一类捕食-食饵模型共存解的存在性
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摘要
研究一类捕食-食饵模型在齐次Dirichlet边值条件下的共存解.首先,利用极值原理和Young不等式得到正平衡态解的先验估计;其次,通过计算不动点指数,结合锥上的拓扑度理论和谱分析方法论讨了平衡态方程存在正解的充分必要条件,以及共存解对参数的依赖性;最后,以食饵的死亡率作为分歧参数,利用局部分歧定理证明了发自半平凡解的局部分支的存在性.
The coexistence solutions of a predator-prey model with homogeneous Dirichlet boundary value conditions are studied.Firstly,by using the principle of extremum and the Young inequality,apriori estimate of positive equilibrium solution is given.Secondly,the sufficient and necessary conditions for the existence of positive solutions to equilibrium equation are discussed through the fixed-point index,topological degree theory and spectral analysis methods.Finally,taking the death rate as the bifurcation parameter,the existence of positive solution to this system is derived by making use of local bifurcation theory.
The coexistence solutions of a predator-prey model with homogeneous Dirichlet boundary value conditions are studied.Firstly,by using the principle of extremum and the Young inequality,apriori estimate of positive equilibrium solution is given.Secondly,the sufficient and necessary conditions for the existence of positive solutions to equilibrium equation are discussed through the fixed-point index,topological degree theory and spectral analysis methods.Finally,taking the death rate as the bifurcation parameter,the existence of positive solution to this system is derived by making use of local bifurcation theory.
引文
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[14]郭改慧,李艳玲.带B-D反应项的捕食-食铒模型的全局分支及稳定性[J].应用数学学报,2008,31(2):220.
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[2]GOEL N S,MAITRA S C,MONTROLL E W.On the Volterra and other nonlinear models of interacting population[J].Reviews of Modern Physics,1971,43(2):231.
[3]郭改慧,吴建华.一类捕食-食铒模型正解的存在性和惟一性[J].武汉大学学报(理学版),2008,54(1):9.
[4]KELLER C,LUI R.Existence of steady-state solutions to predator-prey equations in a heterogeneous environment[J].Journal of Mathematical Analysis and Applications,1987,123(2):306.
[5]LI H,LI Y,YANG W.Existence and asymptotic behavior of positive solutions for a one-prey and twocompeting-predators system with diffusion[J].Nonlinear Analysis:Real World Applications,2016,27:261.
[6]王明新.非线性椭圆方程[M].北京:科学出版社,2010.
[7]王明新.非线性抛物方程[M].北京:科学出版社,1997.
[8]钟承奎,范令先,陈文山原.非线性泛函分析[M].兰州:兰州大学出版社,1998.
[9]叶其孝,李正元,王明新,等.反应扩散方程引论[M].北京:科学出版社,2013:10.
[10]袁海龙,李艳玲.一类捕食-食铒模型共存解的存在性与稳定性[J].陕西师范大学学报(自然科学版),2014,42(1):15.
[11]王妮娅,李艳玲.一类捕食-食铒模型正解的存在性和稳定性[J].纺织高校基础科学学报,2015(1):48.
[12]PENG R,YI F,ZHAO X.Spatiotemporal patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme[J].Journal of Differential Equations,2013,254(6):2465.
[13]陈亚浙,吴兰成.二阶椭圆型方程与椭圆型方程组[M].北京:科学出版社,1997.
[14]郭改慧,李艳玲.带B-D反应项的捕食-食铒模型的全局分支及稳定性[J].应用数学学报,2008,31(2):220.
[15]SMOLLER J.Shock Waves and Reaction-diffusion Equations[M].New York:Springer,1999:64.