# Dynamics in a Delayed Competition System on a Weighted Network

Authors: Yaohua Tong; Xiaoling Wang
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this paper, we study the stability of positive steady states in a delayed competition system on a weighted network, which does not satisfy the comparison principle appealing to classical competitive systems. By introducing some auxiliary equations and constructing proper contracting rectangles, we present some sufficient conditions on the stability of the unique positive steady state. Moreover, some numerical examples are given to explore the complex dynamics of this nonmonotone model, which implies the nontrivial roles of weights and time delays.


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## Body

## 1. Introduction

Competition is one of the most universal phenomena in the natural world due to the limit resources including water and sunshine. When the amount of individuals is concerned in population dynamics, some differential systems modeling both interspecific and intraspecific competitions have been proposed and studied. For example, if only two competitive species are concerned in population dynamics, then one famous model is(1)dN1dt=r1N11−N1−b1N2,dN2dt=r2N21−N2−b2N1,in which all the parameters are positive, and N1,N2 present the densities of two competitive species; we may refer to Murray [1], section 3.5, for the dynamics and biology background of (1). When multispecies and delayed effect [2–4] were considered, Faria [5] and Smith [6] studied the following delayed system:(2)duitdt=riuit1−∑j=1ncij∫−τ0ujt+sdη¯ijs,in which i∈1,2,…,n=:I,t>0, all the parameters are positive, and η¯ijs formulates the history delayed effect and satisfies(3)η¯ijsis nondecreasing on−τ,0,η¯ij0−η¯ij−τ=1,i,j∈I.In some cases, the spatial distribution of individuals must be considered. For example, to control the disease spreading, the spatial distribution of the infected is important. To model the spatial distribution of individuals, many reaction-diffusion systems have been established and studied; see [7–10]. In particular, we refer to He and Ni [11] and references cited therein for some results on the stability of different steady states in the corresponding reaction-diffusion system of (1). Moreover, when the spatial habitat is ℝ, Weinberger et al. [12] studied the invasion dynamics between the invader and the native in the corresponding diffusion model of (1). For the corresponding diffusion model of (2), Martin and Smith [13] proved the stability of steady states in the bounded domain. In the classical reaction-diffusion systems, it is necessary to assume that the density of individuals is continuous in both spatial and temporal variables. However, the spatial continuity is difficult to be satisfied due to the limitation of data in some cases. To overcome the deficiency, one recipe is to study the ordinary differential equations on a weighted network by considering finite patches as habitat of individuals; we may refer to Beretta et al. [14], Faria [5], Gao and Ruan [15], Liao and Lou [16], Mai et al. [17], Sun and Mai [18], Xu and Chen [19], and Zhang and Wang [20] for patch/network models with time delay.Recently, Liu et al. [21] considered the competition dynamics of two competitors on a network or m patches (m>1 and m∈ℕ); their model can be written as(4)dN1jdt=∑i=1mdij1N1i−N1j+r1N1j1−N1j−b1N2j,dN2jdt=∑i=1mdij2N2i−N2j+r2N2j1−N2j−b2N1j,in which j∈1,2,…,m=:J,dijl>0 describes the movement between the i-th and j-th patches of the l-th species, which is called the weight between patches iandj. In [18], Sun and Mai studied the following system on a network:(5)dN1jtdt=∑i=1mdij1N1it−τ1−N1jt+r1N1jt1−N1jt−b1N2jt,dN2jtdt=∑i=1mdij2N2it−τ2−N2jt+r2N2jt1−N2jt−b2N1jt,in which τ1≥0andτ2≥0 are time delays. A natural question is how to further study the dynamics of multicompetitive species with delayed effect on networks. In this paper, we study the following competition system with time delays on the network:(6)duijtdt=∑l=1mdjli∫−τ0uilt+sdρjlisds−uijt+riuijt1−∑k=1ncik∫−τ0ukjt+sdη¯iks,t>0,uijs=vijs≥0,s∈−τ,0,in which all the parameters are positive, j∈J,i∈I,t>0, uijt denotes the population density of the i-th species on the j-th patch, and ρjli satisfies(7)ρjlisis nondecreasing on−τ,0,ρjli0−ρjli−τ=1,i∈I,j,l∈J.That is, (6) is a coupled system of m×n functional differential equations. Evidently, there are m patches in this model, the same species on different patches may move from one to another due to djli>0, and different species on the same patch compete each other. So, (6) is a competitive system on a weighted network or patch environment.To compare the literature results with the complex phenomena in the real world, the stability of mathematical models is important since the disturbance is inevitable. Moreover, the controllability of many mathematical models in finite time is also important, which partly depends on the stability conditions of some states; see a number of examples about switched nonlinear systems [22–24]. Liu et al. [21] and Sun and Mai [18] studied the stability of different steady states of (4) and (5) by using the comparison principle and other techniques, which also include some applications in population dynamics. However, because of the intraspecific delayed effect, (6) does not satisfy the comparison principle of classical competitive systems, and the recipe in [21] does not work directly. Moreover, since there are m×n functional differential equations, the analysis of eigenvalues would be complex if we analyse the stability by the characteristic equation at the positive steady state, which may involve m×n transcendental equations. Therefore, to study the dynamics of (6), some other techniques are necessary.Evidently, (6) may have several different steady states under proper conditions ((1) may have four steady states [1]), which reflect different biological sense in population dynamics. Coexistence of competitive species has been widely studied in many cases due to its important biological backgrounds on the biodiversity [25, 26]. In this paper, by combining the technique in [21] with the contracting rectangles in [6], we study the stability of the positive steady state by giving some sufficient conditions. Moreover, we also show some numerical results to support our theoretical conclusions and present the complex dynamics of this system when our stability conditions do not hold, which implies the nontrivial role of time delays and partial degenerate weights.The rest of this paper is organized as follows. In section2, we present some preliminaries on the stability via contracting rectangles. In Section 3, we show our main results on the stability of the unique positive steady state. Some numerical examples will be given in Section 4. Furthermore, we make a discussion in Section 5.


