Complex dynamics of a discrete predator–prey system with a strong Allee effect on the prey and a ratio-dependent functional response

Abstract

In this paper, we consider the complex dynamics of a discrete predator-prey system with a strong Allee effect on the prey and a ratio-dependent functional response, which is the discrete version of the continuous system in (Nonlinear Anal., Real World Appl. 16:235-249, 2014). First, by giving several examples to display the limitations and errors of the local stability of the equilibrium point p obtained in (Nonlinear Anal., Real World Appl. 16:235-249, 2014), we provide an easily verified and complete discrimination criterion for the local stability of this equilibrium. Then we study some properties of its discrete version, especially for the stability and bifurcation for the equilibrium point E 1 , which has not been considered in any literature to the best of our knowledge. By using the center manifold theorem and bifurcation theory, we consider the flip bifurcation of this system at E 1 and obtain the stability of the closed orbits bifurcated from E 1 . The numerical simulations not only show the correctness of our theoretical analysis, but also we find some new and interesting dynamics of this system.

Bifurcation analysis of a two‐dimensional discrete‐time predator–prey model

“…In order for the map (14) to undergo a period-doubling bifurcation, we require that the following discriminatory quantities are non-zero:…”

“…For example, a discrete-time model with a single species can display chaos and more complex dynamical behavior, but chaos requires at least three species in a continuous-time model (see literature [3][4][5][6][7][8] ). Discrete-time and continuous-time models that regulate population systems can be found in previous studies [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and the references listed therein. Many of these studies have used the logistic map to represent the growth in the prey population.…”

Compensatory and overcompensatory dynamics in prey–predator systems exposed to harvest

“…Moreover, the interplay between the species may lead to dynamics of stunning complexity. There is a vast literature regarding such models and models which also include age or stage structure, see [15][16][17][18][19][20][21][22][23][24][25], and more recently, cf [26][27][28][29] and references therein, the question of how to control and stabilize chaotic behaviour in prey-predator systems has been addressed. The dynamics found in such systems strongly depends on compensatory or overcompensatory recruitment.…”

Bifurcations of a two-dimensional discrete-time predator–prey model

“…Moreover, these models also provide efficient computational results for numerical simulations and provide a rich dynamics as compared to the continuous ones [5][6][7][8][9][10]. In the last few years, many interesting papers have appeared in the literature that discuss the stability, bifurcation and chaos phenomena in discretetime models (see [11][12][13][14][15][16][17][18][19][20] and the references cited therein).…”