Elsevier

Mathematical Biosciences

Volume 318, December 2019, 108271
Mathematical Biosciences

Review
The effect of random dispersal on competitive exclusion – A review

https://doi.org/10.1016/j.mbs.2019.108271Get rights and content

Highlights

  • All known results on the problem, which has been open for more than 30 years, are presented.

  • These results are surprisingly consistent and lead to a conjecture that is tested numerically.

  • Biological background and several possible applications in various fields of biology are provided.

Abstract

Does a high dispersal rate provide a competitive advantage when risking competitive exclusion? To this day, the theoretical literature cannot answer this question in full generality. The present paper focuses on the simplest mathematical model with two populations differing only in dispersal ability and whose one-dimensional territories are spatially segregated. Although the motion of the border between the two territories remains elusive in general, all cases investigated in the literature concur: either the border does not move at all because of some environmental heterogeneity or the fast diffuser chases the slow diffuser. Counterintuitively, it is better to randomly explore the hostile enemy territory, even if it means highly probable death of some individuals, than to “stay united”. This directly contradicts a celebrated result on the intermediate competition case, emphasizing the importance of the competition intensity. Overall, the larger picture remains unclear and the optimal strategy regarding dispersal remains ambiguous. Several open problems worthy of a special attention are raised.

Introduction

The interplay between dispersion and competition is a vast and important problem in theoretical population biology, with applications in ecology but also in evolution (natural selection precisely originates in the interplay between competitive pressure and mutations, namely “dispersion” in the phenotypical space), epidemiology (competition between pathogen strains during spreading epidemics), medicine (populations being in this context cell populations). This interplay leads to qualitative outcomes (displacement, segregation, etc.) that would not appear in the spatially homogeneous, well-mixed counterpart, which makes them difficult to predict. But such predictions are of the utmost importance, especially since these outcomes usually result in some form of spatialized extinction process, which could be a goal or on the contrary something to be avoided, depending on the exact biological problem. An exhaustive overview of the biology and mathematical biology literature on this wide topic is of course impossible; the reader is referred for instance to some recent works and references therein [3], [13], [19], [27], [56], [60], [62], [65].

A common phenomenological mathematical model to study this interplay, inspired by the general population dynamics equation:variationintimeofthepopulationsize=dispersion+birthsdeaths,is the deterministic, diffusive and competitive Lotka–Volterra system [32], [51], [75]:{tN1=·(d1N1)+r1N1(1N1K1)c1N1N2,tN2=·(d2N2)+r2N2(1N2K2)c2N1N2,where N1(t, x) ≥ 0 and N2(t, x) ≥ 0 are two continuous population densities depending on time t (a real variable) and space x (a Euclidean variable x=(x1,x2,)), t=t is the partial derivative with respect to time, =(x1,x2,) is the nabla operator (so that ∇• is the spatial gradient and ∇ · • is the spatial divergence), d1(t, x) and d2(t, x) are the diffusion (dispersal) rates, r1(t, x) and r2(t, x) are the intrinsic (per capita) growth rates, K1(t, x) ≥ 0 and K2(t, x) ≥ 0 are the carrying capacities, c1(t, x) ≥ 0 and c2(t, x) ≥ 0 are the interpopulation competition rates. The dependencies on time and space of the various coefficients account for instance for seasonality or unfavorable regions of space.

Note that in absence of one population, the other grows logistically (no Allee effect) and its density solves the well-known Fisher–Kolmogorov–Petrovskii–Piskunov partial differential equation (PDE) [29], [50], [72]:tNi=·(diNi)+riNi(1NiKi).Note also that the paper adopts an ecology vocabulary for the sake of simplicity but, again, the model is abstract and quite general: “individuals” could be cancer cells or infected hosts, etc.

Assuming for a moment that the environment is spatio-temporally homogeneous (heterogeneous environments will come back later on), all the coefficients become positive constants and space, time and the two population densities can be nondimensionalized to obtain the reduced system:{tu=Δu+u(1u)huv,tv=dΔv+rv(1v)kuv,where d, r, h, k are positive constants that can be estimated using field data and where the Fickian diffusion operator reduces to the simpler spatial Laplacian (corresponding to isotropic Brownian motion of individuals). Such a PDE system has indeed been used and discussed extensively for more than fifty years by modelers in biology. A non-exhaustive list illustrating the variety of biological applications includes for instance studies on the competitive displacement of the red squirrel by the invasive grey squirrel in the British Isles [61], optimization of cancer therapy taking into account the competition between cancer cells that are sensitive to the treatment and those that are resistant to it [20], [21], biodiversity conservation in fire-prone savannas accounting for competition for light and nutrients between trees and grass [78], reproduction–dispersion trade-offs in experimental bacterial invasions used to study the evolution of dispersal [23].

This synthesis is concerned with results investigating whether the population v, in order to outcompete the population u, should have a high or low dispersal rate d, all else being equal. Here, all else being equal means that the two populations only differ in dispersal rate, namely r=1 and h=k:{tu=Δu+u(1u)kuv,tv=dΔv+v(1v)kuv.The symmetry assumption (h=k and r=1) prevents pure reaction-driven extinctions that would strongly perturb the analysis (in other words, cases where, for instance, one competitor feels much less pressure than the other and might prevail despite a poorly chosen dispersal strategy are discarded). This assumption will simplify a lot the forthcoming presentation, although many results actually remain true under specific, yet more general, assumptions on h and r.

