Selection-mutation Model Research Articles

Predicting the evolution of bacterial populations with an epistatic selection-mutation model

Predicting the evolution of bacterial populations with an epistatic selection-mutation model

A Hamilton-Jacobi approach to evolution of dispersal

The evolution of dispersal is a classical question in evolutionary biology, and it has been studied in a wide range of mathematical models. A selection-mutation model, in which the population is structured by space and a phenotypic trait, with the trait acting directly on the dispersal (diffusion) rate, was formulated by Perthame and Souganidis [Math. Model. Nat. Phenom. 11:154–166, 2016] to study the evolution of random dispersal toward the evolutionarily stable strategy. For the rare mutation limit, it was shown that the equilibrium population concentrates on a single trait associated to the smallest dispersal rate. In this paper, we consider the corresponding evolution equation and characterize the asymptotic behaviors of the time-dependent solutions in the rare mutation limit, under mild convexity assumptions on the underlying Hamiltonian function.

Selection–mutation dynamics with asymmetrical reproduction kernels

We study a family of selection–mutation models of a sexual population structured by a phenotypical trait. The main feature of these models is the asymmetric trait heredity or fecundity between the parents: we assume that each individual inherits mostly its trait from the female or that the trait acts on the female fecundity but does not affect male. Following previous works inspired from principles of adaptive dynamics, we rescale time and assume that mutations have limited effects on the phenotype. Our goal is to study the asymptotic behavior of the population distribution. We derive non-extinction conditions and BV estimates on the total population. We also obtain Lipschitz estimates on the solutions of Hamilton–Jacobi equations that arise from the study of the population distribution concentration at fittest traits. Concentration results are obtained in some special cases by using a Lyapunov functional.

Asymptotic analysis of selection-mutation models in the presence of multiple fitness peaks

We study the long-time behaviour of phenotype-structured models describing the evolutionary dynamics of asexual populations whose phenotypic fitness landscape is characterised by multiple peaks. First we consider the case where phenotypic changes do not occur, and then we include the effect of heritable phenotypic changes. In the former case the model is formulated as an integrodifferential equation for the phenotype distribution of the individuals in the population, whereas in the latter case the evolution of the phenotype distribution is governed by a non-local parabolic equation whereby a linear diffusion operator captures the presence of phenotypic changes. We prove that the long-time limit of the solution to the integrodifferential equation is unique and given by a measure consisting of a weighted sum of Dirac masses centred at the peaks of the phenotypic fitness landscape. We also derive an explicit formula to compute the weights in front of the Dirac masses. Moreover, we demonstrate that the long-time solution of the non-local parabolic equation exhibits a qualitatively similar behaviour in the asymptotic regime where the diffusion coefficient modelling the rate of phenotypic change tends to zero. However, we show that the limit measure of the non-local parabolic equation may consist of less Dirac masses, and we provide a sufficient criterion to identify the positions of their centres. Finally, we carry out a detailed characterisation of the speed of convergence of the integral of the solution (i.e. the population size) to its long-time limit for both models. Taken together, our results support a more in-depth theoretical understanding of the conditions leading to the emergence of stable phenotypic polymorphism in asexual populations.

Uniqueness of the viscosity solution of a constrained Hamilton–Jacobi equation

In quantitative genetics, viscosity solutions of Hamilton–Jacobi equations appear naturally in the asymptotic limit of selection-mutation models when the population variance vanishes. They have to be solved together with an unknown function I(t) that arises as the counterpart of a non-negativity constraint on the solution at each time. Although the uniqueness of viscosity solutions is known for many variants of Hamilton–Jacobi equations, the uniqueness for this particular type of constrained problem was not resolved, except in a few particular cases. Here, we provide a general answer to the uniqueness problem, based on three main assumptions: convexity of the Hamiltonian function H(I, x, p) with respect to p, monotonicity of H with respect to I, and BV regularity of I(t).

Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures

In this paper we consider a selection-mutation model with an advection term formulated on the space of finite signed measures on \begin{document}$ \mathbb{R}^d $\end{document} . The selection-mutation kernel is described by a family of measures which allows the study of continuous and discrete kernels under the same setting. We rescale the selection-mutation kernel to obtain a diffusively rescaled selection-mutation model. We prove that if the rescaled selection-mutation kernel converges to a pure selection kernel then the solution of the diffusively rescaled model converges to a solution of an advection-diffusion equation.

Time-asymptotic convergence rates towards discrete steady states of a nonlocal selection-mutation model

This paper is concerned with large time behavior of solutions to a semi-discrete model involving nonlinear competition that describes the evolution of a trait-structured population. Under some threshold assumptions, the steady solution is shown unique and strictly positive, and also globally stable. The exponential convergence rate to the steady state is also established. These results are consistent with the results in [P.-E. Jabin, H. L. Liu. Nonlinearity 30 (2017) 4220–4238] for the continuous model.

