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

We study a scalar reaction-diffusion equation which contains a nonlocal ter m in the form of an integral convolution in the spatial variable and demons trate, using asymptotic, analytical and numerical techniques, that this sca lar equation is capable of producing spatio-temporal patterns. Fisher's equ ation is a particular case of this equation. An asymptotic expansion is obt ained for a travelling wavefront connecting the two uniform steady states a nd qualitative differences to the corresponding solution of Fisher's equati on are noted. A stability analysis combined with numerical integration of t he equation show that tinder certain circumstances nonuniform solutions are formed in the wake of this front. Using global bifurcation theory, we prov e the existence of such non-uniform steady state solutions for a wide range of parameter values. Numerical bifurcation studies of the behaviour of ste ady state solutions as a certain parameter is varied, are also presented.