The Asymptotic Spreading Speed of a Free Boundary Problem With Resource‐Dependent Motility

ABSTRACT This paper is a sequel to the first reference, on the dynamics of an invasive/new species model, in which the diffusion term depends on resource distribution described by an operator of the divergence form, and the spreading front is represented by a free boundary. In the reference, it has been shown that the model is well‐posed, and its long‐time behavior is governed by a spreading–vanishing dichotomy. However, the asymptotic spreading speed was not determined. The main purpose of this paper is to study it when spreading happens. Because the positive stationary solutions to the model play a crucial role in analyses, we first verify that such solutions exist by using the squeezing argument to overcome the analytical difficulties brought by the resource‐dependent motility. Furthermore, the estimate of asymptotic spreading speed is obtained by the upper–lower solutions method and the associated semiwave problem. Compared to the free boundary problem with random dispersal models, the asymptotic spreading speed not only depends on the resource‐dependent motility but also increases in it.