﻿ 一类离散种群模型的动力学性质 Dynamic Properties of a Class of Discrete Population Model

Vol. 08  No. 08 ( 2019 ), Article ID: 31806 , 8 pages
10.12677/AAM.2019.88171

Dynamic Properties of a Class of Discrete Population Model

Yuqing Chen, Yuman Zhang, Jiangming Xu, Xiaoliang Zhou

School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang Guangdong

Received: July 31st, 2019; accepted: August 15th, 2019; published: August 22nd, 2019

ABSTRACT

In this paper, the dynamic behavior of a class of discrete population models is studied. Fixed point types and hyperbolic properties are determined by discussing coefficient parameters. By using central manifold theorem and bifurcation theory, we obtain the conditions of transcritical bifurcation and flip bifurcation at two fixed points.

Keywords:Discrete Population Model, Central Manifold Theorem, Transcritical Bifurcation, Flip Bifurcation

1. 引言

$x\left(t\right)$$y\left(t\right)$ 分别表示在时间 $t=0,1,2,\cdots$ 时活跃种群和被动种群的密度。Newman等人在 [2] 中提出的活跃种群和被动种群之间的离散模型如下

$\left\{\begin{array}{l}x\left(t+1\right)=\left(1-\epsilon \right)f\left(x\left(t\right)\right)+\epsilon y\left(t\right),\\ y\left(t+1\right)=\left(1-\epsilon \right)y\left(t\right)+\epsilon f\left(x\left(t\right)\right),\\ x\left(0\right)\ge 0,y\left(0\right)\ge 0,x\left(0\right)+y\left(0\right)>0,\end{array}$

$\left\{\begin{array}{l}x\left(t+1\right)=\frac{\left(1-\epsilon \right)\gamma x\left(t\right)}{1+kx\left(t\right)}+\epsilon y\left(t\right),\\ y\left(t+1\right)=\left(1-\epsilon \right)y\left(t\right)+\frac{\epsilon \gamma x\left(t\right)}{1+kx\left(t\right)},\end{array}$ (1)

 (2)

2. 双曲和非双曲情形

2.1. 不动点 ${E}_{0}\left(0,0\right)$ 的性质

${\gamma }_{1}^{*}=\frac{1+{\epsilon }_{1}}{1-3{\epsilon }_{1}},{\epsilon }_{1}=1-\epsilon ,{l}_{1}=\left\{\left(\gamma ,{\epsilon }_{1}\right)|0<{\epsilon }_{1}<1,\gamma =1\right\},{l}_{2}=\left\{\left(\gamma ,{\epsilon }_{1}\right)|0<{\epsilon }_{1}<\frac{1}{3},\gamma ={\gamma }_{1}^{*}\right\},$

${B}_{2}=\left\{\left(\gamma ,{\epsilon }_{1}\right)|1<\gamma <{\gamma }_{1}^{*},0<{\epsilon }_{1}<\frac{1}{3}\right\},{B}_{3}=\left\{\left(\gamma ,{\epsilon }_{1}\right)|\gamma >{\gamma }_{1}^{*},0<{\epsilon }_{1}<\frac{1}{3}\right\}.$

1) 不动点 ${E}_{0}$ 是非双曲的，当且仅当 $\left(\gamma ,{\epsilon }_{1}\right)$ 位于直线 ${l}_{1,2}$ 上；

2) 如果 $\left(\gamma ,{\epsilon }_{1}\right)\in {B}_{0}$ ，该不动点是渐近稳定的；

3) 如果 $\left(\gamma ,{\epsilon }_{1}\right)\in {B}_{1,2}$ ，该不动点是一个鞍点；如果，该不动点为不稳定的结点。

