Variational principles and the effect of a cutoff on population pattern size

The relation between pattern size and maximum population density is obtained for the stationary state of populations living in a refuge surrounded by a hostile environment. The population dynamics is described by reaction–diffusion equations whose kinetic terms display a cutoff. The latter takes into account the discreteness of the population when the population density is small. We employ a variational principle for the nonlinear eigenvalue problem to obtain lower bounds for the pattern length. Numerical solutions display excellent agreement with our analytical results.