Under the influence of crowding effects: Stability, bifurcation and chaos control for a discrete-time predator–prey model

Abstract

The interaction between predators and preys exhibits more complicated behavior under the influence of crowding effects. By taking into account the crowding effects, the qualitative behavior of a prey–predator model is investigated. Particularly, we examine the boundedness as well as existence and uniqueness of positive steady-state and stability analysis of the unique positive steady-state. Moreover, it is also proved that the system undergoes Hopf bifurcation and flip bifurcation with the help of bifurcation theory. Moreover, a chaos control technique is proposed for controlling chaos under the influence of bifurcations. Finally, numerical simulations are provided to illustrate the theoretical results. These results of numerical simulations demonstrate chaotic long-term behavior over a broad range of parameters. The presence of chaotic behavior in the model is confirmed by computing maximum Lyapunov exponents.

Bifurcation analysis and chaos control for a plant–herbivore model with weak predator functional response

“…x n α(1 + y 2 n ) + βx n , y n+1 = γ y n (1 + x n ), (1) where x n and y n are population densities for grape vine and apple twig borer, respectively. Moreover, α, β and γ are positive parameters.…”

“…Moreover, α, β and γ are positive parameters. For some earlier investigations related to model (1), we refer to [4][5][6][7]34,35]. Moreover, taking into account strong predator functional response, Din [12] discussed global stability of system (1), and Khan et al [32] investigated Neimark-Sacker bifurcation for system (1) with strong predator functional response.…”

“…• It is proved that model (1) undergoes transcritical bifurcation at its boundary equilibrium by implementing bifurcation theory of normal forms and centre manifold theorem. • Direction and existence criteria for Neimark-Sacker bifurcation are investigated at positive steady-state of plant-herbivore model (1). • Pole-placement and hybrid control methods are implemented in order to discuss chaos control in system (1).…”

“…• Direction and existence criteria for Neimark-Sacker bifurcation are investigated at positive steady-state of plant-herbivore model (1). • Pole-placement and hybrid control methods are implemented in order to discuss chaos control in system (1).…”

“…Moreover, in Section 2 the existence of steady-states and their local asymptotic behaviour is analysed. Section 3 is dedicated to investigate transcritical bifurcation about boundary steady-state of system (1). In Section 4, we discuss Neimark-Sacker bifurcation at positive steady-state of model (1).…”