Qualitative properties of the spreading speed of a population structured in space and in phenotype

We consider a nonlocal Fisher-KPP equation that models a population structured in space and in phenotype. The population lives in a heterogeneous periodic environment: the diffusion coefficient, the mutation coefficient and the fitness of an individual may depend on its spatial position and on its phenotype.We first prove a Freidlin-Gärtner formula for the spreading speed of the population. We then study the behaviour of the spreading speed in different scaling limits (small and large period, small and large mutation coefficient). Finally, we exhibit new phenomena arising thanks to the phenotypic dimension. Our results are also valid when the phenotype is seen as another spatial variable along which the population does not spread

Large deviations and the emergence of a logarithmic delay in a nonlocal Fisher-KPP equation

We study a variant of the Fisher-KPP equation with nonlocal dispersal. Using the theory of large deviations, we show the emergence of a "Bramson-like" logarithmic delay for the linearised equation with step-like initial data. We conclude that the logarithmic delay emerges also for the solutions of the nonlinear equation. Previous papers found very precise results for the nonlinear equation with strong assumptions on the decay of the kernel. Our results are less precise, but they are valid for all continuous symmetric thin-tailed kernels

Large deviations and the emergence of a logarithmic delay in a nonlocal Fisher-KPP equation

We study a variant of the Fisher-KPP equation with nonlocal dispersal. Using the theory of large deviations, we show the emergence of a "Bramson-like" logarithmic delay for the linearised equation with step-like initial data. We conclude that the logarithmic delay emerges also for the solutions of the nonlinear equation

Reaction-diffusion model for a population structured in phenotype and space I - Criterion for persistence

We consider a reaction-diffusion model for a population structured in phenotype. We assume that the population lives in a heterogeneous periodic environment, so that a given phenotypic trait may be more or less fit according to the spatial location. The model features spatial mobility of individuals as well as mutation.We first prove the well-posedness of the model. Next, we derive a criterion for the persistence of the population which involves the generalised principal eigenvalue associated with the linearised elliptic operator. This notion allows us to handle the possible lack of coercivity of the operator. We then obtain a monotonicity result for the generalised principal eigenvalue, in terms of the frequency of spatial fluctuations of the environment and in terms of the spatial diffusivity. We deduce that the more heterogeneous is the environment, or the higher is the mobility of individuals, the harder is the persistence for the species.This work lays the mathematical foundation to investigate some other optimisation problems for the environment to make persistence as hard or as easy as possible, which will be addressed in the forthcoming companion paper

Polymorphic population expansion velocity in a heterogeneous environment

How does the spatial heterogeneity of landscapes interact with the adaptive evolution of populations to influence their spreading speed? This question arises in agricultural contexts where a pathogen population spreads in a landscape composed of several types of crops, as well as in epidemiological settings where a virus spreads among individuals with distinct immune profiles. To address it, we introduce an analytical method based on reaction-diffusion models. We focus on spatially periodic environments with two distinct patches, where the dispersing population consists of two specialized morphs, each potentially mutating to the other. We present new formulas for the speed together with criteria for persistence, accounting for both rapidly and slowly varying environments, as well as small and large mutation rates. Altogether, our analytical and numerical results yield a comprehensive understanding of persistence and spreading dynamics. In particular, compared to a situation without mutations or to a single morph spreading in a heterogeneous landscape, the introduction of mutations to a second morph with reverse specialization, while consistently impeding persistence, can significantly increase speed, even if the mutation rate between the two morphs is very small.Additionally, we find that the amplitude of the spatial fragmentation effect is significantly increased in this case. This has implications for agroecology, emphasizing the higher importance of landscape structure in influencing adaptation-driven population dynamics