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 $w(t,x)$ 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 $D(a)$ and the death rate $d(a)$ are age dependent, and when the birth function $f(w)$ 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 $et\ al$, 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