5 research outputs found
A Global Attractivity in a Nonmonotone Age-Structured Model with Age Dependent Diffusion and Death Rates
In this paper, we investigated the global attractivity of the positive
constant steady state solution of the mature population governed by
the age-structured model: \begin{equation*} \left\{\begin{array}{ll}
\frac{\partial u}{\partial t}+\frac{\partial u}{\partial a}=D(a)\frac{\partial
^2 u}{\partial x^2} - d(a)u, & t\geq t_0\geq A_l,\;a\geq 0,\; 0< x< \pi,\\
w(t,x)=\int_r^{A_l}u(t,a,x)da,& t\geq t_0\geq A_l,\; 0<x<\pi,\\
u(t,0,x)=f(w(t,x)), & t\geq t_0\geq A_l,\; 0<x<\pi,\\
u_x(t,a,0)=u_x(t,a,\pi)=0,\;& t\geq t_0\geq A_l,\; a \geq 0, \end{array}
\right. \end{equation*} when the diffusion rate and the death rate
are age dependent, and when the birth function is nonmonotone. We
also presented some illustrative examples.Comment: 11 page
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
Monotone traveling wavefronts of the KPP-Fisher delayed equation
In the early 2000's, Gourley (2000), Wu et al. (2001), Ashwin et al. (2002)
initiated the study of the positive wavefronts in the delayed
Kolmogorov-Petrovskii-Piskunov-Fisher equation. Since then, this model has
become one of the most popular objects in the studies of traveling waves for
the monostable delayed reaction-diffusion equations. In this paper, we give a
complete solution to the problem of existence and uniqueness of monotone waves
in the KPP-Fisher equation. We show that each monotone traveling wave can be
found via an iteration procedure. The proposed approach is based on the use of
special monotone integral operators (which are different from the usual Wu-Zou
operator) and appropriate upper and lower solutions associated to them. The
analysis of the asymptotic expansions of the eventual traveling fronts at
infinity is another key ingredient of our approach.Comment: 25 pages, 2 figures, submitte