Global analysis of an impulsive delayed Lotka–Volterra competition system

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Abstract

In this paper, a retarded impulsive n-species Lotka–Volterra competition system with feedback controls is studied. Some sufficient conditions are obtained to guarantee the global exponential stability and global asymptotic stability of a unique equilibrium for such a high-dimensional biological system. The problem considered in this paper is in many aspects more general and incorporates as special cases various problems which have been extensively studied in the literature. Moreover, applying the obtained results to some special cases, I derive some new criteria which generalize and greatly improve some well known results. A method is proposed to investigate biological systems subjected to the effect of both impulses and delays. The method is based on Banach fixed point theory and matrix’s spectral theory as well as Lyapunov function. Moreover, some novel analytic techniques are employed to study GAS and GES. It is believed that the method can be extended to other high-dimensional biological systems and complex neural networks. Finally, two examples show the feasibility of the results.

Introduction

Differential equations have been extensively used in the study of population dynamics, ecology and epidemiology. A rudimentary model for population dynamics is the Lotka–Volterra system. One of basic models is the n-species competitive system which can be represented as follows:y˙i(t)=yi(t)ri-j=1naijyj(t),i=1,2,,n.Many results on the (uniform) persistence, the asymptotic stability, bifurcations, chaos and (almost) periodic solutions are obtained for the Lotka–Volterra systems (see, e.g. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25] and the references cited therein). Also, some authors obtained some interesting type of explicit solutions such as traveling waves, solitons, positons, and complexitons to the Toda lattice equations and the Lotka–Volterra lattice equations (see, e.g. [26], [27] and the references cited therein).

However, in the real-world ecosystem is continuously distributed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Of practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In the language of control variables, we call the disturbance functions as control variables. For instance, in some situation, people may wish to change the position of the existing equilibrium but to keep its stability. This is of significance in the control of ecology balance. One of the methods for the realization of it is to alter the system structurally by introducing some feedback control variables so as to get a population stabilizing at another equilibrium. The realization of the feedback control mechanism might be implemented by means of some biological control scheme or by harvesting procedure. (For more discussion on this direction, one could refer to [11], [12], [28] for more details.)

On the other hand, time delays may lead to oscillation, divergence, or instability which may be harmful to a system. May [29] has shown that if a time delay is incorporated into the resource limitation of the logistic equation, then it has destabilizing effect on the stability of the system. For some systems the stability switches can happen many times and the systems will eventually become unstable when time delays increase (see [30], [31]). But sometimes, the delays may be harmless under some restriction and this is more important in some sense (see, e.g. [32], [33], [34]).

As we know, dynamic systems are often classified into two categories of either continuous-time or discrete-time systems, which are both widely studied in the literatures [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36]. However, many real-world phenomena are neither purely continuous-time nor purely discrete-time. This leads to the development of dynamic systems with impulses (see, e.g. the monographs [39], [40], [41] and the works of Li [37], [38], Liu et al. [42], [43], Chu and Nieto [44], Nieto and Rodríguez-López [45], [46], Lakshmikantham et al. [47], [48], Pinto [49], [50], Fenner and Pinto [51]), which display a combination of characteristics of both the continuous-time and discrete-time systems, and hence provide a more natural framework for mathematical modelling of many real-world phenomena. In particular, some impulsive equations have been introduced into population dynamics, for examples, see Robert and Kao [52], Zhang et al. [53], Yan et al. [54], [55], Benchohra et al. [56], Samoilenko and Perestyuk [57], Zavalishchin and Sesekin [58], Nieto et al. [59], [60], [61].

Motivated by the richness of dynamic systems with impulses, I deal with a n-species competition system with feedback controls subject to both the effect of impulses and the effect of delays. I consider the following system of retarded functional impulsive differential equations (IDE):dyi(t)dt=yi(t)ri-j=1naijyj(t)-j=1nbijyj(t-τij)-j=1ncij0Hij(s)yj(t-s)ds-diui(t)-eiui(t-σi)-fi0Pi(s)ui(t-s)ds,t0,ttk,dui(t)dt=hi-αiui(t)+βiyi(t)+γiyi(t-δi)+πi0Qi(s)yi(t-s)ds,t0,ttk,Δyi(tk)=Ik(yi(tk)),t=tk,k=1,2,,Δui(tk)=Jk(ui(tk)),t=tk,k=1,2,,i=1,2,,n,where the impulsive operators Ik, Jk:R  R are continuous; Δyi(tk)=yitk+-yitk-andΔui(tk)=uitk+-uitk- are the impulses at moments tk and 0 < t1 < t2 < ⋯ is a strictly increasing sequence such that limk→∞tk = +∞. As usual in the theory of impulsive differential equations, by a solution of (2) we mean z(t) =  (y1(t), y2(t), …, yn(t), u1(t), u2(t), …, un(t))T  R2n in which yi(·), ui(·) is piecewise continuous on (0, μ) for some μ > 0 such that ztk+andztk- exist and z(·) is differentiable on intervals of the form (tk−1, tk)   (0, μ) and satisfies (2), moreover z(t) is left continuous with ztk-=z(tk). System (2) is supplemented with initial values given byyi(s)=φyi(s),s(-,0],φyi(0)>0,sups(-,0]φyi(s)<+,i=1,2,n,ui(s)=φui(s),s(-,0],φui(0)>0,sups(-,0]φyi(s)<+,i=1,2,,n,where φyi(s)andφui(s) denote the real-valued continuous functions defined on (−∞, 0]. For the point of biological view, the parameters are always assumed that ri, aij, bij, cij, di, ei, fi, hi, βi, γi, πi, (i, j = 1, 2, …, n) are nonnegative and aii, αi is strictly positive; the delays τij, σi, δi  R. On considering the biological significance of Eq. (2), we specify z(0) =  (y1(0), y2(0), …, yn(0), u1(0), u2(0), …, un(0))T > 0. Since I deal with the high dimensional system, set n  2. Assume that the kernels Hij(·), Pi(·), Qi(·) are nonnegative continuous functions defined on [0, ∞) and

