2,087 research outputs found
Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model
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 , we prove
the uniqueness (for a fixed propagation speed, up to translation) of each
monotone front.Comment: 20 pages, submitte
A note on the existence of non-monotone non-oscillating wavefronts
In this note, we present a monostable delayed reaction-diffusion equation
with the unimodal birth function which admits only non-monotone wavefronts.
Moreover, these fronts are either eventually monotone (in particular, such is
the minimal wave) or slowly oscillating. Hence, for the Mackey-Glass type
diffusive equations, we answer affirmatively the question about the existence
of non-monotone non-oscillating wavefronts. As it was recently established by
Hasik {\it et al.} and Ducrot {\it et al.}, the same question has a negative
answer for the KPP-Fisher equation with a single delay.Comment: 11 pages, 3 figures, submitte
Perron Theorem in the Monotone Iteration Method for Traveling Waves in Delayed Reaction-Diffusion Equations
In this paper we revisit the existence of traveling waves for delayed
reaction diffusion equations by the monotone iteration method. We show that
Perron Theorem on existence of bounded solution provides a rigorous and
constructive framework to find traveling wave solutions of reaction diffusion
systems with time delay. The method is tried out on two classical examples with
delay: the predator-prey and Belousov-Zhabotinskii models.Comment: 17 pages. To appear in Journal of Differential Equation
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