Abstract
In this paper, we investigate the stability of equilibrium in the stage-structured and density-dependent predator–prey system with Beddington–DeAngelis functional response. First, by checking the sign of the real part for eigenvalue, local stability of origin equilibrium and boundary equilibrium are studied. Second, we explore the local stability of the positive equilibrium for τ=0 and τ≠0 (time delay τ is the time taken from immaturity to maturity predator), which shows that local stability of the positive equilibrium is dependent on parameter τ. Third, we qualitatively analyze global asymptotical stability of the positive equilibrium. Based on stability theory of periodic solutions, global asymptotical stability of the positive equilibrium is obtained when τ=0; by constructing Lyapunov functions, we conclude that the positive equilibrium is also globally asymptotically stable when τ≠0. Finally, examples with numerical simulations are given to illustrate the obtained results.
Highlights
Functional Response and Harvesting.The dynamical behavior of the predator–prey system is one of the main research topics in mathematical ecology and theoretical biology [1,2,3,4,5,6,7,8,9]
This paper mainly investigates the local and global stability of positive equilibrium in the system (7) on parameter τ, which is organized as follows
Min{τ ∗, τ ∗∗ }, Theorem 6 holds. This implies that the value of time delay determines the global asymptotic stability of the positive equilibrium
Summary
Where p stands for predator density-dependent mortality rate, the predator consumes cx (t)y(t) prey with functional response of Beddington-DeAngelis type m +m x(t)+m y(t) and con f x (t)y(t). Note that compared with the system (1), only bx (t) (which stands for intraspecific competition of prey tributes to its growth with rate the system (4) contains not species), and py (t) (which stands for intraspecific competition of predator species) That is, they consider both the prey density dependence and the predator density dependence in the predator–prey model (4). They consider both the prey density dependence and the predator density dependence in the predator–prey model (4) She and Li [29] studied the dynamics of the system (4) and pointed out the impact of the predator density-dependent mortality rate p on the global attraction and permanence of the system (4).
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