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

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

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

Travelling wave solutions for Kolmogorov-type delayed lattice reaction–diffusion systems

[[abstract]]This work investigates the existence and non-existence of travelling wave solutions for Kolmogorov-type delayed lattice reaction–diffusion systems. Employing the cross iterative technique coupled with the explicit construction of upper and lower solutions in the theory of quasimonotone dynamical systems, we can find two threshold speeds c∗ and c∗ with c∗≥c∗>0. If the wave speed is greater than c∗, then we establish the existence of travelling wave solutions connecting two different equilibria. On the other hand, if the wave speed is smaller than c∗, we further prove the non-existence result of travelling wave solutions. Finally, several ecological examples including one-species, two-species and three-species models with various functional responses and time delays are presented to illustrate the analytical results.[[notice]]補正完畢[[journaltype]]國外[[incitationindex]]SCI[[ispeerreviewed]]Y[[booktype]]紙本[[countrycodes]]GB

Traveling Wave in a Ratio-dependent Holling-Tanner System with Nonlocal Diffusion and Strong Allee Effect

In this paper, a ratio-dependent Holling-Tanner system with nonlocal diffusion is taken into account, where the prey is subject to a strong Allee effect. To be special, by applying Schauder's fixed point theorem and iterative technique, we provide a general theory on the existence of traveling waves for such system. Then appropriate upper and lower solutions and a novel sequence, similar to squeeze method, are constructed to demonstrate the existence of traveling waves for c>c*. Moreover, the existence of traveling wave for c=c* is also established by spreading speed theory and comparison principle. Finally, the nonexistence of traveling waves for c<c* is investigated, and the minimal wave speed then is determined

Traveling waves connecting equilibrium and periodic orbit for reaction–diffusion equations with time delay and nonlocal response

AbstractA class of reaction–diffusion equations with time delay and nonlocal response is considered. Assuming that the corresponding reaction equations have heteroclinic orbits connecting an equilibrium point and a periodic solution, we show the existence of traveling wave solutions of large wave speed joining an equilibrium point and a periodic solution for reaction–diffusion equations. Our approach is based on a transformation of the differential equations to integral equations in a Banach space and the rigorous analysis of the property for a corresponding linear operator. Our approach eventually reduces a singular perturbation problem to a regular perturbation problem. The existence of traveling wave solution therefore is obtained by the application of Liapunov–Schmidt method and the Implicit Function Theorem