2.2. 不动点 ${E}_{1}\left({x}^{\ast },{x}^{\ast }\right)$ 的性质

${\gamma }_{2}^{*}=\frac{3\epsilon -2}{2-\epsilon },{\mathcal{D}}_{1}=\left\{\left(\gamma ,\epsilon \right)|0<\epsilon <1,\gamma =1\right\},{\mathcal{D}}_{2}=\left\{\left(\gamma ,\epsilon \right)|\frac{2}{3}<\epsilon <1,\gamma ={\gamma }_{2}^{*}\right\},$

${\mathcal{K}}_{1}=\left\{\left(\gamma ,\epsilon \right)|0<\epsilon \le \frac{2}{3},\gamma >1\right\},{\mathcal{K}}_{2}=\left\{\left(\gamma ,\epsilon \right)|\frac{2}{3}<\epsilon <1,\gamma >1\right\},$

${\mathcal{K}}_{3}=\left\{\left(\gamma ,\epsilon \right)|0<\epsilon <1,{\gamma }_{2}^{*}<\gamma <1\right\},{\mathcal{K}}_{4}=\left\{\left(\gamma ,\epsilon \right)|\frac{2}{3}<\epsilon <1,0<\gamma <{\gamma }_{2}^{*}\right\}.$

1) 不动点 ${E}_{1}$ 是非双曲的，当且仅当 $\left(\gamma ,\epsilon \right)\in {\mathcal{D}}_{1,2}$

2) 如果 $\left(\gamma ,\epsilon \right)\in {\mathcal{K}}_{1,2}$ ，不动点 ${E}_{1}$ 是渐近稳定的；

3) 如果 $\left(\gamma ,\epsilon \right)\in {\mathcal{K}}_{3}$ ，不动点 ${E}_{1}$ 是鞍点；如果 $\left(\gamma ,\epsilon \right)\in {\mathcal{K}}_{4}$ ，不动点 ${E}_{1}$ 为不稳定结点。

3. 跨临界分岔和Flip分岔

3.1. ${E}_{0}\left(0,0\right)$ 处的跨临界分岔和Flip分岔

${f}_{1}={\epsilon }^{2}-{\epsilon }_{1}{\lambda }_{2};{f}_{2}={\epsilon }^{2}-{\epsilon }_{1}{\lambda }_{1};{f}_{3}={\epsilon }_{1}^{2};{f}_{4}=\frac{{\epsilon }^{2}}{{\left(1+S\right)}^{2}}={\left(1-\epsilon \right)}^{4}.$

$D{F}_{\gamma }\left({E}_{0}\right)=\left[\begin{array}{cc}{\epsilon }_{1}\gamma & \left(1-{\epsilon }_{1}\right)\\ \left(1-{\epsilon }_{1}\right)\gamma & {\epsilon }_{1}\end{array}\right],$

${\lambda }_{1}=\frac{{\epsilon }_{1}\left(1+\gamma \right)-\sqrt{{\left({\epsilon }_{1}+\gamma {\epsilon }_{1}\right)}^{2}-8{\epsilon }_{1}\gamma +4\gamma }}{2},{\lambda }_{2}=\frac{{\epsilon }_{1}\left(1+\gamma \right)+\sqrt{{\left({\epsilon }_{1}+\gamma {\epsilon }_{1}\right)}^{2}-8{\epsilon }_{1}\gamma +4\gamma }}{2},$

${\left(\epsilon ,{\lambda }_{1}-{\epsilon }_{1}\gamma \right)}^{\tau },{\left(\epsilon ,{\lambda }_{2}-{\epsilon }_{1}\gamma \right)}^{\tau },$ (3)

$\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{cc}-\epsilon & -\epsilon \\ {\lambda }_{1}-{\epsilon }_{1}\gamma & {\lambda }_{2}-{\epsilon }_{1}\gamma \end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right],$ (4)

