A delay induced nonlocal free boundary problem

We study the dynamics of a population with an age structure whose population range expands with time, where the adult population is assumed to satisfy a reaction– diffusion equation over a changing interval determined by a Stefan type free boundary condition, while the juvenile population satisfies a reaction–diffusion equation whose evolving domain is determined by the adult population. The interactions between the adult and juvenile populations involve a fixed time-delay, which renders the model nonlocal in nature. After establishing the well-posedness of the model, we obtain a rather complete description of its long-time dynamical behaviour, which is shown to follow a spreading–vanishing dichotomy. When spreading persists, we show that the population range expands with an asymptotic speed, which is uniquely determined by an associated nonlocal elliptic problem over the half line. We hope this work will inspire further research on age-structured population models with an evolving population range

Fisher-KPP equation with free boundaries and time-periodic advections

We consider a reaction-diffusion-advection equation of the form: ut = uxx - β(t)ux + f (t, u) for x ∈ (g(t), h(t)), where β(t) is a T-periodic function representing the intensity of the advection, f (t, u) is a Fisher-KPP type of nonlinearity, T periodic in t, g(t) and h(t) are two free boundaries satisfying Stefan conditions. This equation can be used to describe the population dynamics in time-periodic environment with advection. Its homogeneous version (that is, both β and f are independent of t) was recently studied by Gu et al. (J Funct Anal 269:1714-1768, 2015). In this paper we consider the time-periodic case and study the long time behavior of the solutions. We show that a vanishing-spreading dichotomy result holds when β is small; a vanishing transition-virtual spreading trichotomy result holds when β is a medium-sized function; all solutions vanish when β is large. Here the partition of β(t) depends not only on the "size" β := 1T ∫ T0 β(t)dt of β(t) but also on its "shape" β(t):=β(t)-β