The diffusive farmers and hunter-gatherers model with a free boundary in a heterogeneous environment

Ecologically, the evolutionary advantage of farmers over hunter-gatherers has led to their worldwide spreading. In this paper, we study a free boundary problem of a farmers and hunter-gatherers model featuring a type of prey-predator dynamics. The nonlinear diffusion of farmers is determined by the density of farmers in a way that higher population density accelerates the migration. We establish a spreading-vanishing dichotomy and provide some sufficient conditions in a heterogeneous environment. Moreover, in a homogeneous environment, we achieve exact asymptotic convergence to the constant equilibrium. Thereby, we verify that a strong sedentary lifestyle of farmers is advantageous for the spreading of farmers.