$\left[\begin{array}{c}u\\ v\\ \gamma \end{array}\right]↦\left[\begin{array}{l}{\lambda }_{1}u\\ {\lambda }_{2}v\\ \gamma \end{array}\right]+\frac{1}{{\lambda }_{2}-{\lambda }_{1}}\left[\begin{array}{c}{f}_{1}\gamma \left(u+v\right)+{f}_{3}{\gamma }^{2}\left(u+v\right)-\frac{{f}_{1}\gamma \left(u+v\right)+{f}_{3}{\gamma }^{2}\left(u+v\right)}{1+k\epsilon \left(u+v\right)}\\ -{f}_{2}\gamma \left(u+v\right)-{f}_{3}{\gamma }^{2}\left(u+v\right)+\frac{{f}_{2}\gamma \left(u+v\right)+{f}_{3}{\gamma }^{2}\left(u+v\right)}{1+k\epsilon \left(u+v\right)}\\ 0\end{array}\right].$ (5)

$\left(\gamma ,{\epsilon }_{1}\right)\in {l}_{1}$ 时，有 ${\lambda }_{2}=1,{\lambda }_{1}={\epsilon }_{1}+{\epsilon }_{1}\gamma -1$ ，由文献 [10] 的定理2.1.4可知，令 ${\gamma }_{1}=\gamma -1$ ，在  附近，不动点 $\left(u,v\right)=\left(0,0\right)$ 的稳定性与分岔可通过研究在中心流形上一参数族映射来确定，中心流形有如下形式：

$u=h\left(v,{\gamma }_{1}\right)=a{v}^{2}+bv{\gamma }_{1}+c{\gamma }_{1}^{2}+\mathcal{O}\left(3\right).$

$\begin{array}{l}\mathcal{N}\left(h\left(u,{\gamma }_{1}\right)\right)\\ =h\left(u,{\gamma }_{1}\right)-{\lambda }_{1}h\left(u,{\gamma }_{1}\right)-\frac{1}{{\lambda }_{2}-{\lambda }_{1}}\left\{{f}_{1}\left({\gamma }_{1}+1\right)\left[h\left(v,{\gamma }_{1}\right)+v\right]+{f}_{3}{\left({\gamma }_{1}+1\right)}^{2}\left[h\left(v,{\gamma }_{1}\right)+v\right]\begin{array}{c}\text{ }\\ \text{ }\end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }-\frac{{f}_{1}\left({\gamma }_{1}+1\right)\left[h\left(u,{\gamma }_{1}\right)+v\right]+{f}_{3}{\left({\gamma }_{1}+1\right)}^{2}\left[h\left(u,{\gamma }_{1}\right)+v\right]}{1+k\epsilon \left[h\left(v,{\gamma }_{1}\right)+v\right]}\right\}=0.\end{array}$

$u=h\left(v,{\gamma }_{1}\right)=\mathcal{O}\left(3\right).$

$v↦{\Phi }_{{\gamma }_{1}}\left(v\right)={\lambda }_{2}v+\frac{1}{{\lambda }_{2}-{\lambda }_{1}}\left[-{f}_{2}\left({\gamma }_{1}+1\right)v-{f}_{3}{\left({\gamma }_{1}+1\right)}^{2}v+\frac{{f}_{2}\left({\gamma }_{1}+1\right)v+{f}_{3}{\left({\gamma }_{1}+1\right)}^{2}v}{1+kv}\right]+\mathcal{O}\left(3\right).$

$\frac{\partial {\Phi }_{{\gamma }_{1}}}{\partial {\gamma }_{1}}\left(0,0\right)=0,\frac{\partial {\Phi }_{{\gamma }_{1}}}{\partial v\partial {\gamma }_{1}}\left(0,0\right)\ne 0,\frac{{\partial }^{2}\Phi }{\partial {v}^{2}}\left(0,0\right)\ne 0.$ (6)

$v↦{\Phi }_{{\gamma }_{1}}\left(v\right)={\lambda }_{2}v+\frac{1}{{\lambda }_{2}-{\lambda }_{1}}\left[-{f}_{2}\left({\gamma }_{1}+1\right)v-{f}_{3}{\left({\gamma }_{1}+1\right)}^{2}v+\frac{{f}_{2}\left({\gamma }_{1}+1\right)v+{f}_{3}{\left({\gamma }_{1}+1\right)}^{2}v}{1+kv}\right]$

