Entire solutions in a two-dimensional nonlocal lattice dynamical system

Abstract

This paper is concerned with entire solutions for a two-dimensional periodic lattice dynamical system with nonlocal dispersal. In the bistable case, by applying comparison principle and constructing appropriate upper-and lowersolutions, two different types of entire solutions are constructed. The first type behaves like a monostable front merges with a bistable front and one chases another from the same side; while the other type can be represented by two monostable fronts merge and converge to a single bistable front. In the monostable case, we first establish the existence and properties of spatially periodic solutions which connect two steady states. Then new types of entire solutions are constructed by mixing a heteroclinic orbit of the spatially averaged ordinary differential equations with traveling wave fronts with different speeds. Further, for a class of special heterogeneous reaction function, we establish the uniqueness and continuous dependence of the entire solution on parameters, such as wave speeds and shifted variables. k1 k2 J(k 1 , k 2)u i−k1,j−k2 (t) − u i,j (t) is the nonlocal dispersal and represents transportation due to long range dispersion mechanism. The kernel function J : Z × Z → [0, ∞), is a probability function formulating the dispersal of individuals and satisfies (J): J(•, •) ≥ 0 is even, k1 k2 J(k 1 , k 2) = 1, J(k 1 , k 2) = 0 if |k 1 | > k 0 or |k 2 | > k 0 , where k 0 is a positive constant. The reaction term f i,j (•), i, j ∈ Z satisfies

Entire solutions originating from three fronts to a two-dimensional nonlocal periodic lattice dynamical system

“…Such two types of entire solutions are often called as merging-front entire solutions. In [9], the authors have established the existence of merging-front entire solutions originating from two fronts for the system of (1.1) with both monostable and bistable nonlinearities.…”

“…where g i,j (v) = −f i,j (a − v). Then it follows from [2,9] that there exists a v * (θ) > 0 such that for every v1 , v1 > v * , (1.5) has two increasing pulsating traveling fronts Wi,j (icosθ + jsinθ + v1 t) and Ŵi,j (icosθ + jsinθ + v1 t) which satisfy Wi,j (−∞) = Ŵi,j (−∞) = 0 and Wi,j (+∞) = Ŵi,j (+∞) = a, for i, j ∈ Z.…”

Exponential attractors for two-dimensional nonlocal diffusion lattice systems with delay

“…where h > 0 denotes the delay time; for i ∈ Z 2 , λ i , u i (t), F i (u i (t)), G i (u i (t−h)), ρ i ∈ R; j∈Z 2 [J(j)u i−j (t)−u i (t)] is the nonlocal diffusion. Very recently, entire solutions for a two-dimensional periodic lattice dynamical system with nonlocal dispersal have been established in [8]. Exponential attractors have been introduced in [10] to characterize the long term behavior of solutions for infinite dimensional dynamical systems.…”

Novel entire solutions in a nonlocal 2-D discrete periodic media for bistable dynamics

“…Such two types of entire solutions are often called as merging-front entire solutions. In [13], the authors have established the existence of merging-front entire solutions originating from two fronts for the system of (1) with both monostable and bistable nonlinearities.…”

“…Since the comparison principle can be well applied, we only need to construct a suitable pair of super-and subsolutions by some auxiliary rational functions with certain properties which were developed by Morita and Ninomiya in [24]. This technique had been used to prove some types of entire solutions originating from two fronts of (1) in [13]. Therefore, we would apply the technique to establish some new types of entire solutions originating from three fronts of (1), i.e.…”