Clumped Distribution by Neighbourhood Competition

Abstract The paper studies the spatial distribution of individuals competing for a continuously distributed resource in one-dimensional space. An individual diffuses randomly, and is assumed to suffer a cost of mortality from neighbors. The population dynamics are then described by an extended version of the Lotka–Volterra competition equation with terms of growth and diffusion, and the term of neighborhood competition/interference through an integral kernel. By allowing neighbouring individuals to compete with each other, the pattern of spatial distribution changes drastically from that of the classical models without any neighbourhood competition: in the classical models, any spatial variation in resource availability is smoothed out in the stationary distribution of species utilizing it. However, in the present model, even negligibly small spatial variation in resource availability could trigger a strongly clumped distribution of the species with a characteristic wavelength. Significant amplification of intermediate spatial frequencies occurs if the width of detrimental influence exceeds the mean distance an individual diffuses before it reproduces. In the extreme of no diffusion, the stationary distribution (the ideal free distribution) is strictly discrete (the support of distribution given by a set of points). The model also reveals the condition for a periodic/quasi-periodic travelling wave when the species is expanding in space, and demonstrates complex spatial distributions in two dimension.