A free boundary problem with resource‐dependent motility in a weak heterogeneous environment

Abstract This paper is concerned with a free boundary problem with resource‐dependent motility in a weak heterogeneous environment. The existence and uniqueness of global solutions are discussed first. Next, we establish long‐time behaviors of solutions which is a spreading–vanishing dichotomy. Moreover, we obtain sharp criteria on spreading and vanishing by investigating the associated linearized eigenvalue problem. The theoretical analyses reveal an important biological phenomenon. (i) Resource abundance reduces the motility, providing advantages for individuals to persist in a habitat. (ii) Resource shortage enhances the motility, forcing individuals to expand outward to survive in an environment. Consequently, resource‐dependent motility is more beneficial to the survival of species compared with random dispersal.