Front Propagation and Clustering in the Stochastic Nonlocal Fisher Equation

The nonlocal Fisher equation is a diffusion-reaction equation where the reaction has a linear birth term and a nonlocal quadratic competition, which describes the reaction between distant individuals. This equation arises in evolutionary biological systems, where the arena for the dynamics is trait space, diffusion accounts for mutations and individuals with similar traits compete, resulting in partial niche overlap. It has been found that the (non-cutoff) deterministic system gives rise to a spatially inhomogeneous state for a certain class of interaction kernels, while the stochastic system produces an inhomogeneous state for small enough population densities. Here we study the problem of front propagation in this system, comparing the stochastic dynamics to the heuristic approximation of this system by a deterministic system where the linear growth term is cut off below some critical density. Of particular interest is the nontrivial pattern left behind the front. For large population density, or small cutoff, there is a constant velocity wave propagating from the populated region to the unpopulated region. As in the local Fisher equation, the spreading velocity is much lower than the Fisher velocity which is the spreading velocity for infinite population size. The stochastic simulations give approximately the same spreading velocity as the deterministic simulation with appropriate birth cutoff. When the population density is small enough, there is a different mechanism of population spreading. The population is concentrated on clusters which divide and separate. This mode of spreading has small spreading velocity, decaying exponentially with the range of the interaction kernel. The dependence on the carrying capacity is more complicated, and the log of the velocity scales as a power law of the carrying capacity, where the power is dependent on the kernel. We also discuss the transition between the bulk homogeneous pattern to separated islands, which occurs when the minimal population density is lower than the cutoff in the deterministic model.