Early-stage invasion and spreading speed in a resource-dependent dispersal model

In this paper, we study the dynamics of biological invasion through complementary modeling frameworks in the context of nonlocal resource-driven dispersal. During the very early stage of invasion, when only a few individuals are present, demographic variability is crucial: extinction may occur even under favorable average conditions. To capture this, we use a branching-process approximation that provides explicit formulas for extinction probabilities, survival conditions, and mean extinction times. At larger scales and higher densities, invasion is described by a deterministic system of nonlinear integro-differential equations. For this system, we establish well-posedness and derive lower and upper bounds on the asymptotic spreading speed. A unifying threshold parameter T₀, defined as the spectral radius of a next-generation operator, characterizes invasion outcomes: if T₀≤1, extinction occurs; if T₀>1, the invader persists and spreads. Importantly, the threshold derived from the early-stage approximation coincides with that of the deterministic model, thus providing a consistent criterion for invasion success. Finally, numerical simulations illustrate the transition between extinction and persistence and highlight how resource-driven dispersal shapes invasion speed.