一类竞争扩散系统的稳定性分析
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摘要
基于常微分方程定性与稳定性理论以及分支理论,研究一类具有毒素和收获影响的竞争扩散系统在不同斑块下的扩散性质.证明了该系统在第一卦限内存在一个六面体吸引域;找到系统的一切正解是永久存在的条件;最后利用数值模拟仿真进一步验证定理的准确性,补充了前人的结果.
In this paper,the diffusion properties of a competitive diffusion system with toxin and gain under different patches is studied based on the qualitative and stability theory of differential equation and bifurcation theory. The attraction domain exists in the first quadrant is proved. The positive solutions are permanent existence which is given when the certain conditions are satisfied in the system. We find the system is more accurate than system without the simulation in the last,and the conclusion is renewed.
In this paper,the diffusion properties of a competitive diffusion system with toxin and gain under different patches is studied based on the qualitative and stability theory of differential equation and bifurcation theory. The attraction domain exists in the first quadrant is proved. The positive solutions are permanent existence which is given when the certain conditions are satisfied in the system. We find the system is more accurate than system without the simulation in the last,and the conclusion is renewed.
引文
[1] TAKEUCHI Y,LU Z. Permanence and global stability for competitive Lotka-volterra diffusion systems[J]. Nonlinear Analysis,1995,24:91-104.
[2] TAKEUCHI Y. Conflict between the need to forge and the need to avoid competition,persistence of two-species model[J].Math Biosci,1990,99:181-194.
[3] DAS T,MUKHERJEE R N,CHAUDHURI K S. Harvesting of a prey-preydator fishery in the presence of toxicity[J]. Applied Mathematical Modelling,2009,33:2282-2292.
[4] TAKEUCHI Y. Diffusion-mediated persistence in two-species competition Lotka-volterra model[J]. Math Biosci,1989,95:65-83.
[5] CHEN F D,XIE X D. Permanence and extinction in nonlinear single and multiple species system with diffusion[J]. Applied Mathematics and Computation,2006,177:410-426.
[6]张兴安,陈兰荪.一类捕食与被捕食LV模型的扩散性质[J].系统科学与数学,1999,19(4):407-414.
[7]武海辉,窦霁虹,王秋芬.具有HollingⅡ类反应函数捕食模型的扩散作用[J].西北大学学报,2010,40(2):189-194.
[8]武海辉.一类带毒素和收获项的扩散模型的定性分析[J].河南科学,2014,32(2):143-146.
[9]武海辉,王秋芬.一类L-V捕食扩散模型的局部Hopf分支分析[J].河南科学,2017,35(3):355-359.
[10] ZHANG X A,CHEN L S. The Linear and nonlinear diffusion of the competitive Lotka-volterra model[J]. Nonlinear Analysis,2007,66:2767-2776.
[11]彭书新,窦霁虹,何德明.具有修正因子的两种群微分模型的定性分析[J].西北大学学报,2008,38(5):693-697.
[12] EDELSTEN-KESHET L. Mathmatical model in biology[M]. New York:Rendom House,1988.
[13] MCHICH R,AUGER P,POGGIALE J C. Effect of predator density dependent dispersal of prey on stability of a predator-prey system[J]. Mathematical Biosciences,2007,206:343-356.
[14] GUO S J,YAN S L. Hopf bifurcation in a diffusive Lotka-volterra type system with nonlocal delay effect[J]. Journal of Differential Equations,2016,260:781-817.
[15]倪文杰.两类反应扩散方程组的定性分析[D].哈尔滨:哈尔滨工业大学,2018.
[2] TAKEUCHI Y. Conflict between the need to forge and the need to avoid competition,persistence of two-species model[J].Math Biosci,1990,99:181-194.
[3] DAS T,MUKHERJEE R N,CHAUDHURI K S. Harvesting of a prey-preydator fishery in the presence of toxicity[J]. Applied Mathematical Modelling,2009,33:2282-2292.
[4] TAKEUCHI Y. Diffusion-mediated persistence in two-species competition Lotka-volterra model[J]. Math Biosci,1989,95:65-83.
[5] CHEN F D,XIE X D. Permanence and extinction in nonlinear single and multiple species system with diffusion[J]. Applied Mathematics and Computation,2006,177:410-426.
[6]张兴安,陈兰荪.一类捕食与被捕食LV模型的扩散性质[J].系统科学与数学,1999,19(4):407-414.
[7]武海辉,窦霁虹,王秋芬.具有HollingⅡ类反应函数捕食模型的扩散作用[J].西北大学学报,2010,40(2):189-194.
[8]武海辉.一类带毒素和收获项的扩散模型的定性分析[J].河南科学,2014,32(2):143-146.
[9]武海辉,王秋芬.一类L-V捕食扩散模型的局部Hopf分支分析[J].河南科学,2017,35(3):355-359.
[10] ZHANG X A,CHEN L S. The Linear and nonlinear diffusion of the competitive Lotka-volterra model[J]. Nonlinear Analysis,2007,66:2767-2776.
[11]彭书新,窦霁虹,何德明.具有修正因子的两种群微分模型的定性分析[J].西北大学学报,2008,38(5):693-697.
[12] EDELSTEN-KESHET L. Mathmatical model in biology[M]. New York:Rendom House,1988.
[13] MCHICH R,AUGER P,POGGIALE J C. Effect of predator density dependent dispersal of prey on stability of a predator-prey system[J]. Mathematical Biosciences,2007,206:343-356.
[14] GUO S J,YAN S L. Hopf bifurcation in a diffusive Lotka-volterra type system with nonlocal delay effect[J]. Journal of Differential Equations,2016,260:781-817.
[15]倪文杰.两类反应扩散方程组的定性分析[D].哈尔滨:哈尔滨工业大学,2018.