$\begin{array}{l}v↦{\Phi }_{{\gamma }_{2}}\left(v\right)={\lambda }_{2}v+\frac{1}{{\lambda }_{2}-{\lambda }_{1}}\left[-{f}_{2}\left({\gamma }_{2}+{\gamma }_{1}^{*}\right)v-{f}_{3}{\left({\gamma }_{2}+{\gamma }_{1}^{*}\right)}^{2}v\begin{array}{c}\text{ }\\ \underset{}{\overset{}{\text{ }}}\end{array}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{f}_{2}\left({\gamma }_{2}+{\gamma }_{1}^{*}\right)v+{f}_{3}{\left({\gamma }_{2}+{\gamma }_{1}^{*}\right)}^{2}v}{1+kv}\right]+\mathcal{O}\left(3\right).\end{array}$

$\left(\gamma ,{\epsilon }_{1}\right)\in {l}_{2}$ 时，有，系统(1)在不动点 ${E}_{0}$

${\left[\frac{\partial {\Phi }_{{\gamma }_{2}}}{\partial {\gamma }_{2}}\frac{{\partial }^{2}{\Phi }_{{\gamma }_{2}}}{\partial {v}^{2}}+2\frac{{\partial }^{2}{\Phi }_{{\gamma }_{2}}}{\partial v\partial {\gamma }_{2}}\right]|}_{\left(0,0\right)}={\alpha }_{1}\ne 0,{\left[\frac{1}{2}{\left(\frac{{\partial }^{2}{\Phi }_{{\gamma }_{2}}}{\partial {v}^{2}}\right)}^{2}+\frac{1}{3}\frac{{\partial }^{3}{\Phi }_{{\gamma }_{2}}}{\partial {v}^{3}}\right]|}_{\left(0,0\right)}={\alpha }_{2}\ne 0.$

3.2. ${E}_{1}\left({x}^{\ast },{x}^{\ast }\right)$ 的跨临界分岔和Flip分岔

${j}_{4}=\epsilon {\epsilon }_{1}^{2}-\epsilon {\epsilon }_{1}{\lambda }_{1}\left({\gamma }_{3}+1\right)-{\epsilon }^{3}\left({\gamma }_{3}+1\right),{j}_{5}=\epsilon {\epsilon }_{1}^{2}-\epsilon {\epsilon }_{1}{\lambda }_{2}\left({\gamma }_{4}+{\gamma }_{2}^{*}\right)-{\epsilon }^{3}\left({\gamma }_{4}+{\gamma }_{2}^{*}\right),$

${j}_{6}=\epsilon {\epsilon }_{1}^{2}-\epsilon {\epsilon }_{1}{\lambda }_{1}\left({\gamma }_{4}+{\gamma }_{2}^{*}\right)-{\epsilon }^{3}\left({\gamma }_{4}+{\gamma }_{2}^{*}\right),{j}_{7}={\epsilon }_{1}{\lambda }_{2}{\gamma }_{2}^{*}-{\epsilon }_{1}^{2}+{\epsilon }^{2}{\gamma }_{2}^{*}$

${j}_{8}=1+{e}^{*}\frac{3\epsilon -2}{2-\epsilon },{j}_{9}=\epsilon {\epsilon }_{1}{\lambda }_{2}+{\epsilon }^{3},e=\frac{2{j}_{4}}{\epsilon \left({\lambda }_{1}-{\lambda }_{2}\right)\left({\lambda }_{2}-1\right)},$

${e}^{*}=\frac{2\left(3\epsilon -2\right)\left(2-\epsilon \right){j}_{6}}{\epsilon {\lambda }_{2}\left({\lambda }_{1}-{\lambda }_{2}\right)\left({\lambda }_{2}-1\right){\left(3\epsilon -2\right)}^{2}-\epsilon {j}_{6}\left(8\epsilon -2\right)}.$

