弱耦合三种群反应扩散模型的全局渐近稳定性

吴楚芬

doi:10.6054/j.jscnun.2015.01.001

摘要: 近年来, 随着入侵种的增加, 多种群的反应扩散模型开始应用于入侵生态理论的研究. 多种群的反应扩散模型可将空间、种群间相互作用过程融合进入侵速率的预测之中, 且模型中连续参数的使用不受尺度限制, 适用空间尺度较广. 该文研究了一类非线性三种群弱耦合食饵-捕食者反应扩散模型的初边值问题, 通过构造合适的常数上下解以及相应的迭代方式,得到了该系统在齐次Neumann边值条件下平凡解和非负半平凡解的全局渐近稳定性的充分性条件. 所得结果揭示了如何通过控制种群自身的出生率、种间、种内相互作用率来达到某些种群消失, 某些种群持续生存的现象. 这些稳定性条件易于验证且与扩散系数无关, 因此, 该结论也适用于某个

$d_i=0$ 或所有

$d_i=0$ 的相应的抛物-常微分系统.

关键词:

全局渐近稳定性

Abstract: Recently,with the increase of the invasive species, multi-group reaction diffusion model has been used in invasion ecology theory. The multi-group reaction diffusion model can combine the spatial and population process within the prediction of invasion rate, also the use of continuous parameters in the model made it free to broad spatial scale. To simulate the phenomenon of two kinds of invasive species invading the native specie, Kim and Lin\cite{kl} introduced a nonlinear weakly-coupled reaction diffusion system in the three-species model. The initial boundary value problem of the aforementioned model is concerned.By using the method of upper and lower solutions as well as iteration, sufficient condition is obtained for the global asymptotic stability of the trivial solution and the nonnegative semitrivial solutions with the homogeneous Neumann boundary condition. The results reveal by means of controlling the birth rate, interspecific, intraspecific interaction rate of populations to achieve the goal of certain population disappearance or certain population persistence. These conditions are easy to check, and the independent of the diffusion rates and thus the conclusions are also appropriate for the corresponding parabolic-ordinary differential system(

$d_i=0$ for some or all

$i$ ).

Keywords:

global asymptotic stability

[1]王勤龙, 李百炼.生物入侵模型研究进展[J].科技导报, 2011, 29(10):71-79.
[2] Kim K I, Lin Zhigui.Coexistence in the three species predator-prey model with diffusio[J].Appl Math Comp, 2003, 145:701-716.
[3]林支桂 .三种群捕食-被捕食模型中具时滞的抛物系统 [J].数学学报, 2004, 47:559-568.
[4] Chen Yunlan, Wang Mingxin.Asymptotic behavior of solutions of a three-species predator-prey model with diffusion and time delays[J]. Appl Math Lett, 2004, 17:1403-1408.
[5] Wang Yuanming.Asymptotic behavior of solutions for a class of predator-prey reaction-diffusion systems with time delays[J]. J Math Anal Appl, 2007, 328: 137-150.
[6] Pao C V.Nonlinear Parabolic and Elliptic Equations [M]. New York: Plenum, 1996.
[7] Pao C V.Convergence of solutions of reaction-diffusion systems with time delays [J].
Nonlinear Anal, 2002, 48: 349-362.
[8] Pao C V.Global asymptotic stability of Lotka-Volterra 3-species reaction-diffusion systems with time delays [J].J Math Anal Appl, 2003, 281: 186-204.

期刊类型引用(1)

1.

李彤羚, 徐莉, 罗亚云. 一类慢扩散互惠模型解的动力学性质. 南京大学学报(数学半年刊). 2018(01): 70-84 .

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