Cross-diffusion driven instability for a Lotka-Volterra competitive reaction-diffusion system

In this work we investigate the possibility of the pattern formation for areaction-di®usion system with nonlinear di®usion terms. Through a linear sta-bility analysis we ¯nd the conditions which allow a homogeneous steady state(stable for the kinetics) to become unstable through a Turing mechanism. Inparticular, we show how cross-di®usion e®ects are responsible for the initiationof spatial patterns. Finally, we ¯nd a Fisher amplitude equation which describesthe weakly nonlinear dynamics of the system near the marginal stability.