The relation between pattern size and maximum population density is obtained for the stationary state of populations living in a refuge surrounded by ahostile environment. The population dynamics is described byreaction-diffusion equations whose kinetic terms display a cutoff. Thelatter takes into account the discreteness of the population when thepopulation density is small. We employ a variational principle for thenonlinear eigenvalue problem to obtain lower bounds for the pattern length. Numerical solutions display excellent agreement with our analytical results. © EDP Sciences/Societé Italiana di Fisica/Springer-Verlag 2007.