帶階段結(jié)構(gòu)的擴(kuò)散的捕食者食餌模型解的一致有界性和整體存在性

摘 要 應(yīng)用能量估計(jì)和GagliardoNirenberg型不等式證明了捕食者帶階段結(jié)構(gòu)的具有自擴(kuò)散和交錯(cuò)擴(kuò)散的捕食者-食餌模型解的一致有界性和整體存在性.

關(guān)鍵詞 捕食者-食餌模型；階段結(jié)構(gòu)；交錯(cuò)擴(kuò)散；一致有界；整體解

中圖分類號(hào) O175.26 文獻(xiàn)標(biāo)識(shí)碼 A

Uniform Boundedness and Global Existence of Solutions for a Diffusive PredatorPrey Model with Stage Structure

JIAO Yujuan

（College of Mathematics  Computer Science， Northwest University for Nationalities， Lanzhou，Gansu 730124，China）

Abstract Using the energy estimates and GagliardoNirenberg type inequalities， the uniform boundedness and global existence of the solutions for a predatorprey model with stage structure for the predator with selfand crossdiffusion were proved.

Key words Predatorprey model; stage structure; crossdiffusion; uniform boundedness; global solution

1 Introduction

Population models with stage structure have been investigated by many researchers， and various methods and techniques have been used to study the existence and qualitative properties of solutions［1-6］. In this paper， we investigate the following predatorprey model with stage structure for the predator

(t)=Ax-Bx2-σCxy1-Cxy2，

1(t)=Kxy2-Dy1-My1-e1y21，

2(t)=Dy1-Py2-e2y22， (1)

where x(t) is the population density of the prey，y1(t) and y2(t) are the population densities of the immature and mature predator， respectively. The interaction terms are of LotkaVolterra type， i.e.， based on linear functional response.C denotes the predation rate of the mature predator， σC(0＜σ＜1) is the predation rate of the immature predator which is less than that of the mature predator. M and P are the death rates of the immature and mature predator， respectively. D denotes the rate of transition from the immature predator to the mature predator.

Using the scaling u=Bx/P，v=e1y1/P，w=e1y2/D，dt=dτ/P and redenoting τ by t， we can reduce the system (1) to

dudt=u(a1－u－σ1v－b1w)，dvdt=b2uw－a2v－v2，dwdt=v－w－cw2，（2）



where a1=A/P，a2=(D+M)/P，b1=CD/(e1P)，b2=KD/(BP)，σ1=σC/e1 and c=e2D/(e1P).

To take into account the natural tendency of each species to diffuse， we are led to the following PDE system of reactiondiffusion type

ut－d1Δu=u(a1－u－σ1v－b1w)，x∈Ω，t＞0，vt－d2Δv=b2uw－a2v－v2，x∈Ω，t＞0，wt－d3Δw=v－w－cw2，x∈Ω，t＞0，ηu=ηv=ηw=0，x∈Ω，t＞0，u(x，0)=u0(x)，v(x，0)=v0(x)，

w(x，0)=w0(x)，x∈Ω， (3)



where Ω is a bounded domain in 

瘙 綆 N with smooth boundary Ω， η is the outward unit normal vector on Ω and η=η. u0(x)，v0(x)，w0(x) are nonnegative smooth functions on Ω. The diffusion coefficients di(i=1，2，3) are all positive constants. The homogeneous Neumann boundary condition indicates that the system (3) is selfcontained with zero population flux across the boundary.

Shigesada et al. in their pioneering work［7］ proposed a crossdiffusion model in order to describe spatial segregation of interacting population species in one space dimension. In recent years， more and more attention have been given to the SKT model with other types of reaction term and some generalized threespecies SKT models (see ［7，8］ and the references therein). In this paper， we are led to the following crossdiffusion system