## 2. Preliminaries

In this paper, we use the standard partial ordering inℝn,n∈ℕ. That is, if(8)u=u1,u2,…,un,v=v1,v2,…,vn∈ℝn,then(9)u≤vif and only ifui≤vi,i∈1,2,…,n.Moreover,u<v implies u≤v, but u≠v, and u≪v implies ui<vi,i∈1,2,…,n. CA,ℝm,A⊂ℝn, denote the set of uniform continuous vector functions from A to ℝn.In this part, we recall the stability of the steady state by contracting rectangles [6]. Consider the functional differential system(10)dvitdt=fivt,i∈1,2,…,r≕R,t>0,vis=νis∈C−τ,0,0,∞,i∈R,s∈−τ,0,in which τ>0 is the delay, v=v1,v2,…,vr∈ℝr, and fi:C−τ,0,ℝr⟶ℝ satisfies the following assumptions:(H1) There existsE=E1,E2,…,Er∈ℝr with E≫0 such that fi0^=fiE^=0,i∈R, where ·^ denotes the constant-valued function in C−τ,0,ℝr.(H2) There exists a one-parameter family of ordered intervals given by(11)Σy=a^y,b^y,such thata0≤E≤b0, and for 0≤y1≤y2≤1,(12)0≤a0≤ay1≤ay2≤a1=E=b1≤by2≤by1≤b0,whereay and by are continuous in y∈0,1.(H3) Ifu∈C−τ,0,ℝr and 0≤vs≤b0 for s∈−τ,0, then(13)fiv:C−τ,0,0,b0⟶ℝ,is Lipschitz continuous in the sense of supremum norm for eachi∈R.(H4)
Σy is a strict contracting rectangle, namely, let(14)ay=a1y,a2y,…,ary,by=b1y,b2y,…,bry.Then, for anyy∈0,1 and v0s∈Σy,s∈−τ,0, we have(15)fiv0>0fiv0<0,ifui0=aiyui0=biy,i∈R.By contracting rectangles, we have the following conclusion [6].Lemma 1.
Assume that (H1)–(H4) hold. Ifνis∈aiy0,biy0 for some y0∈0,1 with s∈−τ,0,i∈R, then(16)limt⟶∞vit=Ei,i∈R.


