Positive solutions for Lotka–Volterra competition systems with large cross-diffusion

Abstract

This paper discusses the stationary problem for the Lotka–Volterra competition systems with cross-diffusion under homogeneous Dirichlet boundary conditions. Although some sufficient conditions for the existence of positive solutions are known, the information for their structure is far from complete. In order to get better understanding of the competition system with cross-diffusion, we study the effects of large cross-diffusion on the structure of positive solutions and focus on the limiting behaviour of positive solutions by letting one of the cross-diffusion coefficients to infinity. Especially, it will be shown that positive solutions of the competition system converge to a positive solution of a suitable limiting system. We will also derive some satisfactory results on positive solutions for this limiting system. These results give us important information on the structure of positive solutions for the competition system when one of the cross-diffusion coefficients is sufficiently large.

Keywords:

• cross-diffusion
• coexistence
• limiting structure
• a priori estimates
• bifurcation

AMS Subject Classifications::

• Primary 35J62, 35B40
• Secondary 35K57, 35B32, 35B45

Acknowledgements

The work of K. Kuto was supported by Grant-in-Aid for Young Scientists (B-21740129), Ministry of Education, Science, Sports and Culture in Japan. The work of Y. Yamada was supported by Grant-in-Aid for Scientific Research (C-21540229), Japan Society for the Promotion of Science.