  • (H1):

    There exists positive number λ such that 0Hij(s)ds=1, 0Hij(s)eλsds<+; 0Pi(s)ds=1, 0Pi(s)eλsds<+; 0Qi(s)ds=1, 0Qi(s)eλsds<+.

  • H¯1:

    The kernels satisfy 0Hij(s)ds=1, 0sHij(s)ds<+; 0Pi(s)ds=1, 0sPi(s)ds<+; 0Qi(s)ds=1, 0sQi(s)ds<+.

Our aim is to study the existence and stability of equilibrium of system (2). The approaches are based on employing Banach fixed point, Lyapunov function, matrix theory and its spectral theory. Moreover, some novel analytic techniques are employed to study GAS and GES.

The plan of the paper is as follows. In Section 2, some new sufficient conditions for the existence of a unique equilibrium of system (2) are obtained. Section 3 is devoted to examining the stability of this equilibrium solution. In Section 4, applying the obtained results to some special cases, I derive some new results which generalize the previous ones. Finally, two examples are given to show the feasibility of our results.

Section snippets

Existence and uniqueness of equilibrium solution

This section is to obtain some new sufficient conditions for the existence and uniqueness of equilibrium of system (2). For convenience, firstly, I introduce some notations. x =  (x1, …, xn)T  Rn denotes a column vector, D=(dij)n×n is an n × n matrix, DT denotes the transpose of D, and En is the identity matrix of size n. A matrix or vector D>0 means that all entries of D are greater than zero, likewise for D0. For matrices or vectors D and E, D>E(DE) means that D-E>0(D-E0). We also denote the

Persistence and stability

In this section, we devote ourselves to studying the stability (including the global exponential stability and global asymptotic stability) of the unique positive equilibrium. Let J  R. Denote by PC(J, R) the set of function ψ:J  R which are continuous for t  J, t  tk are continuous from the left for t  J and have discontinuities of the first kind at the point tk  J. Denote by PC1(J, R) the set of function ψ:J  R with a derivative ψ  PC(J, R).

Definition 3.1

Let z=y1,,yn,u1,,unT be the unique positive equilibrium

Application 4.1

When the impulsive operators are absent, system (2) reduces to non-impulsive system.dyi(t)dt=yi(t)ri-j=1naijyj(t)-j=1nbijyj(t-τij)-j=1ncij0Hij(s)yj(t-s)ds-diui(t)-eiui(t-σi)-fi0Pi(s)ui(t-s)ds,dui(t)dt=hi-αiui(t)+βiyi(t)+γiyi(t-δi)+πi0Qi(s)yi(t-s)ds,i=1,2,,n.

As pointed out in Remark 3.1, in this case, condition (H5) and (H6) reduce to H^6. Then applying our main results to system (50), we have the following results.

Theorem 4.1

Assume that (H1) and H^6 hold, then system (50) is persistent.

Theorem 4.2

Assume

Examples

Example 1

Consider the two-species impulsive competitive system with feedback controls.x˙1(t)=x1(t)4-2x1(t)-14x2(t)-14x2(t-1)-14u1(t-1),x˙2(t)=x2(t)8-12x1(t)-12x1(t-1)-2x2(t)-14u2(t-1),u˙1(t)=3-2u1(t)-14x1(t)-14x1(t-1),u˙2(t)=12-2u2(t)-14x2(t)-14x2(t-1),Δx1(t)=-18x1(t),Δx2(t)=-18x2(t),Δu1(t)=-18u1(t),Δu2(t)=-18u2(t).Corresponding to system (2), simple computation shows that Δ1=a11+d1α1-1(h1+β1+γ1)=3916, Γ21=a21+b21=1, Δ2=a22+d2α2-1(h2+β2+γ2)=178, Γ12=a12+b12=12. Thus, K=0817×121639×10=041716390andρK=64

Acknowledgements

The author express his gratitude to the editors and anonymous reviewers for their valuable comments which improve the presentation of this paper.

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