tailieunhanh - Traveling wavefronts in a single species model with nonlocal diffusion and age-structure

This paper is concerned with the existence of monotone traveling wavefronts in a single species model with nonlocal diffusion and age-structure. We first apply upper and lower solution technique to prove the result if the wave speed is larger than a threshold depending only on the basic parameters. | Turk J Math 34 (2010) , 377 – 384. ¨ ITAK ˙ c TUB doi: Traveling wavefronts in a single species model with nonlocal diffusion and age-structure ∗ X. S. Li and G. Lin Abstract This paper is concerned with the existence of monotone traveling wavefronts in a single species model with nonlocal diffusion and age-structure. We first apply upper and lower solution technique to prove the result if the wave speed is larger than a threshold depending only on the basic parameters. When the wave speed equals to the threshold, we show the conclusion by passing to a limit function. Key Words: Age-structure, nonlocal diffusion, traveling wavefront, upper and lower solutions. 1. Introduction Due to the different behavior of individuals with different ages in population dynamics, Aiello and Freedmann [1] first introduced the following single species model with time delay and age-structure u i (t) = αum (t) − rui (t) − αe−rτ um (t − τ ), u m (t) = αe−rτ um (t − τ ) − βu2m (t), () in which all the parameters are positive, ui and um denote the number of immature and mature individuals of a single species, and time delay τ > 0 describes the time taken from birth to maturity. Based on the model (), Gourley and Kuang [5] further considered the spatial inhomogeneity of the individuals distribution and proposed the following reaction-diffusion system with non-local delays: ⎧ 2 ⎨ ∂ui(x,t) = d Δu (x, t) + αu (x, t) − ru (x, t) − αe−rτ √ 1 e− 4dyi τ u (x − y, t − τ )dy, i i m i m ∂t R 4πdi τ y2 − ⎩ ∂um(x,t) = d Δu (x, t) + αe−rτ √ 1 e 4di τ u (x − y, t − τ )dy − βu2 (x, t), ∂x m m R 4πdi τ m () m where di , dm are positive constants accounting for the diffusivity. In view of the background of the Gaussian kernel, the random migration of the individuals of model () is obvious, see also [10, 12, 13, 14, 15]. 2000 AMS Mathematics Subject Classification: 35K57, 35R20, 92D25. by NSF of China (10871085). ∗Supported 377 LI, LIN Recently, Al-Omari and

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