In spatially homogeneous, well-mixed cases, the system (2) is strongly determined by the sign of k1 (e.g., [45, Chapter 7, Section 7.9]). On one hand, in the weak competition case (k < 1), the system is systematically driven to coexistence. On the contrary, in the strong competition case (k > 1), both populations are able to wipe out the other provided they are numerically sufficiently superior. The intermediate case (k=1), corresponding to a competitive pressure exerted completely blindly, is degenerate (all pairs (u, v) satisfying u+v=1 are steady states) and is usually discarded. Note that “blind” competition means here that one individual competes uniformly with all surrounding individuals, independently of the population to which they belong; the population label of competitors is in some sense not seen, not taken into account.

In spatially structured systems, the picture is more complicated. A very important paper of mathematical biology, due to Dockery, Hutson, Mischaikow and Pernarowski [24], established that the blind competition case k=1 becomes relevant in spatially heterogeneous environments with an intrinsic growth rate a(x):{tu=Δu+u(au)uv,tv=dΔv+v(av)uv.Assuming that the domain where the populations evolve is bounded with no-flux boundary conditions (say, an island or a Petri dish), the authors showed that v wipes out u whenever it is the slower diffuser, d < 1. Of course, by symmetry, u wipes out v if d > 1. This was interpreted as follows: because of the interplay between heterogeneity and competition, it is a better strategy to claim favorable areas and to defend them collectively by remaining concentrated there than to randomly explore unfavorable areas, where deaths due to the environment are more likely. In other words, v wins if and only if, compared to u, its individuals remain “united” instead of venturing alone in unknown areas. In the present paper, such a result is referred to as a “Unity is strength”-type result. The analysis of Dockery et al. relied entirely upon the monostability of the system induced by the combination of spatial heterogeneity and difference in diffusion rates: the only stable steady state is the one where the slow diffuser persists while the fast diffuser vanishes. Initial conditions do not matter: even if, initially, the fast diffuser is vastly superior numerically, the slow diffuser will eventually prevail.

But what if both semi-extinct steady states are stable, so that the stability analysis does not suffice to conclude and initial conditions matter? As explained above, bistability is for instance achieved in spatially homogeneous systems (2) with strong competition (k > 1). With such systems, the success of a dispersal strategy is a more delicate notion that can be defined in a few ways.

For instance, the diffusion-induced extinction property could be used to define this success. This criterion uses initial conditions that are exactly calibrated so that neither u nor v takes over in the absence of diffusion or with equal diffusion rates (d=1). Given such initial conditions, what is the outcome when taking the unequal diffusion into account? In homogeneous environments, where this balance condition simply reads u(0,x)=v(0,x) at every x, and provided the habitat is one-dimensional with no-flux boundary conditions and the interpopulation competition rate k is equal to 2, Ninomiya [58] showed that there exist values of d larger than 1 but close to it and carefully chosen initial conditions satisfying the above condition such that v wipes out u. The fast diffuser prevails: in this sense, “Unity is not strength” (one could even say “Disunity is strength”).

Nevertheless, this definition of success is unsatisfying, as it uses very precise initial conditions that are in some sense artificial and would not appear in the nature. Is there a more robust and natural definition (that might a priori disagree with the conclusions of Ninomiya [58] and agree with those of Dockery et al. [24])?

The strong competition case k > 1 is also known in the mathematical ecology literature as the competitive exclusion case [32]: persistence of both species can occur only if the two niches are differentiated. In the setting of this paper, niches are purely geographical, and their differentiation means that, roughly speaking, u is positive where v is close to 0 and vice-versa (note the sharp contradistinction with Ninomiya’s balance condition u(0,x)=v(0,x)). If the territories are segregated, then borders between these territories naturally arise. At these interfaces, the two populations meet frequently and individuals compete fiercely to chase competitors and take over. In this context, it seems natural to track the motion of the interface and to define a dispersal strategy as successfull if it leads to taking over the territory of the opponent, namely to territorial expansion.

This definition agrees with situations studied in the biological literature (e.g., [13], [15], [16], [57]).

Mathematically, this definition translates in homogeneous environments into the study of a particular solution of the system (2), referred to as a traveling wave, that has a constant profile and a constant speed and evolves in the infinite real line (approximating a very large one-dimensional domain where propagation phenomena matter). This solution is illustrated in Fig. 1. Its existence and its uniqueness were confirmed in the ’80s and ’90s [30], [46], [74]. In this context, the success of the dispersal strategy is simply given by the sign of the speed of the wave. However, in contrast with the existence and uniqueness of the wave, this sign is in general a very difficult mathematical problem, that cannot be solved by any standard tool of the analysis of PDEs. Only partial results are known and these are the main topic of this synthesis paper. It turns out that they are all in agreement with Ninomiya [58]: in situations of competitive exclusion due to strong interpopulation competition, “Unity is not strength”.

The paper is organized as follows. Section 2 gives an exhaustive survey of these partial results, the last subsection being devoted to a delicate extension in spatially periodic media. Section 3 lists some open problems that should, in my opinion, attract the attention of the community.

Section snippets

Known results

In this section the known results are presented.

Open problems

In this section, directions of research worthy of a special attention from the community are raised. Although the focus remains on theoretical questions and viewpoints, comparisons with in vitro models and field data are also obviously interesting and important.

Conclusion

All existing results on strongly competing systems in homogeneous environments and with populations differing only in diffusion rate concur: “Disunity is strength”. In other words, the fast dispersers win and chase the slow dispersers. Although the map of rigorous results (Fig. 2) is far from being complete, numerical investigations (Fig. 5) tend to confirm this is always the case: whatever the values of the two parameters (the interpopulation competition rate and the ratio between the

Conflict of interest

The author declares no competing interest.

Acknowledgments

The author thanks Florence Débarre for the attention she paid to this work and two anonymous referees for very valuable suggestions.

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    This work was supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH. This work has been carried out in the framework of the NONLOCAL project (ANR-14-CE25-0013) funded by the French National Research Agency (ANR)

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