Dynamics of a strongly nonlocal reaction-diffusion population model

We study the development of travelling waves in a population that competes with itself for resources in a spatially nonlocal manner. We model this. situation as an initial value problem for the integro-differential reaction-diffusion equation (partial derivativeu)/(partial derivativet) = (partial derivative2u)/(partial derivativex2) + u {1 + alphau - betau(2) - (1 + alpha - beta) integral(-infinity)(infinity) lambdag(lambda(x - y))u(y, t)dy} with g an even function that satisfies g(y)-->0 as y-->+/-infinity, integral(-infinity)(infinity)g(y) dy = 1, alpha > 0, 0 < beta < 1+alpha and lambda > 0. We concentrate on the limit of highly nonlocal interactions lambda much less than 1, focusing on the particular case g(y) = (1)/(2)e(-\y\), which is equivalent to the reaction-diffusion system (partial derivativeu)/(partial derivativet) = (partial derivative2u)/(partial derivativex2) + u {1 + alphau - betau(2) - (1 + alpha - beta)w}, 0 = (partial derivative2w)/(partial derivativex2) + lambda(2)(u-w). Using numerical and asymptotic methods, we show that in different, well-defined regions of parameter space, steady travelling waves, unsteady travelling waves and periodic travelling waves develop from localized initial conditions. A key feature of the system for lambda much less than 1 is the local existence of travelling wave solutions that propagate with speed c < 2, and which, although they cannot exist globally, attract the solution of the initial value problem for an asymptotically long time. By using a Cole-Hopf transformation, we derive a first order hyperbolic equation for the gradient of log u ahead of the wavefront, where u is exponentially small. An analysis of this equation in terms of its characteristics, allowing for the formation of shocks where necessary, explains the dynamics of each of the different types of travelling wave. Moreover, we are able to show that the techniques that we develop for this particular case can be used for more general kernels g(y) and that we expect the same range of different types of travelling wave to be solutions of the initial value problem for appropriate parameter values. As another example, we briefly consider the case g(y) = e(-y2) / rootpi, for which the system cannot be simplified to a pair of partial differential equations.