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本文主要研究下列空间异质环境下Lotka-Volterra交错扩散方程组
其中:Ω是
$\mathbb{R}$ N(N≤3)中的有界区域;c(x)>0和d(x)≥0都是连续函数,ρ(x)>0是光滑函数并且$\partial $ νρ=0,x∈$\partial $ Ω;u和v是被捕食者和捕食者;a,k是正常数,b是实数,a,b代表被捕食者和捕食者的出生率.交错扩散项Δ[ρ(x)vu]=▽[u▽(ρ(x)v)+ρ(x)v▽u]描述u扩散到ρ(x)v浓度低的区域的一种趋势.已有一些文章研究了关于空间异质环境对种群浓度的影响.文献[1-3]对于一些扩散的Lotka-volterra方程组研究了种内之间的退化效果;文献[4-7]研究了对于一些带扩散项的竞争方程组,空间异质环境下的出生率问题;文献[8]对于方程组(1)所对应的平衡解的方程组,证明了当ρ,c,d是常数的时候,方程组由分岔参数b分岔出来的正解的集合Γp形成一个有界的曲线,并且当a和|b|小,k很大时,ρ(x),d(x)使得Γp关于b形成一个型的曲线;文献[9]研究了带两个趋化参数的趋化模型非常数平衡解的存在性.本文主要研究文献[8]中得到的分岔平衡解在分岔点(0,b*,b*)处的稳定性.
The Stability of Bifurcating Solution for a Spatially Heterogeneous Lotka-Volterra Model with Cross Diffusion
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摘要: 主要研究空间异质环境下带交错扩散项的Lotka-Volterra方程组分岔解的局部渐近稳定性.由分岔方向及细致的谱分析,证明了分岔平衡解是局部渐近稳定的.Abstract: In this paper, we concern with the local asymptotical stability of the bifurcating solution for the Lotka-Volterra system with cross diffusion in a spatially heterogeneous environment. By applying a detailed spectral analysis based on the bifurcating direction we prove that the bifurcating steady state solution is locally asymptotically stable.
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Key words:
- spectral analysis /
- stability /
- bifurcating solution .
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