一类具有毒素的非均匀chemostat模型正解的存在性和唯一性

摘要/Abstract

摘要：

研究了一类具有毒素的非均匀chemostat食物链模型.运用稳定性理论分析了平凡解和半平凡解的稳定性，并采用最大值原理和上下解方法给出了正解的先验估计.接着，利用不动点指数理论得到了正解存在的充分条件.最后，通过扰动理论和分歧理论讨论了毒素对动力学行为的影响，得到正解的稳定性和唯一性.结果表明毒素存在时，当微生物u和v的生长率较大时物种能共存.进而当毒素的影响充分大且微生物v的生长率介于一定范围内时系统存在唯一且稳定的正解.

关键词: Chemostat模型, 毒素, 不动点指数理论, 分歧理论, 扰动理论, 唯一性

Abstract:

A food chain model in the unstirred chemostat with toxins is studied. The stability of the trivial solution and semi-trivial solution is analyzed by means of the stability theory, and a priori estimate of positive solution is given by the maximum principle and the super and sub-solution method. Then, by using the fixed point index theory, the sufficient conditions for the existence of positive solutions are achieved. Finally, the effect of the toxins on the dynamic behavior is discussed by virtue of the perturbation theory and bifurcation theory, and the stability and uniqueness of positive solutions are obtained. The results show that the species can coexist when the growth rates of the microorganisms u and v are larger in the presence of the toxins. Furthermore, if the effect of the toxins is sufficiently large, the system has unique stable positive solution when the growth rate of the microorganism v belongs to a certain range.

Key words: Chemostat model, Toxins, Fixed point index theory, Bifurcation theory, Perturbation theory, Uniqueness

中图分类号:

O175.26

李海侠. 一类具有毒素的非均匀chemostat模型正解的存在性和唯一性[J]. 数学物理学报, 2020, 40(5): 1175-1185.

Haixia Li. Existence and Uniqueness of Positive Solutions to an Unstirred Chemostat with Toxins[J]. Acta mathematica scientia,Series A, 2020, 40(5): 1175-1185.

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