Bifurcation Analysis and Chaos Control in a Discrete-Time Predator-Prey System of Leslie Type with Simplified Holling Type IV Functional Response

Abstract

The dynamic behavior of a discrete-time predator-prey system of Leslie type with simplified Holling type IV functional response is examined. We algebraically show that the system undergoes a bifurcation (flip or Neimark-Sacker) in the interior of R 2 + . Numerical simulations are presented not only to validate analytical results but also to show chaotic behaviors which include bifurcations, phase portraits, period 2, 4, 6, 8, 10, and 20 orbits, invariant closed cycle, and attracting chaotic sets. Furthermore, we compute numerically maximum Lyapunov exponents and fractal dimension to justify the chaotic behaviors of the system. Finally, a strategy of feedback control is applied to stabilize chaos existing in the system.

Bifurcation and chaos in a host-parasitoid model with a lower bound for the host

“…Although this kind of model has been investigated by many scholars, little research has been carried out on discrete systems [9][10][11][12][13]. In this study, the discrete host-parasitoid system (1.1) is further investigated in detail.…”

“…Next, we will give attention to recapitulating the conditions of the existence for a Neimark-Sacker bifurcation (discrete Hopf bifurcation) by using the bifurcation theorem [11,12,19]. Considering system (1.2) with arbitrary parameters (a, r 2 , m, k, b, c) ∈ H B , we write system (1.2) in the form y 1 ) is the only positive equilibrium of system (3.7) given by (2.1) and…”

Neimark–Sacker bifurcation and stability analysis of a discrete-time prey–predator model with Allee effect in prey

“…They have studied stability and Neimark-Sacker bifurcation of a discrete-time predator-prey model. The analysis of bifurcations has already received much attention during the last few years [20,[22][23][24][25][26][27][28][29]. Bifurcation and stability analysis are examined in detail in [20,[22][23][24][30][31][32][33][34][35][36][37][38][39][40][41].…”

Dynamics and chaos control in a discrete-time ratio-dependent Holling-Tanner model

“…However, lots of exploratory works have recommend that if population size is small, or population generations are relatively discrete (nonoverlapping), or population changes within certain time intervals, one can consider the difference equation model rather than differential equation model to reveal chaotic dynamics [5][6][7][8][9][10][11][12][13][14][15]. These researches explored many complex properties including flip and Neimark-Sacker bifurcations, stable orbits and chaotic attractors which had been derived either by numerically or by normal form and center manifold theory.…”