Eduardo Henrique Colombo
Abstract
Effects of nonlinear diffusion in biological population spatial patterns
Motivated by the observation that anomalous diffusion is a realistic feature in the dynamics of
biological populations, we investigate its implications in a paradigmatic model for the evolution
of a single species density. The standard model includes growth and competition in a logistic
expression, and spreading is modeled through normal diffusion. Moreover, the competition term
is nonlocal, which has been shown to give rise to spatial patterns. This can be evidenced by the
presence of a characteristic mode in the stationary Fourier distribution which is analytically
predicted. We found that the number of clusters inside the population is inversely proportional
to the range of interaction in competitive processes. Anomalous diffusion is further introduced
generalizing the diffusion term, that becomes dependent on a power of the local density. We
show that for the subdiffusive case the stationary distribution displays a fragmented population,
in contrast with the superdiffusive case where homophilic behavior is observed.
Eventually, as a main result, we analytically obtain the phase diagram (homogenous-patterned)
relating the interaction range to the nonlinear diffusion term in the pattern formation process,
with additional considerations where the population becomes fragmented or connected.