Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation

We study a scalar reaction-diffusion equation which contains a nonlocal term in the form of an integral convolution in the spatial variable and demonstrate, using asymptotic, analytical and numerical techniques, that this scalar equation is capable of producing spatio-temporal patterns. Fisher's equation is a particular case of this equation. An asymptotic expansion is obtained for a travelling wavefront connecting the two uniform steady states and qualitative differences to the corresponding solution of Fisher's equation are noted. A stability analysis combined with numerical integration of the equation show that under certain circumstances nonuniform solutions are formed in the wake of this front. Using global bifurcation theory, we prove the existence of such non-uniform steady state solutions for a wide range of parameter values. Numerical bifurcation studies of the behaviour of steady state solutions as a certain parameter is varied, are also presented.