Density-dependent dispersal and population aggregation patterns.

We have derived reaction-dispersal-aggregation equations from Markovian reaction-random walks with density-dependent jump rate or density-dependent dispersal kernels. From the corresponding diffusion limit we recover well-known reaction-diffusion-aggregation and reaction-diffusion-advection-aggregation equations. It is found that the ratio between the reaction and jump rates controls the onset of spatial patterns. We have analyzed the qualitative properties of the emerging spatial patterns. We have compared the conditions for the possibility of spatial instabilities for reaction-dispersal and reaction-diffusion processes with aggregation and have found that dispersal process is more stabilizing than diffusion. We have obtained a general threshold value for dispersal stability and have analyzed specific examples of biological interest.