–Dulac criterion gives a sufficient condition for the non-existence of periodic orbits. Various techniques have been proposed to construct Dulac functions, which range from algebraic methods for special systems, methods for the construction of Lyapunov functions to techniques involving the solutions of certain partial differential equations (see [3, 4, 6, 10, 13]). The Bendixson–Dulac criterion discards existence of polycycles making it useful in establishing global stability for certain systems. Even though Dulac functions are an important tool in many issues of differential equations, their determination is a difficult task. We give some consequences and examples to illustrate applications of these results
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