Persistence of wavefronts in delayed nonlocal reaction–diffusion equations

Abstract

We develop a perturbation argument based on existing results on asymptotic autonomous systems and the Fredholm alternative theory that yields the persistence of traveling wavefronts for reaction-diffusion equations with nonlocal and delayed nonlinearities, when the time lag is relatively small. This persistence result holds when the nonlinearity of the corresponding ordinary reaction-diffusion system is either monostable or bistable. We then illustrate this general result using five different models from population biology, epidemiology and bio-reactors.

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

“…Regarding the spatial dynamics of a single-species population with age-structure and spatial diffusion such as the Australian blowflies population distribution, there is a class of time-delayed reaction-diffusion equations with nonlocal nonlinearity (see, e.g., [6,12,32,40 Here u(t, x) denotes the total mature population of the species (with age greater than the maturation age τ > 0) at time t and position x, D > 0 is the spatial diffusion rate for the mature population, α > 0 is the total amount of diffusion for the immature species and satisfies α ≤ τD, ε > 0 is the survival rate of the species in time τ period and represents the impact of the death rate of the immature population, and f α (y) is the heat kernel in the form of [12,19,20,27,28,32,40,41,45] In particular, when q = 1, b 1 (u) is just the so-called Nicholson's birth rate function.…”

“…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])…”

Spreading Speeds and Traveling Waves for Nonmonotone Integrodifference Equations

“…The spreading speeds were obtained for some non-monotone continuous-time integral equations and time-delayed reactiondiffusion models in [17,19], and a general result on the nonexistence of traveling waves was also given in [19, Theorem 3.5]. The existence of monostable traveling waves were established for several classes of non-monotone time-delayed reactiondiffusion equations in [22,4,16,14]. For certain types of non-monotone discrete-time integrodifference equation models, non-monotone traveling waves and even traveling cycles were observed in [7] by numerical simulations.…”

Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity

“…There has been significant progress in the study of traveling wave solutions for both bistable and monostable equations; see, for example, Ai [1], Ashwin et al [2], Billingham [4], Faria et al [14,15], Gourley and Kuang [17,18], Liang and Wu [25], Ou and Wu [29], Ruan and Xiao [35], Wang et al [41,43], Wu and Zou [46], Zou [48], and the references cites therein.…”