Elsevier

Ecological Complexity

Volume 21, March 2015, Pages 215-232
Ecological Complexity

Dynamic coexistence in a three-species competition–diffusion system

https://doi.org/10.1016/j.ecocom.2014.05.004Get rights and content

Abstract

From the viewpoint of competitor-mediated coexistence, it is important to study the influence of an exotic species on other native species in terms of exogenous effects, in general, exotic species are weaker than native ones because they have evolved in a different environment. Even if an exotic species is weaker, however, it might cooperate with a native species, after which the competitive relations among native species may have reversed. This motivates us to consider the ecological situation whereby one exotic competing species invades the native system of two strongly competing species. Therefore, we discuss the problem of competitive exclusion or competitor-mediated coexistence using a three-species competition–diffusion system.

Introduction

Studies of species diversity in ecological systems are currently being conducted via both field research and from a theoretical standpoint. One simple but suggestive example is that of competitor-mediated coexistence under the situation whereby one exotic competing species (say, W) invades the environment of two native species (say, U and V) that are strongly competing. If U, V and W move by random dispersal, the possibility of such coexistence in the presence of W can be theoretically discussed by the following three-species reaction–diffusion system of Gause–Lotka–Volterra type (e.g., Petrovskii et al., 2001, Adamson and Morozov, 2012):ut=d1Δu+(r1a1ub12vb13w)u,vt=d2Δv+(r2b21ua2vb23w)v,wt=d3Δw+(r3b31ub32va3w)w,where u(t, x), v(t,x) and w(t,x) denote the population densities of U, V and W at time t and position x, respectively. The parameters di, ri, ai and bij (i, j = 1, 2, 3(i  j)), which are all positive constants, represent the diffusion rates, intrinsic growth rates, intra-specific and inter-specific competition rates, respectively. Eq. (1) is often considered in a bounded domain Ω in N(N=1,2,) with boundary conditionsuν=vν=wν=0,t>0,xΩ,where ν is the outward unit normal vector on the boundary ∂Ω of Ω, and the initial conditions areu(0,x)=u0(x)0,v(0,x)=v0(x)0,w(0)=w0(x)0,xΩ.

We start with a two-competing-species system of (1) in the absence of W, that is,ut=d1Δu+(r1a1ub12v)u,vt=d2Δv+(r2b21ua2v)v,t>0,xΩ,with boundary and initial conditionsuν=vν=0,t>0,xΩandu(0,x)=u0(x)0,v(0,x)=v0(x)0,xΩ,respectively.

We first assumea1b21<r1r2<b12a2,which, in ecological terms, implies that U and V are strongly competing. If Ω is convex, it is already known that any spatially nonconstant equilibrium solutions of (4), (5), (6) with (A1) are unstable, even if they exist (Kishimoto and Weinberger, 1985). Furthermore, the stable attractor of (4), (5), (6) consists of equilibrium solutions (Hirsch, 1982). With these results, one can conclude that any positive solution (u(t,x),v(t,x)) of (4), (5), (6) generically converges to either (r1/a1, 0) or (0, r2/a2). This indicates that competitive exclusion occurs between U and V, as shown in Fig. 1.

We consider (1), (2), (3) to study the problem of competitor-mediated coexistence, under the situation where an exotic competing species W invades the (U, V) system. As an earlier study in this direction, we refer to the paper by Ei et al. (1999).

They first assume the situation where W is strongly competing with both U and V in the sense thata3b13<r3r1<b31a1anda2b32<r2r3<b23a3,respectively. Then, for the diffusionless system corresponding to (1),ut=(r1a1ub12vb13w)u,vt=(r2b21ua2vb23w)v,wt=(r3b31ub32va3w)w,t>0,P1 = (r1/a1, 0, 0), P2 = (0, r2/a2, 0) and P3 = (0, 0, r3/a3) are asymptotically stable, and any positive solution (u(t),v(t),w(t)) of (9) generically converges to one of P1, P2, or P3 (van den Driessche and Zeeman, 1998); that is, competitive exclusion occurs among U, V and W, if the influence of diffusion is disregarded. Under this situation, they addressed the following question for (1), (2), (3): does competitor-mediated coexistence occur when the influence of diffusion cannot be disregarded? To answer this, they used the one-dimensional traveling-wave solutions of any two competing species (U, V), (V, W), or (W, U) among U, V and W. The traveling-wave solution of the (U, V) system is given by (u(z),v(z)) (z = x  θuvt) with velocity θuv satisfyingθuvuz=d1uzz+(r1a1ub12v)u,θuvvz=d2vzz+(r2b21ua2v)v,zand boundary conditionslimz(u(z),v(z))=(0,r2/a2),limz+(u(z),v(z))=(r1/a1,0).

There exists a traveling wave solution (u(z),v(z)) with unique velocity θuv that is asymptotically stable under (A1) (Kan-on and Fang, 1996).