On a non-local selection–mutation model with a gradient flow structure

In this paper, we are interested in an integro-differential model with a non-linear competition term that describes the evolution of a population structured with respect to a continuous trait. Under some assumptions, the steady solution is shown unique and strictly positive, and also globally stable. The exponential convergence rate to the steady state is also established.

Observation and control in models of population genetics

Mathematical systems theory and optimal control have been mostly developed in the context of engineering. In this paper it is shown how these techniques can be applied in population genetics. Based on the classical Fisher's selection model, first a very natural monitoring problem is studied: can the change of the genetic state of a population (described in terms of allele frequencies) be uniquely recovered from the observation of the frequencies of certain phenotypes? We give sufficient conditions for a positive answer to this question in a typical case of heterosis (when mixed genotypes are better than the pure ones, implying stable coexistence of all allele types). The second question is: How to effectively estimate the genetic composition of the population from phenotypic observation? The answer is observer design, which is carried out for two different dominance structures, determining the manifestation of the genetic state. In a model of artificial selection, we show how the population can be steered into equilibrium where maximal mean fitness is attained. Finally, the application of the above methodology is also extended to selection–mutation models, where both fitness parameters and mutation rates are controlled.

Population dynamics under selection and mutation: Long-time behavior for differential equations in measure spaces

We study the long-time behavior of solutions to a measure-valued selection–mutation model that we formulated in [14]. We establish permanence results for the full model, and we study the limiting behavior even when there is more than one strategy of a given fitness; a case that arises in applications. We show that for the pure selection case the solution of the dynamical system converges to a Dirac measure centered at the fittest strategy class provided that the support of the initial measure contains a fittest strategy; thus we term this Dirac measure an Asymptotically Stable Strategy. We also show that when the strategy space is discrete, the selection–mutation model with small mutation has a locally asymptotically stable equilibrium that attracts all initial conditions that are positive at the fittest strategy.

A class of Hamilton–Jacobi equations with constraint: Uniqueness and constructive approach

We discuss a class of time-dependent Hamilton–Jacobi equations, where an unknown function of time is intended to keep the maximum of the solution to the constant value 0. Our main result is that the full problem has a unique viscosity solution, which is in fact classical. The motivation is a selection–mutation model which, in the limit of small diffusion, exhibits concentration on the zero level set of the solution of the Hamilton–Jacobi equation.Uniqueness is obtained by noticing that, as a consequence of the dynamic programming principle, the solution of the Hamilton–Jacobi equation is classical. It is then possible to write an ODE for the maximum of the solution, and treat the full problem as a nonstandard Cauchy problem.

Uniqueness in a class of Hamilton–Jacobi equations with constraints

In this note, we discuss a class of time-dependent Hamilton–Jacobi equations depending on a function of time, this function being chosen in order to keep the maximum of the solution of the constant value 0. The main result of the note is that the full problem has a unique classical solution. The motivation is a selection–mutation model that, in the limit of small diffusion, exhibits concentration on the zero-level set of the solution to the Hamilton–Jacobi equation. The uniqueness result that we prove implies strong convergence and error estimates for the selection–mutation model.

Singular Limits for Reaction-Diffusion Equations with Fractional Laplacian and Local or Nonlocal Nonlinearity

We perform an asymptotic analysis of models of population dynamics with a fractional Laplacian and local or nonlocal reaction terms. The first part of the paper is devoted to the long time/long range rescaling of the fractional Fisher-KPP equation. This rescaling is based on the exponential speed of propagation of the population. In particular we show that the only role of the fractional Laplacian in determining this speed is at the initial layer where it determines the thickness of the tails of the solutions. Next, we show that such rescaling is also possible for models with non-local reaction terms, as selection-mutation models. However, to obtain a more relevant qualitative behavior for this second case, we introduce, in the second part of the paper, a second rescaling where we assume that the diffusion steps are small. In this way, using a WKB ansatz, we obtain a Hamilton-Jacobi equation in the limit which describes the asymptotic dynamics of the solutions, similarly to the case of selection-mutation models with a classical Laplace term or an integral kernel with thin tails. However, the rescaling introduced here is very different from the latter cases. We extend these results to the multidimensional case.

On a discrete selection–mutation model

A discrete-time population model in which individuals are distributed over a discrete phenotypic trait-space is studied. It is shown that, for an irreducible mutation matrix , if mutation is small, then an interior equilibrium exists, that is globally asymptotically stable in , while for arbitrary large mutation, each trait persists uniformly. For the model reduced to only two traits, conditions for the global stability of the interior equilibrium are provided. When structure is introduced in the model, namely when mutation matrix has block-diagonal form, with each diagonal block being irreducible, competitive exclusion among traits is analysed and sufficient conditions are given for one trait to drive all the other traits to extinction.

