ON THE DEPENDENCE OF POPULATION SIZE UPON RANDOM DISPERSAL RATE

This paper concerns the dependence of the population size for a single species on its random dispersal rate and its applications on the invasion of species. The population size of a single species often depends on its random dispersal rate in non-trivial manners. Previous results show that the population size is usually not a monotone function of the random dispersal rate. We construct some examples to illustrate that the population size, as a function of the random dispersal rate, can have at least two local maxima. As an application we illustrate that the invasion of exotic species depends upon the random dispersal rate of the resident species in complicated manners. Previous results show that the total population is maximized at some intermediate random dispersal rate for several classes of local intrinsic growth rates. We find one family of local intrinsic growth rates such that the total population is maximized exactly at zero random dispersal rate. We show that the population distribution becomes flatter in average if we increase the random dispersal rate, and the environmental gradient is always steeper than the population distribution, at least in some average sense. We also discuss the dependence of the population size on movement rates in other contexts and propose some open problems.