Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model

Abstract

We are revisiting the topic of travelling fronts for the food-limited (FL) model with spatio-temporal nonlocal reaction. These solutions are crucial for understanding the whole model dynamics. Firstly, we prove the existence of monotone wavefronts. In difference with all previous results formulated in terms of 'sufficiently small parameters', our existence theorem indicates a reasonably broad and explicit range of the model key parameters allowing the existence of monotone waves. Secondly, numerical simulations realized on the base of our analysis show appearance of non-oscillating and non-monotone travelling fronts in the FL model. These waves were never observed before. Finally, invoking a new approach developed recently by Solar et al, we prove the uniqueness (for a fixed propagation speed, up to translation) of each monotone front. Keywords: food-limited model, monotone wave, nonlocal interaction, existence, uniqueness 2010 MSC: 34K60, 35K57, 92D25 R K(s, y, τ )u(t − s, x − y)dy, and K : R + × R × R + → R + is a function satisfyingI c,τ (λ) := R+×R K(s, y, τ )e −λ(cs+y) dsdy ∈ R, I c,τ (0) = 1, lim λ→−λ0(c) + I c,τ (λ) = +∞,

Nonlinearly determined wavefronts of the Nicholson's diffusive equation: when small delays are not harmless

“…This numerical result complements the discussion in Section 7 of [7]. The Nicholson's blowflies diffusive equation together with the food-limited diffusive equations [22,40] seem to be the first scalar models coming from applications where untypical behavior in the form of non-monotone non-oscillating wavefronts is established analytically and also observed in numerical experiments. Among previous studies, we would like to mention an illustrative example in [10] of the Mackey-Glass type diffusive equation with a single delay and with a piece-wise linear birth function.…”