Similarly, under (7), (8), there exist asymptotically stable traveling wave solutions (v(z),w(z)) (z = x  θvwt) with velocity θvw and (w(z),u(z)) (z = x  θwut) with velocity θwu which are given byθvwvz=d2vzz+(r2a2vb23w)v,θvwwz=d3wzz+(r3b32va3w)w,limz(v(z),w(z))=(0,r3/a3),limz+(v(z),w(z))=(r2/a2,0),z,andθwuwz=d3wzz+(r3b31ua3w)w,θwuuz=d1uzz+(r1a1ub13w)u,limz(w(z),u(z))=(0,r1/a1),limz+(w(z),u(z))=(r3/a3,0),z,respectively. For (10), (11), they assumedθuv<0,that is, U is stronger than V in space, which corresponds to Fig. 1(a). They also assumedθvw<0andθwu<0for (12), (13). Since (14), (15) indicate that U, V and W possess the property of cyclic competition in space, it is numerically shown that dynamic spiral-like coexistence with triple junctions can occur when d1, d2 and d3 are relatively small, as shown in Fig. 2. This result emphasizes that if each species moves by diffusion, dynamic coexistence is possible, even if competitive exclusion occurs for (u(t),v(t),w(t)) in (9). Though a rigorous proof has not been given, such a coexistence can be intuitively expected because of the cyclic competition of U, V and W in space.

In this paper, we consider a different situation from the above, that is, for (V, W) system, we assumea2b32<r2r3andb23a3<r2r3in place of (8), and for (W, U) system,a3b13<r3r1<b31a1,which implies ecologically that W and U are strongly competing. (A1), (7), (8), (9), (10), (11), (12), (13), (14), (15), (A2), (A3) implies that P1 and P2 are asymptotically stable, while P3 is unstable. Moreover, we assume

Eitherthepositivecriticalpointof(9)doesnotexistoritisunstable,evenifitexists.

Then, any positive solution (u(t),v(t),w(t)) of (9) generically tends to either P1 or P2 (Hofbauer and Sigmund, 1998, Zeeman, 1993). This means that the exotic species w(t) is weak in the sense that it does not influence the dynamics of u(t) and v(t) asymptotically when the effect of diffusion is disregarded.

We now consider the problem (1), (2), (3). We first note that (A2) leads to θvw < 0 (Kan-on, 1997). In addition, under (A2), (A3), we assumeθuv<0andθwu<0.

This situation is similar to the previous one by Ei et al. (1999) except W is a weak species, in the sense that U, V and W possess the property of cyclic competition in space.

Under (A2), (A3), (A4), (A5), (A6), we consider the possibility of competitor-mediated coexistence when a weak species W invades the native (U, V) system.

We numerically study the two-dimensional problem of (1), (2), (3) in a square domain Ω = (0, L) × (0, L) with L = 200, where the parameters in (1) are specified asr1=r2=r3=r=28,a1=a2=a3=a=1,b12=2221,b13=4,b21=3721,b31=2621,b32=2221and b23 is taken as a free parameter satisfying0<b23<1,which is required from (A1), (7), (8), (9), (10), (11), (12), (13), (14), (15), (A2), (A3), (A4). Under (16), (17), there are four saddle nodes P3 = (0, 0, 28), P4 = (1764/83, 0, 140/83), P5 = (588/373, 9408/373, 0), and P6 = (1764/(110b23 + 699), (2940b23 + 16464)/(110b23 + 699), 140/(110b23 + 699)) in +3. When b23 = 0.25, for instance, the trajectory of solutions (u(t),v(t),w(t)) of (9) in (u,v,w)-space is shown in Fig. 3.

For the diffusion constants di(i = 1, 2, 3), we simply assumed1=d2=d3=1.

We color the areas of U, V and W as follows. First, by using a certain threshold value ϵ(= (0.05r)/a = 1.4), we divide Ω into the following eleven areas: (I) u,v,wϵ, (II) uv>ϵ>w, (III) v>u>ϵ>w, (IV) v>w>ϵ>u, (V) wv>ϵ>u, (VI) u>w>ϵ>v, (VII) wu>ϵ>v, (VIII) uϵ>v,w, (IX) vϵ>w,u, (X) wϵ>u,v, (XI) ϵ>u,v,w. Because the areas of (II) + (VIII) is mainly occupied by U, we color it light gray. Similarly, as the areas of (III) + (IX) and (V) + (VII) + (X) are mainly occupied by V and W, we color these areas dark gray and black, respectively. Furthermore, to emphasize the areas of the weak species W, we color the areas of (I) + (IV) + (VI) black. Finally, the area of (XI) is colored white.

When b23 = 0.2, two interesting behaviors can be observed, as shown in Fig. 4. First, the domain Ω is eventually occupied by V alone. This indicates that the invasion of W reverses the competitive relation between U and V, because U finally occupies the domain Ω in the absence of W, as was shown in Fig. 1(a). Second, V forms expanding disk-shaped patterns and, in the vicinity of the boundaries of these disks, W survives. This suggests the occurrence of traveling-wave-like behavior in the three species U, V and W.