$D{\stackrel{˜}{F}}_{\gamma }\left({E}_{1}\right)=\left[\begin{array}{cc}\frac{{\epsilon }_{1}}{\gamma }& \epsilon \\ \frac{\epsilon }{\gamma }& {\epsilon }_{1}\end{array}\right],$

${\lambda }_{1}=\frac{\left({\epsilon }_{1}+{\epsilon }_{1}\gamma \right)\gamma -\sqrt{{\left({\epsilon }_{1}\gamma -{\epsilon }_{1}\right)}^{2}+4{\epsilon }^{2}\gamma }}{2\gamma },{\lambda }_{2}=\frac{\left({\epsilon }_{1}+{\epsilon }_{1}\gamma \right)\gamma +\sqrt{{\left({\epsilon }_{1}\gamma -{\epsilon }_{1}\right)}^{2}+4{\epsilon }^{2}\gamma }}{2\gamma },$

$\left[\begin{array}{c}u\\ v\\ \gamma \end{array}\right]↦\left[\begin{array}{c}{\lambda }_{1}u\\ {\lambda }_{2}v\\ \gamma \end{array}\right]+\frac{1}{\epsilon \left({\lambda }_{1}-{\lambda }_{2}\right)}\left[\begin{array}{c}\frac{{j}_{1}\left(u+v\right)}{1-k\epsilon \left(u+v\right)}-\frac{{j}_{1}\left(u+v\right)}{{\gamma }^{2}}\\ -\frac{{j}_{2}\left(u+v\right)}{1-k\epsilon \left(u+v\right)}+\frac{{j}_{2}\left(u+v\right)}{{\gamma }^{2}}\\ 0\end{array}\right].$ (7)

$T=\left[\begin{array}{cc}-\epsilon & -\epsilon \\ {\lambda }_{1}-\frac{{\epsilon }_{1}}{\gamma }& {\lambda }_{2}-\frac{{\epsilon }_{1}}{\gamma }\end{array}\right]$

${j}_{1}=\epsilon {\epsilon }_{1}^{2}-\epsilon {\epsilon }_{1}{\lambda }_{2}\gamma -{\epsilon }^{3}\gamma ,{j}_{2}=\epsilon {\epsilon }_{1}^{2}-\epsilon {\epsilon }_{1}{\lambda }_{1}\gamma -{\epsilon }^{3}\gamma .$

${\gamma }_{3}=\gamma -1$ ，假设在 ${\gamma }_{3}$ 附近的中心流形的形式为

$v=h\left(u,{\gamma }_{3}\right)=d{u}^{2}+eu{\gamma }_{3}+f{\gamma }_{3}^{2}+\mathcal{O}\left(3\right).$

$\mathcal{N}\left(h\left(u,{\gamma }_{3}\right)\right)=h\left(u,{\gamma }_{3}\right)-{\lambda }_{2}h\left(u,{\gamma }_{3}\right)+\frac{1}{\epsilon \left({\lambda }_{1}-{\lambda }_{2}\right)}\left\{\frac{{j}_{4}\left(u+h\left(u,{\gamma }_{3}\right)\right)}{1-k\epsilon \left(u+h\left(u,{\gamma }_{3}\right)\right)}-\frac{{j}_{4}\left(u+h\left(u,{\gamma }_{3}\right)\right)}{{\left({\gamma }_{3}+1\right)}^{2}}\right\}=0.$

$d=0,e=\frac{2{j}_{4}}{\epsilon \left({\lambda }_{1}-{\lambda }_{2}\right)\left({\lambda }_{2}-1\right)},f=0,$

$v=h\left(u,{\gamma }_{3}\right)=eu{\gamma }_{3}+\mathcal{O}\left(3\right).$ (8)

