Equilibriums' linear stability of a diffusive model of pioneer and climax species interaction
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摘要: 研究扩散的先锋-顶级种群竞争模型的平衡态线性稳定性. 所讨论的系统有4至6个平衡点,其中最多可以有2个正平衡点,其动力学性态十分丰富. 文章对系统可能的各个常数平衡点的线性稳定性进行分析并给出每个平衡点稳定的充分条件或充分必要条件,主要的方法是线性化方法和特征根分析技巧,最后举例说明本文结论的应用并进行了数值模拟和讨论.
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关键词:
- 线性稳定性
Abstract: The linear stability of equilibria for a diffused pioneer and climax species competition model is discussed. This system has at least 4 and at most 6 equilibria with two possible positive equilibria, and thus the dynamical behaviors are very rich. All the constant equilibria of the model are given in the second section, and then the linear stabilities of these equilibria are analyzed in the third section. The sufficient condition or sufficient and necessary condition of the stability for every equilibrium is given. The linearization method is used accompanied with the eigenvalue skill. At the last, an example is given to illustrate the application of the results, with numerical simulation and conclusion discussion are given.-
Keywords:
- linear stability
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[2] J.E. Franke, A.A. Yakubu, Pioneer exclusion in a one-hump discrete pioneer-climax competing system[J],,
{J.Math . Biol,}1994, {\bf 32:} 771-787.
[3]J.R. Buchanan, Asymptotic behavior of two interacting pioneer-climax species[J].
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{ Nature. Resource Modeling,}1990, {\bf 4:}215-227.
[5] S. Summer, Hopf bifurcation in pioneer-climax species models[J].
{ Math. Biosci,}1996,{\bf 137:} 1-24.
[6] S. Summer, Stable periodic behavior in pioneer-climax competing species models with constant rate forcing[J].
{ Nature. Resource Modeling, }1998, {\bf 11:} 155-171.
[7] J.X. Liu, J.J. Wei, Bifurcation analysis of a diffusive model of pioneer and climax species interaction[J].
{Advance in Difference Equation,} 2001,{\bf 24:} 11-52.
[8] J. Robert Buchanan, Turning instability in pioneer-climax species interactions[J].
{ Math. Biosci,} 2005,{\bf 194:} 126-199.
[9] 马知恩, 周义仓, 常微分方程定性与稳定性方法[M]. 北京：科学出版社, 2004.
[10] N. Shigesada, K. Kawasaki, Biology Invasions: Theory and Practice[M]. { Oxford University Press}, Oxford, 1997.
[11] {J.D. Murray, Mathematical Biology: I and II[M]. Spriner-Verlag, New York, 2002.}
[1] W.E. Ricker, Stock and recruitment[J],
{ J. Fish. Res. Board Can,} 1954, {\bf 11:}559-623.
[2] J.E. Franke, A.A. Yakubu, Pioneer exclusion in a one-hump discrete pioneer-climax competing system[J],,
{J.Math . Biol,}1994, {\bf 32:} 771-787.
[3]J.R. Buchanan, Asymptotic behavior of two interacting pioneer-climax species[J].
{Fields 1nst.comm,} 1999, {\bf 21:}51-63.
[4]J.F. Selgrade, G. Namkong, Stable periodic behavior in a pioneer-climax models[J].
{ Nature. Resource Modeling,}1990, {\bf 4:}215-227.
[5] S. Summer, Hopf bifurcation in pioneer-climax species models[J].
{ Math. Biosci,}1996,{\bf 137:} 1-24.
[6] S. Summer, Stable periodic behavior in pioneer-climax competing species models with constant rate forcing[J].
{ Nature. Resource Modeling, }1998, {\bf 11:} 155-171.
[7] J.X. Liu, J.J. Wei, Bifurcation analysis of a diffusive model of pioneer and climax species interaction[J].
{Advance in Difference Equation,} 2001,{\bf 24:} 11-52.
[8] J. Robert Buchanan, Turning instability in pioneer-climax species interactions[J].
{ Math. Biosci,} 2005,{\bf 194:} 126-199.
[9] 马知恩, 周义仓, 常微分方程定性与稳定性方法[M]. 北京：科学出版社, 2004.
[10] N. Shigesada, K. Kawasaki, Biology Invasions: Theory and Practice[M]. { Oxford University Press}, Oxford, 1997.
[11] {J.D. Murray, Mathematical Biology: I and II[M]. Spriner-Verlag, New York, 2002.}
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