The evolutionary limit for models of populations interacting competitively via several resources (original) (raw)
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Time fluctuations in a population model of adaptive dynamics
Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2013
We study the dynamics of phenotypically structured populations in environments with fluctuations. In particular, using novel arguments from the theories of Hamilton-Jacobi equations with constraints and homogenization, we obtain results about the evolution of populations in environments with time oscillations, the development of concentrations in the form of Dirac masses, the location of the dominant traits and their evolution in time. Such questions have already been studied in time homogeneous environments. More precisely we consider the dynamics of a phenotypically structured population in a changing environment under mutations and competition for a single resource. The mathematical model is a non-local parabolic equation with a periodic in time reaction term. We study the asymptotic behavior of the solutions in the limit of small diffusion and fast reaction. Under concavity assumptions on the reaction term, we prove that the solution converges to a Dirac mass whose evolution in time is driven by a Hamilton-Jacobi equation with constraint and an effective growth/death rate which is derived as a homogenization limit. We also prove that, after long-time, the population concentrates on a trait where the maximum of an effective growth rate is attained. Finally we provide an example showing that the time oscillations may lead to a strict increase of the asymptotic population size.
Comptes Rendus Mathematique
In this note, we characterize the solution of a system of elliptic integro-differential equations describing a phenotypically structured population subject to mutation, selection and migration. Generalizing an approach based on Hamilton-Jacobi equations, we identify the dominant terms of the solution when the mutation term is small (but nonzero). This method was initially used, for different problems from evolutionary biology, to identify the asymptotic solutions, while the mutations vanish, as a sum of Dirac masses. A key point is a uniqueness property related to the weak KAM theory. This method allows to go further than the Gaussian approximation commonly used by biologists and is an attempt to fill the gap between the theories of adaptive dynamics and quantitative genetics. Résumé Méthode Hamilton-Jacobi pour décrire deséquilibresévolutives dans les environnements hétérogènes et avec des mutations non-évanescentes. Dans cette note, nousétudions un système d'équations intégrodifférentielles elliptiques, décrivant une population structurée par trait phénotypique soumiseà des mutations,à la sélection età des migrations. Nous généralisons une approche basée sur deséquations de Hamilton-Jacobi pour détérminer les termes dominants de la solution lorsque les effets des mutations sont petits (mais non-nuls). Cette méthodeétait initialement utilisée, pour différents problèmes venant de la biologieévolutive, pour identifier les solutions asymptotiques, lorsque les effets des mutations tendent vers 0, sous forme des sommes de masses de Dirac. Un point clé est une propriété d'unicité en rapport avec la théorie de KAM faible. Cette méthode nous permet d'aller au-delà des approximations Gaussiennes habituellement utilisées par les biologistes et contribue ainsià relier les théories de la dynamique adaptative et de la génétique quantitative.
Equations for biological evolution
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1995
In this paper we consider mathematical models inspired by the mechanisms of biological evolution. We take populations which are subject to interaction and mutation. In the cases we consider, the interaction is through competition or through a prey-predator relationship. The models consider the specific characteristics as taking values in real intervals and the equations are of the integro—differential type. In the case of competition, thanks to the fact that some of the equations have solutions which are quite explicit, we succeed in proving the existence of attracting stationary solutions. In the case of prey and predator, using techniques of dynamical systems in infinite-dimensional spaces, we succeed in showing the existence of a global attractor, which in some instances reduces to a point. Our analysis takes into account having δ distributions, corresponding to all individuals having the same characteristics, as possible populations.
Some Competition Phenomena in Evolution Equations
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This paper is concerned with some nonlinear evolution equations governed by tow competitive operators. We treat the problem in a general setting and show how to apply the results to the particular situation of reaction diusion equation with large
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In Chapter 1, we consider a cell population where the individuals live in the same environmental conditions for some fixed period of time where they compete for nutrients among themselves, considering that offspring has the same trait as their parents, we were defining a fitness function that is trait and density dependent, assuming there were a unique trait best adapted at fixed environmental conditions. We modeled this phenomenon using a Transport Equation. The main result have been obtaining a Dirac mass concentration like solutions for the asymptotic behavior, incorporating a parameter, which is biologically sustained. We applied the classical framework to obtain this result. First, we give the apriori estimates and existence result to the simplified problem, next we add terms to have a more realistic model, then we study an approximate problem given some regularity and properties at solutions, finally we obtain this limit. We used tools as BV convergence properties, Anzats, sub...
Asymptotics of steady states of a selection–mutation equation for small mutation rate
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2013
We consider a selection–mutation equation for the density of individuals with respect to a continuous phenotypic evolutionary trait. We assume that the competition term for an individual with a given trait depends on the traits of all the other individuals, therefore giving an infinite-dimensional nonlinearity. Mutations are modelled by means of an integral operator. We prove existence of steady states and show that, when the mutation rate goes to zero, the asymptotic profile of the population is a Cauchy distribution.
Concentration phenomenon in some non-local equation
Discrete and Continuous Dynamical Systems - Series B
We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the following integro-differential equation ∂tu(t, x) = a(x) −ˆΩ k(x, y)u(t, y) dy u(t, x) +ˆΩ m(x, y)[u(t, y) − u(t, x)] dy (t, x) ∈ R+ × Ω, together with the initial condition u(0, •) = u0 in Ω. Such a problem is used in population dynamics models to capture the evolution of a clonal population structured with respect to a phenotypic trait. In this context, the function u represents the density of individuals characterized by the trait, the domain of trait values Ω is a bounded subset of R N , the kernels k and m respectively account for the competition between individuals and the mutations occurring in every generation, and the function a represents a growth rate. When the competition is independent of the trait, we construct a positive stationary solution which belongs to the space of Radon measures on Ω. Morever, when this "stationary" measure is regular and bounded, we prove its uniqueness and show that, for any non negative initial datum in L ∞ (Ω) ∩ L 1 (Ω), the solution of the Cauchy problem converges to this limit measure in L 2 (Ω). We also construct an example for which the measure is singular and non-unique, and investigate numerically the long time behaviour of the solution in such a situation. These numerical simulations seem to reveal some dependence of the limit measure with respect to the initial datum.
A diffusion model in population genetics with dynamic fitness
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We analyze a degenerate diffusion equation with singular boundary data, modeling the evolution of a polygenic trait under selection and drift. The equation models the contributions of a large but finite number of loci (genes) to the trait and at the same time allows the population trait mean to vary in a way that affects the strength of selection at individual loci; in this respect it differs from other population-genetic models that have been rigorously analyzed. We present existence, uniqueness and stability results for solutions of the system. We also prove that the genetic variance in the system tends to zero in the long time limit, and relate the dynamics of the trait mean to the variance.
Persistence and global stability in a selection–mutation size-structured model
Journal of Biological Dynamics, 2011
We analyse a selection–mutation size-structured model with n ecotypes competing for common resources. Uniform persistence and robust uniform persistence are established, when the selection–mutation matrix Γ is irreducible, i.e. individuals of one ecotype may contribute directly or indirectly to individuals of other ecotypes. Similar results are also presented for a particular reducible form of Γ. In the case of pure selection in which the offspring of one ecotype belong to the same ecotype, i.e. Γ=I, the identity matrix, we prove that the boundary equilibrium that describes competitive exclusion, with the fittest being the winner ecotype, is globally asymptotically stable. We show that small perturbations of the pure selection matrix lead to the existence of globally asymptotically stable interior equilibria. For the case when the selection–mutation matrix is reducible, we present and discuss the outcome of a series of numerical simulations.