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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.

In [

Now we consider the counterpart of (

First, we consider possible steady-state (equilibrium point)

Let

Using Lemma

Polynomial

Due to above analysis, we have the following result about the existence and uniqueness of positive equilibrium point of system (

Under the conditions of Lemma

Next, the Jacobian matrix for system (

For system (

The trivial equilibrium point

The equilibrium point

(i) The Jacobian matrix for system (

(ii) The Jacobian matrix for system (

The unique positive equilibrium

The Jacobian matrix for system (

In order to study the Neimark-Sacker bifurcation in system (

Consider an

Eigenvalue assignment:

Transversality condition:

Nonresonance condition:

The following result shows that system (

The unique positive equilibrium point of system (

According to Lemma

Controlling chaos in discrete-time models is a topic of great interest for many researchers in recent times, and practical methods can be used in many fields such as biochemistry, cardiology, communications, physics laboratories, and turbulence [

First, we study chaos controlling technique based on pole-placement methodology introduced by Romeiras et al. [

Next in order to control Neimark-Sacker bifurcation in system (

First, we take

Furthermore, Figure

Bifurcation diagrams and MLE for system (

Bifurcation diagram for

Bifurcation diagram for

Bifurcation diagram for

Maximum Lyapunov exponents

Next we take

Furthermore, Figure

Next, we fix the value of

Furthermore, the phase portraits of system (

Bifurcation diagrams and MLE for system (

Bifurcation diagram for

Bifurcation diagram for

Bifurcation diagram for

Maximum Lyapunov exponents

Plots for system (

Plot of

Plot of

Plot of

Phase portrait of system (

Phase portraits of system (

Phase portrait of system (

Phase portrait of system (

Phase portrait of system (

Phase portrait of system (

We choose the parametric values

Finally, choosing the parametric values

Bifurcation diagrams for the controlled system (

Bifurcation diagram of

Bifurcation diagram of

Bifurcation diagram of

Plots for the controlled system (

Plot of

Plot of

Plot of

Phase portrait of system (

We study the qualitative behavior of a 3-dimensional discrete-time fractional-order plant-herbivore model, and thus we obtain the mathematical results related to existence and uniqueness of positive steady state and conditions for local asymptotic stability of positive equilibrium. In order to confirm complexity in system (

The authors declare that they have no conflicts of interest regarding the publication of this paper.