Biased movement and the ideal free distribution in some free boundary problems

Some invasive species can persist in their habitat but eventually spread very slow in a nonlinear fashion to expand their habitat range. In order to capture this phenomenon, we consider reaction–diffusion–advection models with a free boundary modeling the spreading and the biased movement of species in one-dimensional spatially heterogeneous environments. Under a condition of low resource quality, we find that large advection can lead to the spreading of the species but the spreading speed goes asymptotically to zero. Moreover, we investigate the effect of the resource on the dynamics of the current problem. Finally, we bring the notion of an ideal free distribution (IFD) into free boundary problems to understand the mechanism such that the species can eventually match the environmental quality perfectly. Under the current problem setting, the IFD may not hold even if the population plays an ideal free strategy. We then provide a sufficient condition for the IFD to be reached when using an ideal free strategy.