## 3. Main Results

In this section, we study the long-time behavior of (6). We denote(17)ai=η¯ii0−η¯ii0−,i∈I,and set(18)ηiis=η¯iis,s∈−τ,0,η¯ii0−,s=0,ηiks=η¯iks,i,k∈I,i≠k.In population dynamics,ai>0 implies the existence of the instantaneous self-limitation effect.By the basic theory of functional differential equations [6, 13], we have the following global existence of (6), in which the unknown function is(19)u=u11,u12,…,u1m,u21,…,un−1m,un1,…,unm.Lemma 2.
Assume thatai>0,i∈I. If(20)vijs∈0,∞,s∈−τ,0,i∈I,j∈J,then (6) has a global solution such that(21)uijt∈0,∞,t>0,i∈I,j∈J.Moreover, if(22)∑j∈Icijcjj<2,i∈I,then (6) has a positive steady state(23)u∗=u1∗,…,u1∗︷m,u2∗,…,un−1∗,un∗,…,un∗︷m≫0.Once the global existence is obtained, we shall prove the following result on the global asymptotic stability of the positive steady state, which also implies the uniqueness of the positive steady state of (6).Theorem 1.
Assume thatai>0,i∈I, such that (20) holds and(24)∑j∈Icijcjjaj<2,i∈I.
For eachi∈I, if there exists ji∈J such that viji0>0, then(25)limt⟶∞uijt=ui∗,i∈I,j∈J.
In the following, we shall prove Theorem1 by several lemmas. The first is the following result on the positivity of solutions.Lemma 3.
Assume thatai0>0 for some i0∈I. If(26)vi0j00>0,for somej0∈J,then ui0jt>0,t>0,j∈J, and(27)limsupt⟶∞uijt≤ciiai−1,i=i0,j∈J.Proof.
In the proof, we fixi=i0 for simplicity. By djli>0,j,l∈J, it is clear that uijt>0,t>0,j∈J. We now prove (27) by Lemma 2. From the positivity, we have(28)duijtdt≤∑l=1mdjli∫−τ0uilt+sdρjlisds−uijt+riuijt1−ciiaiuijt,t>0,uijs=vijs≥0,vij0s>0,for somej0∈J,j∈J,s∈−τ,0.
Further consider(29)du¯ijtdt=∑l=1mdjli∫−τ0u¯ilt+sdρjlisds−u¯ijt+riu¯ijt1−ciiaiu¯ijt,t>0,u¯ij0=vij0≥0,vij00>0,for somej0∈J,j∈J,i∈I,s∈−τ,0.
Then, the comparison principle leads to(30)u¯ijt≥uijt>0,j∈J,t>0,since djli>0,j,l∈J, implies that (29) is a cooperative, irreducible system. By the theory of monotone dynamical systems [6], we have(31)limt⟶∞u¯ijt=ciiai−1,j∈J.
Using the comparison principle of cooperative systems, we have (27) for such a fix i∈I. We complete the proof.
Consider the initial value problem(32)dwitdt=riwit1−∑k=1ncik∫−τ0wkt+sdη¯iks,t>0,i∈I,wis=ωis>0,s∈−τ,0,i∈I.
After direct calculation, we have the following conclusion by Lemma1 (we also refer to Smith [6], Lemma 7.4, and Appendix).Lemma 4.
Assume that (24) holds. Define(33)aiy=yui∗,biy=yui∗+1−yciiai−1+ε,y∈0,1,ay=a1y,a2y,…,any,by=b1y,b2y,…,bny,y∈0,1,with ε>0. If there exists ε0>0 such that(34)ωis∈0,ciiai−1+ε0,s∈−τ,0,i∈I,and ay,by with ε=ε0 satisfy (H3) and (H4) in (32), then(35)limt⟶∞wit=ui∗,i∈I.Lemma 5.
Assume that (24) holds and ε0>0 satisfies Lemma 4. Then, there exist t0>0andy0>0 such that(36)uijt0+s∈aiy0,biy0,i∈I,J∈J,s∈−τ,0,ε=ε0.Proof.
By (27), we have t0>τ such that(37)uijt+s<ciiai−1+ε0,i∈I,j∈J,s∈−τ,0,t>t0.
Due to the positivity of solutions in Lemma3, the result is true by selecting y0>0 small enough. The proof is complete.
By what we have done, we now give the proof of Theorem1.Proof.
Define(38)Aijy=aiy,Bijy=biy,i∈I,j∈J,y∈0,1,with ε0>0 small enough. By direct analysis, we can verify that Aijy,Bijy satisfy (H3) and (H4) for (6) if (24) holds (we present the calculation in Appendix). By Lemmas 1 and 5, we complete the proof.


