Stability and Hopf bifurcation for a stage-structured predator–prey model incorporating refuge for prey and additional food for predator

Abstract

In this paper, we study a stage-structured predator-prey model incorporating refuge for prey and additional food for predator. By analyzing the corresponding characteristic equations, we investigate the local stability of equilibria and the existence of Hopf bifurcation at the positive equilibrium taking the time delay as a bifurcation parameter. Furthermore, we obtain the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions applying the center manifold theorem and normal form theory. Numerical simulations are illustrated to verify our main results.

Hopf Bifurcation Analysis of a Synthetic Drug Transmission Model with Time Delays

“…In conclusion, we can obtain the following results based on the fundamental results about Hopf bifurcation in the literature [32]. Complexity T 2 > 0 (T 2 < 0), then the period of the bifurcating periodic solutions increases (decreases).…”

“…is assumption seems not to be realistic since the synthetic drugs are addictive easily and it usually needs a period to give up drugs. Time delays have been incorporated into dynamical models about some other fields by many scholars [27][28][29][30][31][32][33]. Generally speaking, delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay could cause the equilibrium of a dynamical model to lose its stability.…”

Asymptotic Analysis of Impulsive Dispersal Predator-Prey Systems with Markov Switching on Finite-State Space

“…Kuang et al investigated the relationship between the stability of the population and the dispersal rate of the prey species in [3]. When two species interact, one as a predator and the other as prey, then the Lotka-Volterra predator-prey model is frequently used to describe the dynamics of the biological system [8][9][10][11][12]. Cui [4] explored a nonautonomous dispersal predator-prey system and the sufficient conditions for persistence were obtained.…”

A non-autonomous Leslie–Gower model with Holling type IV functional response and harvesting complexity

“…In spite of a lot of works focused on the global dynamics and bifurcation analysis of the ecological systems (e.g., ), in realistic environment, ecological systems are usually affected by the seasonable perturbations or other unpredictable disturbances (e.g., see [24][25][26][27][28][29][30][31][32][33][34][35][36]). Thus the time-varying parameters are more reasonable when we try to consider the periodic environment.…”