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In this paper, a discrete space-time Lotka–Volterra model with the periodic boundary conditions and feedback control is proposed. By means of a discrete version of comparison theorem, the boundedness of the nonnegative solution of the system is proved. By the combination of the Volterra-type and quadratic Lyapunov functions, the global asymptomatic stability of the unique positive equilibrium is investigated. Finally, numerical simulations are presented to verify the effectiveness of the main results.

It is well known that the ecosystem in the real world is often distributed by unpredictable forces or interference factors, such as natural disturbances (floods, fires, disease outbreaks, and droughts), human-caused interference factors (oil spills), and slowly changing long-term stresses (nutrient enrichment), which may result into changes in the biological parameters such as survival rates [

For population dynamical systems with feedback controls, an important and interesting subject is to study the effects of feedback controls to the persistence, permanence, and extinction of species, the stability, and dynamical complexity of systems [

The discrete-time models governed by difference equation are more realistic than the continuous ones when the populations have nonoverlapping generations or the population statistics are compiled from given time intervals and not continuously. Moreover, discrete-time models can also provide efficient computational models of continuous models for numerical simulations. Therefore, it is reasonable to study discrete-time models governed by difference equations, and there has been some work done on the study of the persistence, permanence, and global stability for various discrete-time nonlinear population systems with feedback when the effect of spatial factors is not considered [

It is a fact that spatial heterogeneity and dispersal play an important role in the dynamics of populations, which has been the subject of much research, both theoretical and experimental, such as the role of dispersal in the maintenance of patchiness or spatial population variation. If the spatial factors are added, more dynamics will occur. The diffusion-driven instability may emerge if the steady-state solution is stable to small spatial perturbations in absence of diffusion, but unstable when diffusion is present [

Motivated by above discussions, the main purpose of this paper is to study the global asymptomatic stability of an one-dimensional spatially discrete reaction diffusion Lotka–Volterra model with the periodic boundary conditions and feedback control. So the organization of this paper is as follows. In the Section

It is well known that a Lotka–Volterra system can be described in the form of

A corresponding discrete model for the system (

It is believed that the diffusion of individuals can play an important role in determining collective behavior of the population. Space factors can be taken into account in all fundamental aspects of ecological organization, and we can get a one-dimensional discrete reaction-diffusion model as follows:

This also indicates the coupling or diffusion from the units or individuals to the left and the right, respectively. The following periodic boundary conditions are considered:

Systems (

To the best of our knowledge, no work on global asymptomatic stability of the positive equilibrium of systems (

By simple computation, systems (

If

To discuss the global asymptomatic stability of the unique positive equilibrium, the following assumptions and preparations are essential.

From the view point of biology, we only need to discuss the positive solution of system (

For our purpose, we first introduce the following lemma which can be obtained easily by comparison theorem of difference equation.

(see [

(see [

Applying the above lemmas, we can obtain the following result.

The solution of (

From the first equation of system (

Similarly, from the second equation of system (

If

Next, we will show the boundedness of the solutions.

From Lemma

Similarly, we can also obtain

From Lemma

In this section, we devote ourselves to studying the global asymptotic stability of the unique positive equilibrium

Denote

Assume

Assume

Let

Then, we can obtain

Let

Then, we can obtain

Let

Then, we can obtain

Let

Then, we can obtain

Let

Then,

If

In the following example, we will show the feasibility of our main results and discuss the effects of feedback controls. Take

To illustrate our purposes, the parameter values are chosen as follows (the choice of parameter values is hypothetical with appropriate units and not based on data):

Dynamic behaviors of systems (

Different initial values for

1 | 0.22 | 0.35 | 0.27 | 0.22 | 0.40 | 0.46 | 0.45 | 0.52 |

2 | 0.33 | 0.26 | 0.23 | 0.28 | 0.47 | 0.42 | 0.48 | 0.43 |

3 | 0.25 | 0.23 | 0.21 | 0.30 | 0.43 | 0.41 | 0.53 | 0.56 |

To explore clearly the dynamical behavior of systems (

Dynamic behaviors of

Dynamic behaviors of

This paper investigates the global asymptomatic stability of the unique positive equilibrium of a discrete diffusion model with the periodic boundary conditions and feedback control. The condition to ensure the nonnegativity and boundedness of the solutions of the discrete model is discussed, and the globally asymptotical stability of the positive equilibrium is proved. Through comparing numerical simulations, we notice that when we improve the feedback control coefficients, the solutions will converge more slowly. It follows that we can adjust the rate of convergence by choosing suitable values of feedback control variables. Such work may also be applied to other discrete diffusion models.

It should be noted that there is only a basic condition obtained to guarantee the existence of positive solution. Judgement sentence

In this study, all of the coefficients of the model system are constant; in many situations, they can be assumed to be nonconstant bounded nonnegative sequences such as periodic positive sequences which can reflect the seasonal fluctuations [

Different control schemes, such as switching control, constraint control, and sliding control, can be applied to the system models. Many interesting results can be obtained. Based on the backstepping recursive technique, a neural network-based finite-time control strategy is proposed for a class of non-strict-feedback nonlinear systems [

It is well known that noise disturbance is unavoidable in real systems, and it has an important effect on the stability of systems. Also, noise can be used to stabilize a given unstable system or to make a system even more stable when the system is already stable which reveals that the stochastic feedback control can stabilize and destabilize the deterministic systems [

No data were used to support this study.

The authors declare that they have no conflicts of interest.

This research was supported by the Applied Study Program (grant nos. 171006901B, 60204, and WH18012).