Abstract
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.
1. 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 [4–9], and models allowing the dispersal of two species have appeared in [10–13]. See also [14–16] 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. The Two-Patch Competition Model
In this paper, we assume that the two patches are identical and the two competing species move randomly between two identical patches. Moreover, the dispersal period is relatively 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 and . and denote the densities of two competing species and in patch (), respectively. and are the intrinsic growth rates of species and , respectively. and are the carrying capacities of species and , respectively. and measure the competition strengths of on and on , respectively. and are the dispersal rates of species and , respectively. and denote the dispersal periods of species and from one patch to the other, respectively. All parameters are assumed to be positive.
By rescaling , , , , , , , and , we obtain the dimensionless form of system (1)where and .
For the case with no dispersal (i.e.,), the dynamics of the competition model in a single patch is given below (See [2]).
Theorem 1. Consider a single patch competition model governed byThe following conclusions hold: (1)There are three trivial equilibria: , , and .(2)The coexistence equilibrium exists if and only if , or , , where and .(3)If and , then the two boundary equilibria and are both stable.(4)The coexistence equilibrium is stable if and and is a saddle point if and .(5)The boundary equilibrium is stable if and unstable if .(6)The boundary equilibrium is stable if and unstable if .
3. Stability Analysis of the Symmetric Equilibria
Since the two patches under consideration are assumed to be identical, system (2) admits four symmetric equilibria as follows:
: extinction of both species in each patch;
: extinction of species and persistence of species in each patch;
: extinction of species and persistence of species in each patch;
: coexistence of both species in each patch, which exists as long as , or , .
Linearizing system (2) at a symmetric equilibrium , we obtain the associated characteristic equation given by with where and
The symmetric equilibrium is locally asymptotically stable if all characteristic roots have negative real parts [20]. Next we will study the stability of all symmetric equilibria by analyzing the associated characteristic equation .
3.1. The Trivial Equilibrium
Note that Thus, at the trivial equilibrium , the characteristic equations consist of the following transcendental equations:If , then the characteristic roots are , . Clearly, there exist at least two positive characteristic roots and . Thus is unstable.
According to the results of [21–23], the change of stability at the equilibrium can happen only if characteristic roots appear on or cross the imaginary axis as increases. We assume that and to ensure that zero is not a root of (5) and (7), respectively.
The following lemma will be used later in verifying the transversality condition.
Lemma 2. Suppose, at certain , the characteristic equationadmits a pair of purely imaginary roots (). Then
Proof. For (9), it is easy to get Thus Consequently, . The proof is complete.
Next we look for a pair of purely imaginary roots of characteristic equations (5)-(8) by setting with . Substituting into (5)-(6), we obtain Thus if , then (5)-(6) admit no purely imaginary roots; while if , then there is a pair of purely imaginary roots for (5) and also for (6). Further we have by Lemma 2. This implies that if , the characteristic roots of (5) and (6) cross the imaginary axis through at from left to right and the number of characteristic roots with positive real parts is increased by , as crosses the critical value.
Similarly, for (7) and (8), we can show that there are no imaginary roots if ; while if , then there exists a pair of purely imaginary roots with . Note that is unstable for . Thus the above analysis shows that the trivial equilibrium remains unstable for and and hence we have the following result on the stability of the trivial equilibrium .
Theorem 3. Consider system (2). The trivial equilibrium is unstable for and .
3.2. The Boundary Equilibrium
At the boundary equilibrium , the characteristic equations read asWhen , (), the characteristic roots of equations (12)-(15) are . Here we have two cases to consider: Case (1): , all roots are negative, and the equilibrium is locally asymptotically stable; and Case (2): , there are at least one positive root, and the equilibrium is unstable.
We also assume that to ensure that is not a characteristic root of (14). Substituting with into (12)-(15), we find that if (12) or (13) admits purely imaginary roots , then ; and if (14) or (15) admits purely imaginary roots , then However, does not exist since . If , then can never exist; while if and , then can never exist either; only if and , there are purely imaginary roots for (14) and (15). However, it follows from Lemma 2 that . Consequently, the boundary equilibrium remains unstable for all and .
Summarizing the above analysis, we have the following result.
Theorem 4. For the boundary equilibrium of system (2), we have the following conclusions: (i)If , then is locally asymptotically stable for and .(ii)If , then is unstable for and .
3.3. The Boundary Equilibrium
The stability of the boundary equilibrium can be dealt with similarly as that of and we have the following result.
Theorem 5. For the boundary equilibrium of system (2), we have the following conclusions: (I)If , then is locally asymptotically stable for and .(II)If , then is unstable for and .
3.4. The Symmetric Coexistence Equilibrium
In this subsection, we assume that , or , to ensure the existence of the symmetric coexistence equilibrium . At the coexistence equilibrium , the associated characteristic equations reduce towhere and . If , , or , , then and .
When , the characteristic equations become Therefore, the symmetric coexistence equilibrium is locally asymptotically stable if , , while it is unstable if .
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.
4. 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 , , and . This set of parameter values gives the symmetric coexistence equilibrium , which is locally asymptotically stable when since and . For sets of different values of , , , and , we obtain numerical solutions in Figure 1, where we only plotted the numerical solutions in one patch. For all 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 , , and . The symmetric coexistence equilibrium is and is unstable for . 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 .
5. 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 symmetric equilibria: the trivial equilibrium , two boundary equilibria and , and the symmetric coexistence equilibrium . For , , and , 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, 17–19], 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.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (nos. 11526183 and 11371313), the National Natural Science Foundation of Shanxi Province (no. 201801D221035), and the Foundation of Yuncheng University (nos. YQ-2017003 and XK-2018033).