Exploration on dynamics in a discrete predator–prey competitive model involving feedback controls

In this work, we set up a new discrete predator–prey competitive model with time-varying delays and feedback controls. By virtue of the difference inequality knowledge, a sufficient condition which guarantees the permanence of the established discrete predator–prey competitive model with time-varying delays and feedback controls is derived. Under some appropriate parameter conditions, we have proved that the periodic solution of the system without delay exists and globally attractive. To verify the correctness of the derived theoretical fruits, we give two examples and execute computer simulations. Our obtained results are novel and complement previous known results.


Introduction
In recent few decades, population dynamical models have received considerable attention [13,19,31,43,57].In the research of population dynamical models, competitive systems play an important role in describing the interaction among the multi-species.A lot of competitive systems have been explored by numerous scholars.For example, Hou [16] analysed the permanence of a competitive Lotka-Volterra system with delays, Balbus [2] addressed the attractivity and stability in a competitive system of PDEs of Kolmogorov type, Shi et al. [39] focused on the extinction of a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls, Liu and Wang [32] discussed the asymptotic behaviour of a stochastic nonautonomous Lotka-Volterra competitive system with impulsive perturbations, Kulenović and Nurkanović [20] made a theoretical discussion on the global behaviour of a two-dimensional competitive system of difference equations with stocking.For more detailed publications on the permanence and global attractivity behaviour of predator-prey models, one can see [4, 6, 7, 9, 11, 12, 15, 17, 18, 21, 23-25, 28-30, 33-36, 40, 41, 44, 45, 47-52, 55, 56].
CONTACT Changjin Xu xcj403@126.comIn 1992, Gopalsamy [14] explored the following competitive model on predator species and prey species: where w 1 (t), w 2 (t) stand for the densities of two competing species at time t, respectively.α 1 , α 2 denote the intrinsic growth rates of predator species and prey species, β 1 , δ 1 , β 2 , δ 2 represent the effects of intra-specific competition, and γ 1 , γ 2 are the effects of inter-specific competition.Considering that the parameters, in natural world, often vary due to the change of surroundings, we think that it is important to introduce the time-varying parameters into the predator-prey model.Varying parameters will lead to much more different dynamical behaviour in predator-prey models than that of predator-prey models with fixed parameters.Motivated by this idea, we can modify system (1) as the form where α i (t), β i (t), γ i (t), δ i (t)(i = 1, 2) are functions with respect to the time t.Since predator species and prey species live in a real fluctuating environment and many exploitation activities of people might lead to abrupt vary, Tan et al. [42] set up the following impulsive competitive model: where w 1 (0 + ) = w 1 (0) > 0, w 2 (0 + ) = w 2 (0) > 0 and N is the set of positive integers, all the coefficients α i (t), β i (t), γ i (t), δ i (t)(i = 1, 2) are all continuous almost periodic functions which are bounded above and below by positive constants, ϑ 1k > −1 and Considering that two species are constantly in the competition, and when a species suffers damage from another one by competition, another one could benefit, the duration time of density for species would also play an important role, we modify system (1) as the where ρ(t) ≥ 0 stands for the hunting delay.Many scholars [1,3,5,8,10,22,26,27,37,38,53,54] argue that discrete dynamical models which are described by difference equations are often regarded as more suitable tools to depict the dynamical relationship among different species than continuous ones since the species owns non-overlapping generations.Discrete dynamical model is also thought to be a very useful tool to carry out numerical simulations for the continuous ones.In addition, we would like to point out that external force often make the parameters of biological systems vary.For example, the competition and cooperation model of two enterprises is usually interfered by manpower, material resources, financial resources and so on.What we are interested in is how to real the dynamical behaviour of biological systems which are affected by external force.In control language, the external force can be called as control variables.From a mathematical point of view, it is of great interest to investigate the effect of control variables on the dynamics for biological systems.Stimulated by the analysis above, we can modify system (4) as follows where w 1 (n) and w 2 (n) denote the densities of two competing species at the generation, respectively, and ) are bounded nonnegative sequences and ρ(n) are integer-valued sequences.As far as I know, it is first time to probe into system (5) involving feedback control.We believe that this work on the permanence and global attractivity of the discrete competitive model has significant theoretical meaning and tremendous potential application in preserving population coexistence and maintaining ecological balance.
The chief task of this study is to probe into the permanence and global attractivity of system (5).In order to set up the key conclusions, the following assumptions are needed: For system (5), we give the initial value as follows: It is not difficult to see that solutions of ( 5) and ( 6) are well defined for all n ≥ 0 and satisfy We plan the structure of this work as follows.Section 2 presents the necessary definitions and lemmas and states the permanence result of system (5) Section 3 deals with the existence of periodic solution and global attractivity of periodic solution for system (5) without delay.Section 4 carries out computer simulation to illustrate the feasibility and effectiveness of our results derived is Sections 2 and 3. A brief conclusion is drawn in Section 5.

