On the competitive exclusion principle for continuously distributed populations

We investigate the competitive exclusion principle in the case of a continuously distributed initial population. We introduce an ordinary differential equations model of species competing for a single resource, whose efficiency is encoded by a continuous variable (the “trait variable”) living in a Euclidean space. The differential equation is solved in a measure space to allow the observation of the natural concentration of the species distribution on specific traits. We show the concentration of the distribution on the maximizing set of the fitness function in the sense of the Kantorovitch-Rubinstein metric. In the case when the initial weight of the maximizing set is positive, we give a precise description of the convergence of the orbit, including a formula for the asymptotic distribution. We also investigate precisely the case of a finite number of regular global maxima and show that the initial distribution may have an influence on the support of the eventual distribution. In particular, the natural process of competition is not always selecting a unique species, as it would be predicted by the competitive exclusion principle, but several species may coexist as long as they maximize the fitness function. In many cases it is possible to compute the eventual distribution of the surviving competitors. In some configurations, species that maximize the fitness may still get extinct depending on the shape of the initial distribution and on an other parameter of the model, and we provide a way to characterize when this unexpected extinction happens. In this paper we investigate the large time behavior of the system    St(t) = Λ− θS(t)− S(t) ∫ RN α(x)γ(x)I(t, dx), x ∈ R , It(t, dx) = ( α(x)S(t) − 1 ) γ(x)I(t, dx), (1) equipped with suitable non-negative initial data. This system of equations intervenes in theoretical ecology, in epidemiology and in genetics. In ecology, it describes the evolution of a population of individuals with density I that compete for a single limited resource with density S = S(t). In epidemiology, this system describes the evolution of the number of susceptible S and of the density of infected individuals I who have been contaminated by different pathogen variants of some disease. In both cases, the density of individuals I is represented by a measure on the space R , for some integer N > 0, of phenotypic traits. Representing the density I by a measure is natural in this context as one expects concentration properties of the density on one or several optimal traits as time goes to infinity. Note also that this model does not take into account mutations but only pure competition (or selection). In the context of ecology, phenotypical traits are related to the competitors and how they interact with the resource via the continuous functions α and γ, while the last two parameters Λ > 0 and θ > 0 are positive constants which encode the renewal and disappearance rates of the resource. The “Competitive exclusion principle” states that “Complete competitors cannot coexist”, which in particular means that given a number of species competing for the same resource in the same place, only one can survive in the long run. This idea was already present to some extent in the book of Darwin, and is sometimes referred to as Gause’s law [19]. The aim of the present paper is to investigate this principle in the case of a continuously distributed initial population. We will show in particular that, while it is true that the species have to maximize the fitness function in order to survive, the natural process of competition is not selecting a unique species but several species may coexist as long as they maximize the fitness function. In many cases it is possible to compute the eventual repartition of the surviving competitors. In some cases, species that maximize the fitness may still get extinct if the initial population is not sufficient, and we provide a way to characterize when this unexpected extinction happens. Considering a situation where R0 has more than one maximum at the same exact level may appear artificial but is not without biological interest. Indeed, the long-time behavior that we observe in these borderline cases can AMS subject classifications (2020). 34D05, 92D25, 34A06, 28A33