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.
Highlights
Introduction and PreliminariesIn [1], authors proposed a mathematical model governed by ordinary differential equations related to the interaction between a plant and an insect
We study the qualitative behavior of a 3-dimensional discretetime fractional-order plant-herbivore model, and we obtain the mathematical results related to existence and uniqueness of positive steady state and conditions for local asymptotic stability of positive equilibrium
We show that system (16) undergoes Neimark-Sacker for wide range of bifurcation parameter β
Summary
In [1], authors proposed a mathematical model governed by ordinary differential equations related to the interaction between a plant and an insect. Using Lemma 1, we have the following result for existence of unique positive real root of polynomial P(t) given in (20). Under the conditions of Lemma 2, system (16) has unique positive equilibrium point if the following condition holds: η]b2 − ηφ 2λd. Applying the Jury condition [11], the unique positive equilibrium point (x∗, y∗, z∗) is locally asymptotically stable if the following conditions are satisfied:. The unique positive equilibrium point of system (16) undergoes Neimark-Sacker bifurcation if the following conditions hold:. According to Lemma 6, for n = 3, we have in (32) the characteristic polynomial of system (16) evaluated at its unique positive equilibrium, we obtain the following equalities and inequalities: Δ−2 (μ) = 1 − A2 + A3 (A1 − A3) = 0, Δ+2 (μ) = 1 + A2 − A3 (A1 + A3) > 0,.
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