Propagation through a barrier: Numerical analysis of a reaction-diffusion model with free boundary

We try to better understand how a spatial barrier may affect the spreading of an invading species via numerical analysis of some variations of a free boundary model in [1, 2] (where only homogeneous environment was considered). Here we incorporate a spatial barrier by replacing a bistable growth term f(u) in the model with f(x,u)=u(r(x)-u)(u-θ), where θ ∈ (0, 1/2) and r(x)=1 except in the barrier region [x₀,x₀+l], in which r(x) becomes negative away from its boundary, representing the biological assumption that the environment becomes hostile to the species inside the barrier. A parameter α > 0 in the expression of r(x) is used to characterize the severity of the environmental hostility. We find that when all the other parameters are fixed there exists a critical value l* of the barrier length l such that successful spreading is continued past the barrier region when l < l*, and the propagation is blocked when l > l*. Similarly we show numerically that when all the other parameters are fixed, there is a critical value α* of the barrier severity α such that propagation can be continued when α < α*, but it is blocked when α > α*. The dependence of l* (respectively α*) on the other parameters are also analysed. To include temporal fluctuations of the environment, we further replace r(x) by a(t)r(x) with a(t) a positive time-periodic function of average 1, to represent the periodic modulation of the environment. Our numerical simulations suggest that increasing the magnitude of temporal variation enhances the ability of species invasion, while increasing the frequency of such variation reduces this ability. To see how Allee effect may influence the invasion with a barrier, our results based on a bistable f discussed above are compared with that for a model obtained from a standard monostable function (no Allee effect), namely f=u[r(x)-u] with the same r(x). A parallel numerical analysis shows that qualitatively everything is the same in the monostable case, including the numerical results incorporating seasonal changes (with r(x) replaced by a(t)r(x)). However, these numerical simulations indicate that in the bistable case (Allee effect included) the invasion is more likely to cross the barrier than in the monostable case (no Allee effect), suggesting the counter intuitive conclusion that Allee effect may increase the chance of invading across a barrier. Dedicated to Professor Shigui Ruan for his 60th birthday.