ReviewThe effect of random dispersal on competitive exclusion – A review☆
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:is the deterministic, diffusive and competitive Lotka–Volterra system [32], [51], [75]: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 ), is the partial derivative with respect to time, 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]: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: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 and :The symmetry assumption ( and ) 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 (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 (), corresponding to a competitive pressure exerted completely blindly, is degenerate (all pairs (u, v) satisfying 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 becomes relevant in spatially heterogeneous environments with an intrinsic growth rate a(x):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 (). Given such initial conditions, what is the outcome when taking the unequal diffusion into account? In homogeneous environments, where this balance condition simply reads 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 ). 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.
References (80)
- et al.
Existence and stability of time periodic traveling waves for a periodic bistable Lotka–Volterra competition system
J. Differ. Equ.
(2013) - et al.
Habitat structure mediates spatial segregation and therefore coexistence
Landsc. Ecol.
(2014) - et al.
Evolution at the edge of expanding populations
Am. Nat.
(2019) - et al.
The evolution of slow dispersal rates: a reaction diffusion model
J. Math. Biol.
(1998) Analysis of a population model with strong cross-diffusion in unbounded domains
Proc. Roy. Soc. Edinburgh Sect. A
(2008)- et al.
Global models of growth and competition
Proc. Natl. Acad. Sci.
(1973) - et al.
Creeping fronts in degenerate reaction-diffusion systems
Nonlinearity
(2005) Asymptotic behavior and stability of solutions of semilinear diffusion equations
Publ. Res. Inst. Math. Sci.
(1979)- et al.
Fortune favours the brave: movement responses shape demographic dynamics in strongly competing populations
J. Theor. Biol.
(2017) - et al.
Biological Invasions: Theory and Practice
(1997)
Traveling wave solutions of parabolic systems
Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media
J. Statist. Phys.
Propagation phenomena for a reaction–advection–diffusion competition model in a periodic habitat
J. Dyn. Differ. Equ.
Travelling waves in near-degenerate bistable competition models
Math. Model. Nat. Phenom.
Reversing invasion in bistable systems
J. Math. Biol.
Competitive coexistence in spatially structured environments: a synthesis
Ecol. Lett.
Spatial waves of advance with bistable dynamics: cytoplasmic and genetic analogues of Allee effects.
Am. Nat.
Intermittent search strategies
Rev. Mod. Phys.
Front blocking and propagation in cylinders with varying cross section
Calc. Var. Partial Differential Equ.
Front propagation in periodic excitable media
Commun. Pure Appl. Math.
Generalized travelling waves for reaction-diffusion equations
Perspectives in Nonlinear Partial Differential Equations
Generalized transition waves and their properties
Commun. Pure Appl. Math.
Analysis of the periodically fragmented environment model. I. Species persistence
J. Math. Biol.
Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating travelling fronts
J. Math. Pures Appl. (9)
Predator-prey models with competition: the emergence of territoriality
Am. Nat.
Competitive exclusion after invasion?
Biol. Invasions
Mechanisms of competitive exclusion between two species of chipmunks
Ecology
Dynamics of populations with individual variation in dispersal on bounded domains
J. Biol. Dyn.
Spatial Ecology
Optimization of an in vitro chemotherapy to avoid resistant tumours
J. Theoret. Biol.
Spreading speeds for a two-species competition-diffusion system
J. Differ. Equ.
The limit equation for the Gross–Pitaevskii equations and S. Terracini’s conjecture
J. Funct. Anal.
Distance-limited dispersal promotes coexistence at habitat boundaries: reconsidering the competitive exclusion principle
Ecol. Lett.
The wave of advance of advantageous genes
Ann. Eugen.
Existence and stability of travelling wave solutions of competition models: a degree theoretic approach
J. Differ. Equa.
The propagation of concentration waves in periodic and random media
Dokl. Akad. Nauk SSSR
The Struggle for Existence
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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)