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讲准字385号:共生模型的稳定性结论与有界性

发布时间:2017-12-07|浏览次数:

题目:共生模型的稳定性结论与有界性

主讲:Paul

时间:12月8日 10:20

地点:理学院206

主办:理学院


主讲简介:Paul,罗马尼亚雅西技术大学教授。研究专长:生物数学。发表学术论文40余篇。JBS的编委。

主讲内容:Many models of mutualism have been proposed and studied individually. Also, while most existing models of mutualism are bi-dimensional, ecosystems are seldom limited to groups of only two species interactions. To overcome these issues, we develop a general model of a mutualistic interaction with an arbitrary number of species which encompasses several classic two-species models, even when extended to their respective multidimensional versions. Our model is defi ned in terms of consistency hypotheses introduced ad hoc, the focus being on obtaining conditions for the boundedness and unboundedness of solutions, respectively, in terms of threshold parameters which are similar in scope to the basic reproduction number from Mathematical Epidemiology. Our threshold parameters, however, are computed in very di fferent conditions. That is, they are not computed in a near-extinction situation, as it is the case in Mathematical Epidemiology and, to a lesser extent, in Ecology, but at high population densities, under given species proportionality. The reason is that, as far as the validity of the model is concerned, what is important is not the extinction of species, but their blow-up. Also, a model of mutualism has an entirely di fferent structure, not exhibiting the asymmetry which is characteristic to disease propagation models and predator-prey models. A single threshold parameter, based on the dynamics of a single species or compartment, may consequently not be enough to describe the behavior of solutions for a model of mutualism, and we employ one reproductive ratio per species to introduce our boundedness conditions. We also observe that for a representative class of models, the boundedness condition can be expressed in terms of eigenvalues for a certain matrix of coe fficients, which represents an useful algebraic test for boundedness. We then discuss particular cases in which there is a single threshold parameter separating boundedness from unboundedness. The situation in which the unboundedness is caused by a particular subset of species is also of concern. For the particular case of two-species mutualisms, using mild assumptions on the growth and self-limiting functions, we prove the global stability of a unique coexistence equilibrium whenever it exists. Our results also allow each of the mutualists to be subject to a weak Allee e ffect. Moreover, we fi nd that if one of the interacting species is subject to a strong Allee effect, then the mutualism can overcome it and cause a unique coexistence equilibrium to be globally stable.


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