In this paper, we are interested in an integro-differential model that describe the evolution of a population structured with respect to a continuous trait. Under some assumption, we are able to find an entropy for the system, and show that some steady solutions are globally stable. The stability conditions we find are coherent with those of Adaptive… (More)
We study the large-time behaviour of a non-local evolution equation for the density of particles or individuals subject to an external and an interaction potential. In particular, we consider interaction potentials which are singular in the sense that their first derivative is discontinuous at the origin. For locally attractive singular interaction… (More)
In this paper, we are interested in the long-time behavior of solutions to a non-local interaction equationin dimension 1. We show that up to an extraction , the solution converges to a steady-state. Then, we study the structure of stable steady-states.
We study a generalized system of ODE's modeling a finite number of biological populations in a competitive interaction. We adapt the techniques in  and  to prove the convergence to a unique stable equilibrium. Résumé. Nousétudions un système généralisé d'´ equations différentielles modélisant un nombre fini de populations biologiques en interaction… (More)
In this paper, we are interested in the nonlinear stability of Dirac-type steady solutions to an integro-differential equation appearing in the study of populations which are structured with respect to a quantative (continuous) trait. We show that stability conditions of adaptive dynamics extend to this model.
Invasion fronts in ecology are well studied but very few mathematical results concern the case with variable motility (possibly due to mutations). Based on an apparently simple reaction-diffusion equation, we explain the observed phenomena of front acceleration (when the motility is unbounded) as well as other quantitative results, such as the selection of… (More)
In this paper, we consider a long time and vanishing mutations limit of an integro-differential model describing the evolution of a population structured with respect to a continuous phenotypic trait. We show that the asymptotic population is a steady-state of the evolution equation without mutations, and satisfies an evolutionary stability condition.
We study sexual populations structured by a phenotypic trait and a space variable, in a non-homogeneous environment. Departing from an infinitesimal model, we perform an asymptotic limit to derive the system introduced in Kirkpatrick and Barton (1997). We then perform a further simplification to obtain a simple model. Thanks to this simpler equation, we can… (More)
We prove the nonlinear local stability of Dirac masses for a kinetic model of alignment of particles on the unit sphere, each point of the unit sphere representing a direction. A population concentrated in a Dirac mass then corresponds to the global alignment of all individuals. The main difficulty of this model is the lack of conserved quantities and the… (More)