scholarly journals Spreading speeds of KPP-type lattice systems in heterogeneous media

2018 ◽  
Vol 22 (01) ◽  
pp. 1850083 ◽  
Author(s):  
Xing Liang ◽  
Tao Zhou
Keyword(s):  
Heterogeneous Media ◽  
Diffusion Rate ◽  
Reaction Diffusion ◽  
Lattice System ◽  
Reaction Diffusion Equations ◽  
Almost Periodic ◽  
Spreading Speeds ◽  
Lattice Systems ◽  
One Dimensional ◽  
Type Lattice

In this paper, we investigated spreading properties of the solutions of the Kolmogorov–Petrovsky–Piskunov–type (KPP-type) lattice system [Formula: see text] Motivated by the work in [H. Berestycki and G. Nadin, Spreading speeds for one-dimensional monostable reaction–diffusion equations, J. Math. Phys. 53(11) (2012) 115619, 23 pp.], we develop some new discrete Harnack-type estimates and homogenization techniques for the lattice system [Formula: see text] to construct two speeds [Formula: see text] such that [Formula: see text] for any [Formula: see text], and [Formula: see text] for any [Formula: see text]. These speeds are characterized by two generalized principal eigenvalues of the linearized systems of [Formula: see text]. In particular, we derive the exact spreading speed when the coefficients are random stationary ergodic or almost periodic (where [Formula: see text]). Finally, in the case where [Formula: see text] is almost periodic in [Formula: see text] and the diffusion rate [Formula: see text] is independent of [Formula: see text], we show that the spreading speeds in the positive and negative directions are identical even if [Formula: see text] is not invariant with respect to the reflection.

A reaction–diffusion epidemic model with incubation period in almost periodic environments

2020 ◽  
pp. 1-24
Author(s):  
LIZHONG QIANG ◽  
BIN-GUO WANG ◽  
ZHI-CHENG WANG
Keyword(s):  
Time Delay ◽  
Spatial Heterogeneity ◽  
Incubation Period ◽  
Epidemic Model ◽  
Reaction Diffusion ◽  
Fixed Time ◽  
Reaction Diffusion Equations ◽  
Threshold Dynamics ◽  
Almost Periodic ◽  
Periodic Reaction

In this paper, we propose and study an almost periodic reaction–diffusion epidemic model in which disease latency, spatial heterogeneity and general seasonal fluctuations are incorporated. The model is given by a spatially nonlocal reaction–diffusion system with a fixed time delay. We first characterise the upper Lyapunov exponent $${\lambda ^*}$$ for a class of almost periodic reaction–diffusion equations with a fixed time delay and provide a numerical method to compute it. On this basis, the global threshold dynamics of this model is established in terms of $${\lambda ^*}$$ . It is shown that the disease-free almost periodic solution is globally attractive if $${\lambda ^*} < 0$$ , while the disease is persistent if $${\lambda ^*} < 0$$ . By virtue of numerical simulations, we investigate the effects of diffusion rate, incubation period and spatial heterogeneity on disease transmission.

scholarly journals Competition and boundary formation in heterogeneous media: Application to neuronal differentiation

2015 ◽  
Vol 25 (13) ◽  
pp. 2477-2502 ◽  
Author(s):  
Benoît Perthame ◽  
Cristóbal Quiñinao ◽  
Jonathan Touboul
Keyword(s):  
Gene Expression ◽  
Heterogeneous Media ◽  
Blow Up ◽  
General Setting ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Developmental Phase ◽  
Inhomogeneous System ◽  
Small Diffusion ◽  
Drive Gene

We analyze an inhomogeneous system of coupled reaction–diffusion equations representing the dynamics of gene expression during differentiation of nerve cells. The outcome of this developmental phase is the formation of distinct functional areas separated by sharp and smooth boundaries. It proceeds through the competition between the expression of two genes whose expression is driven by monotonic gradients of chemicals, and the products of gene expression undergo local diffusion and drive gene expression in neighboring cells. The problem therefore falls in a more general setting of species in competition within a nonhomogeneous medium. We show that in the limit of arbitrarily small diffusion, there exists a unique monotonic stationary solution, which splits the neural tissue into two winner-takes-all parts at a precise boundary point: on both sides of the boundary, different neuronal types are present. In order to further characterize the location of this boundary, we use a blow-up of the system and define a traveling wave problem parametrized by the position within the monotonic gradient: the precise boundary location is given by the unique point in space at which the speed of the wave vanishes.

scholarly journals Averaging Principles for Nonautonomous Two-Time-Scale Stochastic Reaction-Diffusion Equations with Jump

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-22
Author(s):  
Yong Xu ◽  
Ruifang Wang
Keyword(s):  
Reaction Diffusion ◽  
Evolution Family ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Averaging Principle ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Random Measures ◽  
Fast System ◽  
Stochastic Reaction

In this paper, we aim to develop the averaging principle for a slow-fast system of stochastic reaction-diffusion equations driven by Poisson random measures. The coefficients of the equation are assumed to be functions of time, and some of them are periodic or almost periodic. Therefore, the Poisson term needs to be processed, and a new averaged equation needs to be given. For this reason, the existence of time-dependent evolution family of measures associated with the fast equation is studied and proved that it is almost periodic. Next, according to the characteristics of almost periodic functions, the averaged coefficient is defined by the evolution family of measures, and the averaged equation is given. Finally, the validity of the averaging principle is verified by using the Khasminskii method.

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