When b23 = 0.4, the behavior of solutions arising in Fig. 5(a)–(c) is similar to that in Fig. 4(a)–(c), respectively. Subsequently, the behavior becomes drastically different from the case of b23 = 0.2. Fig. 5(f)–(h) illustrates the very complex spatio-temporal coexistence of U, V and W.

When b23 = 0.6, spatio-temporal coexistence can still be observed, but the behavior is quite regular, as shown in Fig. 6. This exhibits a pair of rotating spirals and target waves, which resemble those observed in the Belousov–Zhabotinsky reaction.

When b23 = 0.8, the behavior is simple, as in Fig. 7. The asymptotic behavior of solutions is similar to the case when b23 = 0.2, that is, competitive exclusion occurs.

Our purpose is to reveal the mechanism whereby either competitive exclusion or competitor-mediated coexistence appears, depending on values of b23, as shown in Fig. 4, Fig. 5, Fig. 6, Fig. 7.

The remainder of this paper is organized as follows: In Section 2, motivated by Fig. 4(e)–(g), we consider three-species traveling-wave solutions (u(z),v(z),w(z)) (z = x  θt) with velocity θ, which provide essential information about the occurrence of either competitive exclusion or the competitor-mediated coexistence of U and V in the presence of W. In Section 3, we discuss the one-dimensional interaction of two- and/or three-species traveling-wave solutions for various values of b23. In Section 4, we discuss the two-dimensional interaction of these traveling waves. In Section 5, we focus on the complex spatio-temporal coexistence of U, V and W shown in Fig. 5, and reveal the mechanism behind such dynamic coexistence. In Section 6, we discuss the occurrence of steadily rotating spirals, and finally, in Section 7, present some concluding remarks on our results.

Section snippets

Three-species traveling-wave solutions

In Fig. 4, Fig. 7, we can observe expanding disk-shaped patterns of V and surviving regions of W in the vicinity of the disk boundaries. This behavior suggests the occurrence of three-species traveling-wave solutions of the form (u(z),v(z),w(z)) (z = x  θt) with velocity θ, satisfyingθuz=d1uzz+(r1a1ub12vb13w)u,θvz=d2vzz+(r2b21ua2vb23w)v,θwz=d3wzz+(r3b31ub32va3w)w,z,andlimz(u(z),v(z),w(z))=0,r2a2,0,limz+(u(z),v(z),w(z))=r1a1,0,0.

For the problem of (19), (20), it is already known

One-dimensional interaction of traveling waves

To understand the two-dimensional dynamics of solutions to (1), (2), (3) in Fig. 4, Fig. 5, Fig. 6, Fig. 7, we numerically consider the one-dimensional interaction of the two-species and stable three-species traveling waves in Fig. 8.

Two-dimensional interaction of two-species and stable three-species traveling waves

In the previous section, we discussed the one-dimensional interaction of two types of traveling-wave solutions. In this section, we consider the two-dimensional interaction of these traveling waves in the square domain Ω = (0, L) × (0, L) with L = 200.

We first numerically confirmed that the two-species traveling-wave solution is planarly stable for any d1 and d2, whereas the stable three-species traveling wave solution does not necessarily possess this property, even if it exists. In fact, if d1 = d2 = d3

Occurrence of an irregular spatio-temporal pattern

When b23 = 0.4, Fig. 22 shows the occurrence of a complex spatio-temporal pattern in which many clusters appear dynamically and irregularly. To understand the reason why such small clusters are generated, we should note that there are two different types of one-dimensional interaction between the traveling waves. The first is the interaction of the two- and three-species traveling waves, in which a new three-species traveling wave appears and moves in the opposite direction, as if it had been

Occurrence of steadily rotating spirals

When b23 = 0.6, Fig. 15 shows a homoclinic-type traveling-wave solution of (19), (23). As b23 increases to 0.75…, the width of the pulse of w in the homoclinic-type traveling wave gradually increases, and when b23 reaches about 0.75…, the wave no longer exists. On the other hand, when b23 decreases to around 0.49, it destabilizes in an oscillatory manner. Then, we can expect that the branch of stable homoclinic-type traveling-wave solutions of (19), (23) exists in some interval of b23

Concluding remarks

We have discussed the possibility of competitor-mediated coexistence using the three-species competition–diffusion system given by (1). Specifically, we considered the situation where one weak exotic competing species invades a system in which two species are strongly competing. We found that competitor-mediated coexistence occurs, despite the exotic species being “weak.” However, solutions to (1), (2), (3) are very sensitive to the parameters in (1). Taking b23 as a free parameter, we

Acknowledgements

We would like to thank Hideo Ikeda for his suggestion on the stability condition (21). We are grateful for the support of the Global COE program (G14) “Formation and Development of Mathematical Science based on Modeling and Analysis.”

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