On the dependence of population size upon random dispersal rate

Abstract

This paper concerns the dependence of the population sizefor a single species onits random dispersal rate and its applications on the invasion of species.The population size of a single species often depends on its random dispersal ratein non-trivial manners. Previous results show thatthe population size is usuallynot a monotone function ofthe random dispersal rate. We construct some examples to illustrate thatthe population size, as a function ofthe random dispersal rate, can have at least two local maxima.As an applicationwe illustrate that the invasion of exotic species depends uponthe random dispersal rate of the resident species in complicated manners.Previous results show that the total population ismaximized at some intermediate random dispersal rate forseveral classes of local intrinsic growth rates.We find one family of local intrinsic growth rates such thatthe total population is maximized exactly at zero random dispersal rate.We show that thepopulation distribution becomes flatter in average if we increase therandom dispersal rate, and the environmentalgradient is always steeper than the population distribution, at least in some average sense.We also discuss the dependence of the population size on movement ratesin other contexts and propose some open problems.