Stability Analysis of a Two-Patch Competition Model with Dispersal Delays

In this paper, we study a Lotka-Volterra competition model with two competing species moving randomly between two identical patches. A constant dispersal delay is incorporated into the dispersal process for each species. We show that the dispersal delays do not affect the stability and instability of all four symmetric equilibria. Numerical simulations are presented to demonstrate the effect of dispersal delays on the stability and instability of the symmetric coexistence equilibrium.


Introduction
As one of the basic relationships in ecological relationships, competition has been extensively studied. For competition models governed by differential equations of Lotka-Volterra type, we refer to [1,2] and the references therein. Dispersal of species between patches is very common in nature. For example, many zooplankton species move downward into the darkness to reduce the predation risk by fish, while at night time, they move upward to consume the phytoplankton [3]. It is of great importance to examine how the dispersal affects local and global dynamics of the resulting metapopulations. To this purpose, many mathematical models incorporating the dispersal of species over patches have been proposed and studied. For instance, the movement of a single species has been considered in [4][5][6][7][8][9], and models allowing the dispersal of two species have appeared in [10][11][12][13]. See also [14][15][16] for studies on models with competition and predator-prey interactions in patchy environments.
We note that, in the above-mentioned work, dispersal is often assumed to be instantaneous. However, in reality, it always takes time for species dispersing from one patch to another. As a result, dispersal delays do exist and should be incorporated into modeling the dispersal process. With this in mind, recently, Zhang et al. considered a two-patch predator-prey model with delayed dispersal of prey only and showed that such dispersal delay exhibits a stabilizing role on the stability of the coexistence equilibrium [17]. Later, Mai et al. generalized the model considered in [17] into the case with an arbitrary number of patches and showed that the dispersal delay can indeed induce stability switches [18]. Sun et al. considered a two-patch predator-prey model with two dispersal delays and showed that the dispersal delays can destabilize and stabilize the coexistence equilibrium [19].
Motivated by above work, in this paper, we consider a two-species competition model and assume both species move randomly between two patches with two dispersal delays. Our objective is to explore the effects of delays on the stability and instability of symmetric equilibria of the resulting metapopulation system. The rest of the paper is organized as follows. We present our two-patch competition model with two dispersal delays in Section 2. We carry out detailed stability analysis for the resulting symmetric equilibria in Section 3. Numerical simulations are reported in Section 4 to demonstrate the effect of dispersal delays on the stability and instability of the symmetric coexistence equilibrium. A brief summary of our work is given in last section.
2 Discrete Dynamics in Nature and Society short compared to the lifespan of each species, and no mortality will be included in the dispersal. Therefore, our two-patch competition model with delayed dispersal of two competing species are described by the following system: where , ∈ {1, 2} and ̸ = .
and denote the densities of two competing species and in patch ( = 1,2), respectively. 1 and 2 are the intrinsic growth rates of species and , respectively. 1 and 2 are the carrying capacities of species and , respectively. 12 and 21 measure the competition strengths of on and on , respectively. 1 and 2 are the dispersal rates of species and , respectively. 1 and 2 denote the dispersal periods of species and from one patch to the other, respectively. All parameters are assumed to be positive.
Linearizing system (2) at a symmetric equilibrium ( , , , ), we obtain the associated characteristic equation The symmetric equilibrium ( , , , ) is locally asymptotically stable if all characteristic roots have negative real parts [20]. Next we will study the stability of all 4 symmetric equilibria by analyzing the associated characteristic equation det Δ = 0.
The following lemma will be used later in verifying the transversality condition.
Summarizing the above analysis, we have the following result.
. . e Boundary Equilibrium 2 . The stability of the boundary equilibrium 2 can be dealt with similarly as that of 1 and we have the following result. (I) If > 1, then 2 is locally asymptotically stable for 1 ≥ 0 and 2 ≥ 0.
As the characteristic equations (16) have two distinct discrete delays, the analysis becomes very challenging though there has been some excellent work (see, for example, [24,25]). We will mainly use numerical simulations to explore if the dispersal delays would affect the stability and instability of the symmetric coexistence equilibrium * in next section.

Numerical Simulations
In this section, we present some numerical simulations for model (2) to explore the effect of the dispersal delays on the stability and instability of the symmetric coexistence equilibrium * .
We first take parameter values = 0.5, = 0.25, and = 2. This set of parameter values gives the symmetric coexistence equilibrium * = (0.57, 0.86, 0.57, 0.86), which is locally asymptotically stable when = 0 ( = 1, 2) since < 1 and < 1. For 12 sets of different values of 1 , 2 , 1 , and 2 , we obtain numerical solutions in Figure 1, where we only plotted the numerical solutions in one patch. For all 12 sets of parameters values, we find that species and approach * and * , respectively. This is a good indication that the dispersal delays do not affect the stability of the symmetric coexistence equilibrium * .
Next we take parameter values = 2, = 4, and = 2. The symmetric coexistence equilibrium is * = (0.14, 0.43, 0.14, 0.43) and is unstable for = 0, ( = 1, 2). As illustrated in Figures 2 and 3, for different initial conditions, the solutions converge to different boundary equilibria. This implies that the outcome of the competition is initial condition dependent and the dispersal delays do not change the instability of the symmetric coexistence equilibrium * .

Summary
In this paper, we have incorporated dispersal of both species between two patches into a two-species competition model. We have shown that the resulting two-patch competition model admits 4 symmetric equilibria: the trivial equilibrium 0 , two boundary equilibria 1 and 2 , and the symmetric coexistence equilibrium * . For 0 , 1 , and 2 , we have analytically proven that the dispersal delays do not affect their stability and instability. For the symmetric coexistence equilibrium * , though we were not able to provide analytical analysis due to the complexity arisen from two delays, we have numerically demonstrated that the dispersal delays are also harmless in the sense that they do not affect the stability and instability of the symmetric coexistence equilibrium.   This conclusion differs from that of predator-prey models discussed in [9,[17][18][19], where the dispersal delays can induce stability switches.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.