Monotone travelling fronts in an age-structured reaction-diffusion model of a single species

Abstract

We consider a partially coupled diffusive population model in which the state variables represent the densities of the immature and mature population of a single species. The equation for the mature population can be considered on its own, and is a delay differential equation with a delay-dependent coefficient. For the case when the immatures are immobile, we prove that travelling wavefront solutions exist connecting the zero solution of the equation for the matures with the delay-dependent positive equilibrium state. As a perturbation of this case we then consider the case of low immature diffusivity showing that the travelling front solutions continue to persist. Our findings are contrasted with recent studies of the delayed Fisher equation. Travelling fronts of the latter are known to lose monotonicity for sufficiently large delays. In contrast, travelling fronts of our equation appear to remain monotone for all values of the delay.

Traveling fronts in diffusive and cooperative Lotka–Volterra system with nonlocal delays

“…Britton [4,5] considered these two factors and introduced the so-called spatio-temporal delay or nonlocal delay, that is, a delay term involves a weighted spatio-temporal average over the whole infinite domain and for previous times. Since then, great progress has been made on the existence of traveling wave fronts in reactiondiffusion equations with nonlocal delays, see Ashwin et al [1], Al-Omari and Gourley [2], Billingham [3], Gourley [7], Gourley and Britton [8], Gourley et al [9], Gourley and Kuang [10], Gourley and Ruan [11], Liang and Wu [13], So et al [15], Wang et al [16], Xu et al [18] and the references cited therein. There are three methods which have been used to prove existence of traveling wave solutions in these works.…”

“…There are three methods which have been used to prove existence of traveling wave solutions in these works. The first one is the perturbation theory of ordinary differential Wang ZAMP equations coupled with the Fredholm alternative, see Al-Omari et al [2] for an age-structured reaction-diffusion model with nonlocal delay and Gourley [7] for a nonlocal Fisher equation. The second one is the geometric singular perturbation theory of Fenichel [6], see Ashwin et al [1], Gourley and Ruan [11], etc.…”

Global Stability of Monostable Traveling Waves For Nonlocal Time-Delayed Reaction-Diffusion Equations

“…If we further assume that the immature species is almost nonmobile, i.e., the impact factor α of spatial diffusion for the immature population is sufficiently close to zero, by using the property of the heat kernel f α (y) = On the other hand, if we take d(u) = δu 2 , δ > 0 and εb(u) = pe −γτ u, p > 0, γ > 0, then (1.1) reduces to the following nonlocal age-structured population model (see, e.g., [1,2,3,6,11,12,32,41,44])…”

The Diffusive Spread of Alleles in Heterogeneous Populations

“…The work of Neubert and Caswell (2000), although ecological in emphasis, is most relevant to my present treatment of wave speeds. My results concerning a population structured with juveniles and adults are also applicable to a similar model studied by Al-Omari and Gourley (2002). Following Lui (1989a), Neubert and Caswell (2000) executed the change of variables y ϭ x Ϫ ct, identified exponential-type solutions, and computed wave speeds using an optimization procedure involving the principal eigenvalue of the resulting matrix of moment-generating functions and demographic parameters.…”