# Lotka-Volterra竞争模型的动力学研究The Dynamics of Lotka Volterra Competition Model

DOI: 10.12677/AAM.2019.812230, PDF, HTML, XML, 下载: 312  浏览: 567

Abstract: In this paper, an improved Lotka-Volterra model is established. Through the dynamic analysis of the model, the stability of the equilibrium point and the occurrence conditions of several kinds of bifurcations are explored.

1. 引言

$\begin{array}{l}\frac{\text{d}u}{\text{d}t}=u\left({r}_{1}-u-\frac{{c}_{1}u}{1+v}-{c}_{2}v\right)\\ \frac{\text{d}v}{\text{d}t}=v\left({r}_{2}-v-\frac{{b}_{1}v}{1+u}-{b}_{2}u\right)\end{array}$ (1)

${r}_{1}$${r}_{2}$ ：物种的内在增长率；

${c}_{1}$ ：物种u的被捕食系数；

${b}_{1}$ ：物种v的被捕食系数；

${c}_{2}$ ：物种v对u的竞争系数；

${b}_{2}$ ：物种u对v的竞争系数。

2. 系统的平衡点极其稳定性分析

2.1. 平衡点

1) 在任意参数下， ${E}_{0}=\left(0,0\right)$ 都是该模型的一个平衡点；

2) 当u或v有一方灭绝时， ${E}_{1}=\left(\frac{{r}_{1}}{1+{c}_{1}},0\right)$${E}_{2}=\left(0,\frac{{r}_{2}}{1+{b}_{1}}\right)$ 也是该系统边界平衡点，此时一个物种灭绝，另一物种的种内竞争与内在增长率相同；

3) 非零平衡点，即共存平衡点 ${E}_{3}=\left({u}^{\text{*}},{v}^{*}\right)$ ，其中 ${u}^{*}=\frac{{r}_{1}+{r}_{1}{v}^{*}-{c}_{2}{v}^{*}-{c}_{2}{\left({v}^{*}\right)}^{2}}{1+{v}^{*}+{c}_{1}}$${r}_{2}-{v}^{*}-\frac{{b}_{1}{v}^{*}}{1+{u}^{*}}-{b}_{2}{u}^{*}=0$${u}^{*}\ne 0$${v}^{*}\ne 0$

2.2. 稳定性分析

1) 若矩阵A所有特征值实部为负，则系统在平衡点 ${x}_{e}$ 处是渐进稳定的；

2) 若矩阵A的特征值中有一个具有正实部，则系统在平衡点 ${x}_{e}$ 处是不稳定的；

3) 若矩阵A的特征值中有一个实部为零，则系统的稳定性与高阶项有关。

$A=\left(\begin{array}{cc}{r}_{1}-2u-\frac{2{c}_{1}u}{1+v}& \frac{{c}_{1}u}{{\left(1+v\right)}^{2}}-{c}_{2}u\\ {r}_{2}-2v-\frac{2{b}_{1}v}{1+u}& \frac{{b}_{1}v}{{\left(1+u\right)}^{2}}-{b}_{2}v\end{array}\right)$ (2)

$A=\left(\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right)$ (3)

${\lambda }^{2}+{m}_{1}\lambda +{m}_{2}=0$ (4)

3. 几类分岔的条件

3.1. N-S分岔

$\frac{\text{d}x}{\text{d}t}=\frac{{x}_{n+1}-{x}_{n}}{{t}_{n+1}-{t}_{n}}$ (5)

$\left(\begin{array}{l}{u}_{n+1}\\ {v}_{n+1}\end{array}\right)=\left(\begin{array}{l}{u}_{n}+\tau u\left({r}_{1}-u-\frac{{c}_{1}u}{1+v}-{c}_{2}v\right)\\ {v}_{n}+\tau v\left({r}_{2}-v-\frac{{b}_{1}v}{1+u}-{b}_{2}u\right)\end{array}\right)$ (6)

N-S分岔的条件：

3.2. flip分岔

 [1] Baretta-Bekker, J.G., Baretta, J.W. and Rasmussen, E.K. (1995) The Microbial Food Web in the european Regional Seas Ecosystem Model. Netherlands Journal of Sea Research, 33, 363-379. https://doi.org/10.1016/0077-7579(95)90053-5 [2] Becks, L., Hilker, F.M., Malchow, H., Jürgens, K. and Arndt, H. (2005) Experimental Demonstration of Chaos in a Microbial Food Web. Nature, 435, 1226-1229. https://doi.org/10.1038/nature03627 [3] Kooi, B.W., Kuijper, L.D.J. and Boer, M.P. (2002) Numerical Bifurcation Analysis of a Tri-Trophic Food Web with Omnivory. Mathematical Biosciences, 177-178, 201-228. https://doi.org/10.1016/S0025-5564(01)00111-0 [4] Perhar, G. and Arhonditsis, G.B. (2009) The Effects of Seston Food Quality on Planktonic Food Web Patterns. Ecological Modelling, 220, 805-820. https://doi.org/10.1016/j.ecolmodel.2008.12.019 [5] Crowley, D.M., Jones, D.E. and Greenberg, M.T. (2012) Resource Consumption of a Diffusion Model for Prevention Programs: The PROSPER Delivery System. Journal of Adolescent Health Official Publication of the Society for Adolescent Medicine, 50, 256-263. https://doi.org/10.1016/j.jadohealth.2011.07.001 [6] Room P.M. (1990) Ecology of a Simple Plant-Herbivore System: Biological Control of Salvinia. Trends in Ecology & Evolution, 5, 74-79. https://doi.org/10.1016/0169-5347(90)90234-5 [7] Sarkar, R.R., Pal, S. and Chattopadhyay, J. () Role of Two Toxin-Producing Plankton and Their Effect on Phytoplankton-Zooplankton System—A Mathematical Study Supported by Experimental Findings. Biosystems, 80, 11-23. https://doi.org/10.1016/j.biosystems.2004.09.029 [8] Riegman, R. and Kuipers, B.R. (1994) Resource Competition and Selective Grazing of Plankton in a Multispecies Pelagic Food Web Model*. Marine Ecology, 15, 153-164. https://doi.org/10.1111/j.1439-0485.1994.tb00050.x [9] Matsuda, H., Ogita, N. and Sasaki, A. (1992) Statistical Mechanics of Population: The Lattice Lotka-Volterra Model. Progress of Theoretical Physics, 88, 1035-1049. https://doi.org/10.1143/ptp/88.6.1035 [10] Prasad, A., Dana, S.K., Karnatak, R., et al. (2008) Universal Occurrence of the Phase-Flip Bifurcation in Time-Delay Coupled Systems. Chaos, 18, Article ID: 023111. https://doi.org/10.1063/1.2905146