Invasion front dynamics in disordered environments

Abstract

Invasion occurs in environments that are normally spatially disordered, however, the effect of such a randomness on the dynamics of the invasion front has remained less understood. Here, we study Fisher’s equation in disordered environments both analytically and numerically. Using the Effective Medium Approximation, we show that disorder slows down invasion velocity and for ensemble average of invasion velocity in disordered environment we have bar{v}=v_0 (1-|xi |^2/6) where |xi | is the amplitude of disorder and v_0 is the invasion velocity in the corresponding homogeneous environment given by v_0=2sqrt{RD_0}. Additionally, disorder imposes fluctuations on the invasion front. Using a perturbative approach, we show that these fluctuations are Brownian with a diffusion constant of: D_{C}= dfrac{1}{8} xi ^2sqrt{RD_0 (1-|xi |^2/3)}. These findings were approved by numerical analysis. Alongside this continuum model, we use the Stepping Stone Model to check how our findings change when we move from the continuum approach to a discrete approach. Our analysis suggests that individual-based models exhibit inherent fluctuations and the effect of environmental disorder becomes apparent for large disorder intensity and/or high carrying capacities.