## 4. Numerical Simulation

In this part, we simulate the dynamics of (6) when two patches and three species are involved by selecting some parameters, which may show the plentiful dynamics of such a delayed system without monotonicity. Since ρjlis in (6) do not change the stability in our results, we shall not consider the delayed effect modeling the spatial movement among different patches.Example 1.
We first try to simulate the dynamics of a system satisfying Theorem1. For the purpose, we consider the following delayed system:(39)du11tdt=u12t−u11t+u11t1−u11t−0.2u11t−1−0.1u21t−1−0.1u31t−1,du12tdt=2u11t−u12t+u12t1−u12t−0.2u12t−1−0.1u22t−1−0.1u32t−1,du21tdt=u22t−u21t+u21t1−0.1u11t−1−u21t−0.1u21t−1−0.1u31t−1,du22tdt=2u21t−u22t+u22t1−0.1u12t−1−u22t−0.1u22t−1−0.1u32t−1,du31tdt=u32t−u31t+u31t1−0.3u11t−1−0.1u21t−1−u31t−0.1u31t−1,du32tdt=2u31t−u32t+u32t1−0.3u12t−1−0.1u22t−1−u32t−0.1u32t−1,with t>0 and initial condition(40)ui1s=0.05,s∈−1,0,ui2s=0,s∈−1,0,i=1,2,3.
Evidently, (39) satisfies (24) and has a unique positive steady state, of which the approximate value is(41)0.7143,0.7857,0.6429,0.7143,0.7857,0.6429.
From our main result, we see that the positive steady state is asymptotic stable, which is also illustrated in Figure1.Figure 1
Simulation of (39) with (40).Example 2.
We now consider the possible extinction of some unknown functions when Theorem1 does not hold and study the dynamics of the following delayed system:(42)du11tdt=u12t−u11t+u11t1−u11t−0.2u11t−1−u21t−1−u31t−1,du12tdt=2u11t−u12t+u12t1−u12t−0.2u12t−1−u22t−1−u32t−1,du21tdt=u22t−u21t+u21t1−0.1u11t−1−u21t−0.1u21t−1−0.1u31t−1,du22tdt=2u21t−u22t+u22t1−0.1u12t−1−u22t−0.1u22t−1−0.1u32t−1,du31tdt=u32t−u31t+u31t1−0.3u11t−1−0.1u21t−1−u31t−0.1u31t−1,du32tdt=2u31t−u32t+u32t1−0.3u12t−1−0.1u22t−1−u32t−0.1u32t−1,with t>0 and initial condition (40). Evidently, this system does not satisfy (24), and its dynamics may be different from our Theorem 1, which is presented in Figure 2.Figure 2
Simulation of (42) with (40).Example 3.
The weights among the same species partly play the role of irreducibility, which leads to the possible synchronism. What will happen if the condition does not hold? We consider the dynamics of the following delayed system:(43)du11tdt=d1u12t−u11t+u11t1−u11t−0.2u11t−1−0.1u21t−1−0.1u31t−1,du12tdt=u12t1−u12t−0.2u12t−1−0.1u22t−1−0.1u32t−1,du21tdt=d2u22t−u21t+u21t1−0.1u11t−1−u21t−0.1u21t−1−0.1u31t−1,du22tdt=u22t1−0.1u12t−1−u22t−0.1u22t−1−0.1u32t−1,du31tdt=d3u32t−u31t+u31t1−0.3u11t−1−0.1u21t−1−u31t−0.1u31t−1,du32tdt=d4u31t−u32t+u32t1−0.3u12t−1−0.1u22t−1−u32t−0.1u32t−1,with t>0 and initial condition (40), and d1,…,d4 are positive.
In this system, the weights may be zero, and it does not satisfy our condition. By simulation in Figures3 and 4, we may observe different persistence-extinction phenomena with partially degenerate weights.Figure 3
Simulation of (43) with (40) by fixing d1,d2,d3,d4=1,1,1,2.Figure 4
Simulation of (43) with (40) by fixing d1,d2,d3,d4=0.1,0.1,0.1,0.1.Example 4.
Finally, we try to explore the influence of delay size. In the literature, some nontrivial results of the delay size have been reported even in scalar equations [2]. We consider the dynamics of the following delayed system:(44)du11tdt=u12t−u11t+u11t1−0.2u11t−u11t−τ−0.1u21t−τ−0.1u31t−τ,du12tdt=2u11t−u12t+u12t1−0.2u12t−u12t−τ−0.1u22t−τ−0.1u32t−τ,du21tdt=u22t−u21t+u21t1−0.1u11t−τ−0.1u21t−u21t−τ−0.1u31t−τ,du22tdt=2u21t−u22t+u22t1−0.1u12t−τ−0.1u22t−u22t−τ−0.1u32t−τ,du31tdt=u32t−u31t+u31t1−0.3u11t−τ−0.1u21t−τ−0.1u31t−u31t−τ,du32tdt=2u31t−u32t+u32t1−0.3u12t−τ−0.1u22t−τ−0.1u32t−u32t−τ,with τ>0,t>0, and initial value condition(45)u1s=u2s=u3s=0.05,s∈−τ,0,u4s=u5s=u6s=0,s∈−τ,0.
By our simulations in Figure5, we find that large delay may lead to the instability of the positive steady state, while the small delay may be harmless to the stability of the positive steady state.Figure 5
Simulation of (44) with (45) by taking different time delays.


