Spatial dynamics of invasion in sinusoidally varying environments

Abstract

Range expansions of invading species in homogeneous environments have been extensively studied since the pioneer works by Fisher (Ann Eugen 7:255–369, 1937 ) and Skellam (Biometrika 38:196–218, 1951 ). However, environments for living organisms are often fragmented by natural or artificial habitat destruction. Here we address how such environmental heterogeneity affects the range expansion of invading species. We consider a single‐species invasion in heterogeneous environments whose habitat parameters vary in a sinusoidal or quasi‐sinusoidal manner. Accordingly, Fisher's model is modified to make the intrinsic growth rate and diffusion coefficient spatially variable. By numerically solving the model, we examine the spatio‐temporal pattern of propagating waves, and predict the speed as a function of the amplitude and the wave length of the diffusion coefficient and the intrinsic growth rate. Firstly, the results demonstrate that in the sinusoidally varying environment, if the intrinsic growth rate solely oscillates, the speed increases with increases in the amplitude of oscillation. Conversely, if the diffusion coefficient solely oscillates, the speed decreases with increases in the amplitude of oscillation. When both the intrinsic growth rate and diffusion coefficient oscillate, the speed is synergistically accelerated if the oscillations are in anti‐phase, whereas it is decelerated if the oscillations are in same phase. Secondly, the increase in the wave length in either the intrinsic growth rate or the diffusion coefficient leads to decreases in the speed. Thirdly, in the irregularly varying environment, the irregularity in the amplitude of the intrinsic growth rate enhances the speed, while that of the diffusion coefficient attenuates the speed.