# Non local Lotka-Volterra system with cross-diffusion in an heterogeneous medium

@article{Fontbona2015NonLL,
title={Non local Lotka-Volterra system with cross-diffusion in an heterogeneous medium},
author={Joaqu{\'i}n Fontbona and Sylvie M{\'e}l{\'e}ard},
journal={Journal of Mathematical Biology},
year={2015},
volume={70},
pages={829-854}
}
• Published 16 March 2013
• Mathematics
• Journal of Mathematical Biology
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## References

SHOWING 1-10 OF 31 REFERENCES
Analysis of a Multidimensional Parabolic Population Model with Strong Cross-Diffusion
• Mathematics
SIAM J. Math. Anal.
• 2004
The global existence of a nonnegative weak solution to a multidimensional parabolic strongly coupled model for two competing species is proved. The main feature of the model is that the diffusion
Invasion and adaptive evolution for individual-based spatially structured populations
• Mathematics
Journal of mathematical biology
• 2007
This work considers a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the phenotypic trait of individuals, and shows the important role of parameter scalings on clustering and invasion.
The non-local Fisher-KPP equation: travelling waves and steady states
• Mathematics
• 2009
We consider the Fisher–KPP equation with a non-local saturation effect defined through an interaction kernel (x) and investigate the possible differences with the standard Fisher–KPP equation. Our
Pattern and Waves for a Model in Population Dynamics with Nonlocal Consumption of Resources
• Mathematics, Economics
• 2006
We study a reaction-diffusion equation with an integral term describing nonlocal consumption of resources in population dynamics. We show that a homogeneous equilibrium can lose its stability
On the Entropic Structure of Reaction-Cross Diffusion Systems
• Mathematics
• 2014
This paper is devoted to the study of systems of reaction-cross diffusion equations arising in population dynamics. New results of existence of weak solutions are presented, allowing to treat systems
Global Well-Posedness of a Conservative Relaxed Cross Diffusion System
• Mathematics
SIAM J. Math. Anal.
• 2012
This work proves global existence in time of solutions to relaxed conservative cross diffusion systems governed by nonlinear operators of the form $u_i\to \partial_tu_i-\Delta(a_i(\tilde{u})u_ i)$, and proves existence of global weak solutions is obtained in any space dimension.
A microscopic probabilistic description of a locally regulated population and macroscopic approximations
• Computer Science, Mathematics
• 2004
This work generalizes a discrete model describing a locally regulated spatial population with mortality selection by adding spatial dependence and gives a path-wise description in terms of Poisson point measures.