How Phenological Variation Affects Species Spreading Speeds

Allee effects introduced by density dependent phenology

We consider a hybrid model of an annual species with the timing of a stage transition governed by density dependent phenology. We show that the model can produce a strong Allee effect as well as overcompensation. The density dependent probability distribution that describes how population emergence is spread over time plays an important role in determining population dynamics. Our extensive numerical simulations with a density dependent gamma distribution indicate very rich population dynamics, from stable/unstable equilibria, limit cycles, to chaos.

Invading and Receding Sharp-Fronted Travelling Waves

Biological invasion, whereby populations of motile and proliferative individuals lead to moving fronts that invade vacant regions, is routinely studied using partial differential equation models based upon the classical Fisher-KPP equation. While the Fisher-KPP model and extensions have been successfully used to model a range of invasive phenomena, including ecological and cellular invasion, an often-overlooked limitation of the Fisher-KPP model is that it cannot be used to model biological recession where the spatial extent of the population decreases with time. In this work, we study the Fisher-Stefan model, which is a generalisation of the Fisher-KPP model obtained by reformulating the Fisher-KPP model as a moving boundary problem. The nondimensional Fisher-Stefan model involves just one parameter, [Formula: see text], which relates the shape of the density front at the moving boundary to the speed of the associated travelling wave, c. Using numerical simulation, phase plane and perturbation analysis, we construct approximate solutions of the Fisher-Stefan model for both slowly invading and receding travelling waves, as well as for rapidly receding travelling waves. These approximations allow us to determine the relationship between c and [Formula: see text] so that commonly reported experimental estimates of c can be used to provide estimates of the unknown parameter [Formula: see text]. Interestingly, when we reinterpret the Fisher-KPP model as a moving boundary problem, many overlooked features of the classical Fisher-KPP phase plane take on a new interpretation since travelling waves solutions with [Formula: see text] are normally disregarded. This means that our analysis of the Fisher-Stefan model has both practical value and an inherent mathematical value.