$\frac{\partial {\Phi }_{{\gamma }_{3}}}{\partial {\gamma }_{3}}\left(0,0\right)=0,\frac{\partial {\Phi }_{{\gamma }_{3}}}{\partial u\partial {\gamma }_{3}}\left(0,0\right)\ne 0,\frac{{\partial }^{2}{\Phi }_{{\gamma }_{3}}}{\partial {u}^{2}}\left(0,0\right)\ne 0.$

${e}^{*}=\frac{2\left(3\epsilon -2\right)\left(2-\epsilon \right){j}_{6}}{\epsilon {\lambda }_{2}\left({\lambda }_{1}-{\lambda }_{2}\right)\left({\lambda }_{2}-1\right){\left(3\epsilon -2\right)}^{2}-\epsilon {j}_{6}\left(8\epsilon -2\right)}.$

${\Phi }_{{\gamma }_{4}}\left(u\right)={\lambda }_{1}u+\frac{1}{\epsilon \left({\lambda }_{1}-{\lambda }_{2}\right)}\left[\frac{{j}_{5}\left(u+{e}^{*}u{\gamma }_{4}\right)}{1-k\epsilon \left(u+{e}^{*}u{\gamma }_{4}\right)}-\frac{{j}_{5}\left(u+{e}^{*}u{\gamma }_{4}\right)}{{\left({\gamma }_{4}+\frac{3\epsilon -2}{2-\epsilon }\right)}^{2}}\right]+\mathcal{O}\left(3\right).$

${\left[\frac{\partial {\Phi }_{{\gamma }_{4}}}{\partial {\gamma }_{4}}\frac{{\partial }^{2}{\Phi }_{{\gamma }_{4}}}{\partial {u}^{2}}+2\frac{{\partial }^{2}{\Phi }_{{\gamma }_{4}}}{\partial u\partial {\gamma }_{4}}\right]|}_{\left(0,0\right)}={\alpha }_{3}\ne 0,{\left[\frac{1}{2}{\left(\frac{{\partial }^{2}{\Phi }_{{\gamma }_{4}}}{\partial {u}^{2}}\right)}^{2}+\frac{1}{3}\frac{{\partial }^{3}{\Phi }_{{\gamma }_{4}}}{\partial {u}^{3}}\right]|}_{\left(0,0\right)}={\alpha }_{4}\ne 0.$

$\begin{array}{l}{\alpha }_{3}=-\frac{2}{\epsilon \left({\lambda }_{1}-{\lambda }_{2}\right)}\left\{2\epsilon {j}_{2}^{2}+k\epsilon {j}_{8}^{2}\left({j}_{9}+\epsilon {j}_{7}{j}_{8}\right)+{j}_{8}\left(2k{e}^{*}{\epsilon }^{2}{j}_{7}+{j}_{9}\right)+\frac{{j}_{9}-2{\epsilon }^{3}}{{\left(3\epsilon -2\right)}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }-\frac{{\left(2-\epsilon \right)}^{2}\left(3{e}^{*}\epsilon -\epsilon -{e}^{*}+2\right)}{{\left(3\epsilon -2\right)}^{2}}+\frac{\left(2-\epsilon \right)\left[2\epsilon {\epsilon }_{1}^{2}+{e}^{*}\left(2-\epsilon \right)\right]+{j}_{9}}{{\left(3\epsilon -2\right)}^{3}}-\frac{\epsilon {\epsilon }_{1}^{2}\left(2-\epsilon \right)}{{\left(3\epsilon -2\right)}^{4}}\right\},\end{array}$

${\alpha }_{4}=\frac{2{k}^{2}{\epsilon }^{2}}{{\lambda }_{1}-{\lambda }_{2}}{j}_{8}^{2}\left[\frac{{j}_{7}^{2}}{{\lambda }_{1}-{\lambda }_{2}}-{j}_{5}{j}_{8}\right].$

4. 总结

Dynamic Properties of a Class of Discrete Population Model[J]. 应用数学进展, 2019, 08(08): 1463-1470. https://doi.org/10.12677/AAM.2019.88171

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