Neimark-Sacker Bifurcation and Chaos Control in a Fractional-Order Plant-Herbivore Model

Abstract

This work is related to dynamics of a discrete-time 3-dimensional plant-herbivore model. We investigate existence and uniqueness of positive equilibrium and parametric conditions for local asymptotic stability of positive equilibrium point of this model. Moreover, it is also proved that the system undergoes Neimark-Sacker bifurcation for positive equilibrium with the help of an explicit criterion for Neimark-Sacker bifurcation. The chaos control in the model is discussed through implementation of two feedback control strategies, that is, pole-placement technique and hybrid control methodology. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long-term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the model.

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

“…where all partial derivatives in (21) are evaluated at (x * , y * , γ 0 ). From (20) and (21), we have the following unique solution of pole-placement problem:…”

“…Li et al [37] discussed period-doubling and Neimark-Sacker bifurcations for a plant-herbivore model incorporating plant toxicity in the functional response of plant-herbivore interactions. Similarly, for some other discussions related to qualitative behaviour of plant-herbivore models, the interested reader is referred to [2,10,21,[25][26][27]29,33,45,47,52] and references are therein.…”

Bifurcation and chaos control in a discrete-time predator–prey model with nonlinear saturated incidence rate and parasite interaction

“…Here we investigate the parametric conditions for which the positive steady state for discrete-time system (1.3) undergoes a Neimark-Sacker (Hopf ) bifurcation. For such an investigation an explicit criterion of Hopf bifurcation is used without computing the eigenvalues for the variational matrix of system under consideration (see also [17,18]). For this purpose, an explicit criterion for Hopf bifurcation is given below.…”

Controlling Chaos and Neimark–Sacker Bifurcation in a Host–Parasitoid Model

“…In this section, we want to investigate the conditions for existence and direction of Neimark-Sacker bifurcation at positive equilibrium point of system (5). For a similar type of discussion related to the existence and direction of Neimark-Sacker bifurcation, we refer the interested reader to [1,23,24,28,[32][33][34][35][36][39][40][41] and references therein. Notice that, the roots of characteristic polynomial (16) are conjugate complex numbers if the following condition is satisfied:…”

“…In this section, we study two feedback control strategies in order to move the unstable trajectory towards the stable one. For similar types of investigations we refer to [1,[32][33][34][35][36][37][38][39][40][41] for controlling chaos in discrete-time population models. For some other applications related to chaos control, see also [51][52][53][54][55][56][57][58][59][60][61][62].…”