Permanence
In this section, we firstly introduce the definition of permanence and several lemmas which is needed to set up the key results.
Definition 2.1: We say that system ( 5) is permanent provided that ∃ two positive constants M and m such that for each positive solution (w Give the single species discrete model as follows: where {α(n)} and {β(n)} are strictly positive sequences of real numbers defined for n ∈ N = {0, 1, 2, . ..} and 0 < α l ≤ α u , 0 < β l ≤ β u .Similarly to the proofs of Propositions 1 and 3 in [3], one can obtain the following result.
Lemma 2.1: Every solution of system (7) with initial value N(0 where Give the first order difference equation where A and B are positive constants.Following Theorem 6.2 of Wang and Wang [46, page 125], one has the following result.

Lemma 2.2 ([46]
): Assume that |A| < 1, for any initial value v(0), there exists a unique solution v(n) of (8) which can be expressed as follows: where v * = B 1−A .Thus, for any solution {v(n)} of system (8), For any fixed n, h(n, r) is a nondecreasing function with respect to r, and for n ≥ n 0 , the following inequalities hold: Proposition 2.1: For system (5), assume that (H1) holds, then where .
be any positive solution of system (5) with the initial condition (w 1 (0), w 2 (0), μ 1 (0), μ 2 (0)).It follows from system (5) that Set is equivalent to By virtue of (10), one has Then By means of (12) and system (5), one gets It follows from ( 13) and Lemma 2.1 that In view of the third and fourth equations of the system (5), one can easily get Then Applying Lemmas 2.2 and 2.3, we have Setting ε → 0, we get which completes the proof.
Theorem 2.1: Let M i and U i be defined by ( 14) and (19), respectively.Assume that (H1) and (H2) Proof: By applying Proposition 2.1, we are easy to see that to end the proof of Theorem 2.1, it is enough to show that under the conditions of Theorem 2.1, It follows from system ( 5) and ( 20) that Set In view of (22), we obtain ) Applying ( 25) and ( 5), one has In view of Lemmas 2.1 and 2.3, we get where By Lemmas 2.1 and 2.2, we get Setting ε → 0 in the above inequality leads to which completes the proof.

Existence and stability of periodic solution of system (5) without delay
In this section, we are to discuss the stability of system ( 5) with ρ i (n) = 0(i = 1, 2), that is, we discuss the following system Throughout this section we always assume that 2) are all bounded negative periodic sequences with a common periodic ω and satisfy Also it is assumed that the initial values of (36) are given by Applying the similar approach, under some conditions, we can obtain the permanence of system (36).We still let M i and U i be the upper bound of {w i (n)} and {μ i (n)}, and m i and U l i be the lower bound of {w i (n)} and {μ i (n)}.

Remark 3.1:
In [14,42], the authors dealt with the continuous or pulsing competitive model without time delays and feedback controls.In this paper, we consider the practical situation and introduce time delays and feedback controls.Based on this viewpoint, the acquired outcomes of our work are new and replenish the outcomes of [14,42].

Remark 3.2:
We do not investigate the existence of periodic solution and global attractivity of system (5) since the introduction of delay leads to the difficulties in analysis methods.
We leave this aspect for future work.

Examples
In this section, we will execute computer simulations via Matlab software to confirm the rationality of our derived key conclusions.
Example 4.1: Consider the following system where We can verify that all the hypotheses in Theorem 2.1 are true.Thus one can easily know that system (55) is permanent.The computer simulation figures are clearly presented in Figures 1-4.

Example 4.2: Consider the following system
where

Conclusions
In this current work, we propose a new discrete predator-prey competitive model involving delays and feedback controls.By virtue of the difference inequality knowledge, a novel sufficient condition guaranteeing the permanence of the system is set up.We find that under a   suitable parameter conditions, two species will keep a state of coexistence.The study reveals that feedback control effect and time delays play a vital role in remaining the co-existence of two species In addition, we also obtain the sufficient conditions which ensure the existence and stability of unique globally attractive periodic solution of the system without time delays.The derived results own significant theoretical guiding value in keeping a balance of biological population.Meanwhile, our results are new and supplement the existed results in [14,42].

Figure 1 .
Figure 1.Computer simulation figure of system (55): the relation between the time k and the variable w 1 .

Figure 2 .
Figure 2. Computer simulation figure of system (55): the relation between the time k and the variable w 2 .

Figure 3 .
Figure 3. Computer simulation figure of system (55): the relation between the time k and the variable μ 1 .

Figure 4 .
Figure 4. Computer simulation figure of system (55): the relation between the time k and the variable μ 2 .

Figure 5 .
Figure 5. Computer simulation figure of system (56): the relation between the time k and the variable w 1 .

Figure 6 .
Figure 6.Computer simulation figure of system (56): the relation between the time k and the variable w 2 .

Figure 7 .
Figure 7. Computer simulation figure of system (56): the relation between the time k and the variable μ 1 .

Figure 8 .
Figure 8. Computer simulation figure of system (56): the relation between the time k and the variable μ 2 .