Positive Steady States of Evolution Equations with Finite Dimensional Nonlinearities

We study the question of existence of positive steady states of nonlinear evolution equations. We recast the steady state equation in the form of eigenvalue problems for a parametrised family of unbounded linear operators, which are generators of strongly continuous semigroups; and a fixed point problem. In case of irreducible governing semigroups we consider evolution equations with non-monotone nonlinearities of dimension two, and we establish a new fixed point theorem for set-valued maps. In case of reducible governing semigroups we establish results for monotone nonlinearities of any finite dimension $n$. In addition, we establish a non-quasinilpotency result for a class of strictly positive operators, which are neither irreducible nor compact, in general. We illustrate our theoretical results with examples of partial differential equations arising in structured population dynamics. In particular, we establish existence of positive steady states of a size-structured juvenile-adult and a structured consumer-resource population model, as well as for a selection-mutation model with distributed recruitment process.

Steady states of a selection-mutation model for an age structured population

We introduce a selection-mutation model for an age and age at maturity structured population. The age at maturity is considered an evolutionary trait on which the vital rates depend, subject to selection by competition, and to mutation modelled by a birth term which takes the form of a convolution-like operator. We prove well-posedness of the initial value problem and the existence of non-trivial steady states under suitable hypotheses which yield uniqueness when the interaction variable is one-dimensional.

Small populations corrections for selection-mutation models

We consider integro-differential models describing the evolution of a population structured by a quantitative trait. Individuals interact competitively, creating a strong selection pressure on the population. On the other hand, mutations are assumed to be small. Following the formalism of [20], this creates concentration phenomena, typically consisting in a sum of Dirac masses slowly evolving in time. We propose a modification to those classical models that takes the effect of small populations into accounts and corrects some abnormal behaviours.

Sympathy and similarity: The evolutionary dynamics of cooperation

The advantage of mutual help is threatened by defectors, who exploit the benefits provided by others without providing benefits in return. Cooperation can only be sustained if it is preferentially channeled toward cooperators and away from defectors. But how? A deceptively simple idea is to distinguish cooperators from defectors by tagging them. It clearly is in the interest of cooperators to use some distinctive cue to assort with their like. Such an assortment, however, conflicts with the interests of the cheaters, who have every incentive to also acquire that tag. This makes for an inherently unstable situation. The history of evolutionary thinking on this issue is long. An article in this issue of PNAS by Antal et al. (1) opens new ground by providing an in-depth analysis of a selection-mutation model.

Comparison between stochastic and deterministic selection-mutation models

We present a deterministic selection-mutation model with a discrete trait variable. We show that for an irreducible selection-mutation matrix in the birth term the deterministic model has a unique interior equilibrium which is globally stable. Thus all subpopulations coexist. In the pure selection case, the outcome is known to be that of competitive exclusion, where the subpopulation with the largest growth-to-mortality ratio will survive and the remaining subpopulations will go extinct. We show that if the selection mutation matrix is reducible, then competitive exclusion or coexistence are possible outcomes. We then develop a stochastic population model based on the deterministic one. We show numerically that the mean behavior of the stochastic model in general agrees with the deterministic one. However, un like the deterministic one, if the differences in the growth-to-mortality ratios are small in the pure selection case, it cannot be determined a priori which subpopulation will have the highest probability of surviving and winning the competition.

Adaptive dynamics via Hamilton–Jacobi approach and entropy methods for a juvenile-adult model

We consider a nonlinear system describing a juvenile-adult population undergoing small mutations. We analyze two aspects: from a mathematical point of view, we use an entropy method to prove that the population neither goes extinct nor blows-up; from an adaptive evolution point of view, we consider small mutations on a long time scale and study how a monomorphic or a dimorphic initial population evolves towards an Evolutionarily Stable State. Our method relies on an asymptotic analysis based on a constrained Hamilton–Jacobi equation. It allows to recover earlier predictions in Calsina and Cuadrado [À. Calsina, S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics, J. Math. Biol. 48 (2004) 135; À. Calsina, S. Cuadrado, Stationary solutions of a selection mutation model: the pure mutation case, Math. Mod. Meth. Appl. Sci. 15(7) (2005) 1091.] that we also assert by direct numerical simulation. One of the interests here is to show that the Hamilton–Jacobi approach initiated in Diekmann et al. [O. Diekmann, P.-E. Jabin, S. Mischler, B. Perthame, The dynamics of adaptation: an illuminating example and a Hamilton–Jacobi approach, Theor. Popul. Biol. 67(4) (2005) 257.] extends to populations described by systems.