## 5. Discussion

In this paper, we studied a delayed model on a weighted network, which may be regarded as a delayed competition model on patches. We established the global stability of the unique positive steady state by constructing contracting rectangles under condition (24). Moreover, we gave some numerical examples to model the dynamics of (6). From Examples 1–4, we found that the dynamics of this nonmonotone system may be very complex, which includes the stability of different steady states. In particular, if the weight coefficients among some patches are zero, the uniform convergence on different patches may be false even if (24) holds, and we observed this from Example 2. That is, weights among different patches may affect the dynamics of this system, and the role of weights should be further studied.Moreover, if (24) does not hold, then the stability of the positive steady state may depend on the size of time delay, which is partially motivated by the dynamics of the classical delayed logistic model [27]. From Example 4, we observed different convergence results with different delays. To further show the effect of time delay in such a delayed model is an interesting question. However, with the increase in the number of unknown functions, the analysis of eigenvalues becomes complex, and this is a challenging question. We shall further consider these questions in our future works.Finally, in many other mathematical models including uncertain discrete-time descriptor systems with multiple time-varying delays and parametric uncertainties [28], nonholonomic systems with time-varying delays and nonlinear disturbances [29], and uncertain chaotic systems with nonlinearity and time delay [30], it is important to estimate the dynamics in finite time. Due to the effect of time-varying delays, this is not a trivial work and deserves further investigation. Because the contracting rectangles could reflect certain uniform convergence, it is possible to consider the finite-time dynamics of these important models by contracting rectangles, at least when the initial data satisfy proper uniform conditions in the delayed interval.



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