International audienc

We investigate spreading properties of solutions of a large class of twocomponent reaction-diffusion systems, including prey-predator systems as a special case. By spreading properties we mean the long time behaviour of solution fronts that start from localized (i.e. compactly supported) initial data. Though there are results in the literature on the existence of travelling waves for such systems, very little has been known -at least theoretically -about the spreading phenomena exhibited by solutions with compactly supported initial data. The main difficulty comes from the fact that the comparison principle does not hold for such systems. Furthermore, the techniques that are known for travelling waves such as fixed point theorems and phase portrait analysis do not apply to spreading fronts. In this paper, we first prove that spreading occurs with definite spreading speeds. Intriguingly, two separate fronts of different speeds may appear in one solution -one for the prey and the other for the predatorin some situations.

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In this paper, we construct a model to describe the evolutionary epidemiology of spore producing asexual plant pathogens in a homogeneous host population. By considering the evolution in the space of the pathogen phenotypic values, we derive an integro-differential equation with nonlocal mutation terms. Our first main result is concerned with the existence and uniqueness of the endemic steady state of the model. Next assuming that the mutation kernel depends on a small parameter [Formula: see text] (the variance of the dispersion into the space of the pathogen phenotypic values), we investigate the concentration properties of the endemic steady state in the space of phenotypic values. In the context of this work, several Evolutionary Attractors (EAs) (as defined in classical adaptive dynamics) may exist. However, in rather general situations, our results show that only one EA persists when the populations are at equilibrium and when [Formula: see text] is small enough. Our analysis strongly relies on a refined description of the spectral properties of some integral operator with a highly concentrated kernel. We conclude the paper by presenting some numerical simulations of the model to illustrate this concentration phenomenon.

International audienc

Abstract. Intra-specific competition in population dynamics can be described by integro-differential equations where the integral term corresponds to nonlocal consumption of resources by individuals of the same population. Already the single integro-differential equation can show the emergence of nonhomogeneous in space stationary structures and can be used to model the process of speciation, in particular, the emergence of biological species during evolution [6], [7]. On the other hand, competition of two different species represents a well known and well studied model in population dynamics. In this work we study how the intra-specific competition can influence the competition between species. We will prove the existence of travelling waves for the case where the support of the kernel of the integral is sufficiently narrow. Numerical simulations will be carried out in the case of large supports.

In this paper we propose an age-structured malaria within-host model taking into account multi-strains interaction. We provide a global analysis of the model depending upon some threshold T 0. When T 0 ≤ 1, then the disease free equilibrium is globally asymptotically stable and the parasites are cleared. On the contrary if T 0 > 1, the model exhibits the competition exclusion principle. Roughly speaking, only the strongest strain, according to a suitable order, survives while the other strains go to extinct. Under some additional parameter conditions we prove that the endemic equilibrium corresponding to the strongest strain is globally asymptotically stable.