Natural populations often exhibit leptokurtic dispersal patterns, with a higher probability of long range dispersal events than predicted by Gaussian diffusion. Heterogeneity in individual dispersal behavior is often observed and has long been suspected to cause such leptokurtosis; this has been confirmed by recent theoretical analysis. However, a general understanding of the effect of dispersal heterogeneity on the wavespeed of an advancing population is lacking. We developed and analyzed a one-dimensional integro-difference model to examine the effects of heterogeneity on invasion speed and persistence in an advective environment such as a stream or river. In the model, each individual’s diffusion coefficient is chosen at random from a trait distribution (independent of the trait of its parent); this diffusion coefficient determines the variance of that individual’s dispersal distribution. We develop mathematical relationships between the moments of the trait distribution and the moments of the population dispersal distribution. Through formal analysis and numerical simulation we show how dispersal heterogeneity affects wavespeed and the critical net reproductive rate required for persistence in an advective environment.
Results/Conclusions
The kurtosis of the population dispersal distribution is directly proportional to the variance of individual diffusivity within the population, so that increasing the variance in diffusivity naturally increases the kurtosis of the population dispersal distribution. Additionally, each population dispersal moment is proportional to a moment of the trait distribution. As variance in diffusivity is increased, the wavespeed is increased, but this is not only due to the dispersal distribution’s increased kurtosis; other distributional moments play an important role as well. When the net reproductive rate is small, the lower distributional moments have the greatest influence on the magnitude of wavespeed; the characteristics of the center of the distribution are more important than the mathematical form of its tail. However, at sufficiently large reproductive rates, high order moments become important giving tail decay rate a strong influence on wavespeed. Increased heterogeneity also decreases the critical reproductive rate required for persistence under advection. A heterogeneous population will spread faster and be more resilient in the face of advective forces than a homogeneous population with the same mean diffusivity. If selection acts to increase dispersal heterogeneity of an invading species, spread rates may be much higher than predicted from estimates of mean dispersal ability.