Invasion Waves in a Higher-Dimensional Lattice Competitive System with Stage Structure
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Invasion Waves in a Higher-Dimensional Lattice Competitive System with Stage Structure Kun Li1 Received: 20 July 2019 / Revised: 10 December 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, we use Schauder’s fixed point theorem to establish the existence of invasion waves in a stage-structured competitive system on higher-dimensional lattices. To illustrate our results, we construct a pair of upper and lower solutions. Keywords Higher-dimensional lattice · Stage structure · Traveling wave solution · Schauder’s fixed point theorem · Upper and lower solutions Mathematics Subject Classification 37L60 · 34K10 · 39A10
1 Introduction In this paper, we consider the existence of traveling wave solutions of a higherdimensional lattice competitive system with stage structure ⎧ ⎪ J1 (ξ )e−γ1 τ1 u 1(η−ξ ) (t − τ1 ), ⎪ ⎪ v1η (t) = d1 (n v1 )η + α1 u 1η (t) − γ1 v1η (t) − α1 ⎪ n ⎪ ξ ∈Z ⎪ ⎪ ⎪ ⎪ J1 (ξ )e−γ1 τ1 u 1(η−ξ ) (t − τ1 ) − a1 u 21η (t) − b1 u 1η (t)u 2η (t), u (t) = D1 (n u 1 )η + α1 ⎪ ⎪ ⎨ 1η n ξ ∈Z (t) = d ( v ) + α u (t) − γ v (t) − α ⎪ v J2 (ξ )e−γ2 τ2 u 2(η−ξ ) (t − τ2 ), ⎪ 2 n 2 η 2 2η 2 2η 2 2η ⎪ ⎪ n ⎪ ξ ∈Z ⎪ ⎪ ⎪ ⎪ ⎪ J2 (ξ )e−γ2 τ2 u 2(η−ξ ) (t − τ2 ) − b2 u 1η (t)u 2η (t) − a2 u 22η (t), u (t) = D2 (n u 2 )η + α2 ⎪ ⎩ 2η n ξ ∈Z
(1.1)
Communicated by See Keong Lee. Supported by the National Natural Science Foundation of China (Grant No. 11971160) and the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 18B472).
B 1
Kun Li [email protected] School of Mathematics and Computational Science, Hunan First Normal University, Changsha 410205, Hunan, People’s Republic of China
123
K. Li
where t > 0, (n w)η = |ξ −η|=1,ξ ∈Zn wξ −2nwη , η ∈ Zn , |·| denotes the Euclidean norm in Rn , n ∈ Z+ , ξ ∈Zn Ji (ξ ) = 1, i = 1, 2. System (1.1) is the spatially discrete version of stage-structured reaction–diffusion competitive system ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂v1 (x,t) ∂t ∂u 1 (x,t) ∂t
+ α1 u 1 (x, t) − γ1 v1 (x, t) − α1 e−γ1 τ1 g1 (y)u 1 (x − y, t − τ1 )dy, Rn 2 = D1 ∂ u∂1x(x,t) + α1 e−γ1 τ1 g1 (y)u 1 (x − y, t − τ1 )dy 2 = d1 ∂
2 v (x,t) 1 ∂x2
Rn
−a1 u 21 (x, t) − b1 u 1 (x, t)u 2 (x, t),
∂v2 (x,t) ∂t ∂u 2 (x,t) ∂t
+ α2 u 2 (x, t) − γ2 v2 (x, t) − α2 e−γ2 τ2 g2 (y)u 2 (x − y, t − τ2 )dy, Rn 2 = D2 ∂ u∂2x(x,t) + α2 e−γ2 τ2 g2 (y)u 2 (x − y, t − τ2 )dy 2 = d2 ∂
2 v (x,t) 2 ∂x2
Rn
−b2 u 1 (x, t)u 2 (x, t) − a2 u 22 (x, t),
(1.2) n, g (y)dy = where all the parameters are positive constants, t > 0, x ∈ R Rn i 1, vi , u i , di , Di , αi , ai , bi , γi , τi , αi Rn e−γi τi gi (y)u i (x − y, t − τi )dy, i = 1, 2, denote the densities of the immature population, the densities of the mature population, the diffusive rate of the immature population, the diffusive rate of the mature population, the birth rate, the mature death and overcrowding rate, the rate of competition, the death rate of the immature population, the mature period, the number born at the location y a
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