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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 consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed c * > 0, and prove the existence… (More)

- Gaël Raoul
- 2009

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.

- Gaël Raoul
- 2015

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 consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed c * > 0, and prove the… (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)

- Gaël Raoul
- 2015

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 [8] and [2] 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)

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)

- Sepideh Mirrahimi, Gaël Raoul
- 2012

We study sexual populations structured by a phenotypic trait and a space variable, in a non-homogeneous environment. Departing from a structured population equation we perform a hydrodynamic-type limit to derive a model close to an existing model of theoretical biology. We then perform a further simplification to obtain a model depending on only one… (More)