#### 期刊菜单

A Stock Price Dynamic Model with Reverse Parameter and General Function of Belief
DOI: 10.12677/AAM.2019.81006, PDF, HTML, XML, 下载: 1,121  浏览: 5,529

Abstract: In this paper, a general function of expectation beliefs based on stock price deviation is introduced to the chartists. And considering the influence of the reverse investors, we add the reverse param-eter, which represents the proportion of the reverse investors in the chartists. Using difference equation to analyse the equilibrium solutions, stability and bifurcations of the system with general beliefs function, we discuss the influence of main parameters. Through analysis, it is verified that the reverse investors has an important role in the market.

1. 引言

2. 建立模型

2.1. 模型的假设

${W}_{t+1}=\left(1+r\right){W}_{t}+\left({p}_{t+1}+{y}_{t+1}-\left(1+r\right){p}_{t}\right){z}_{t}$

${E}_{h,t}$${V}_{h,t}$ 分别表示投资者h的条件期望和条件方差。假定投资者都是均值-方差最大化者，即

$\underset{{z}_{t}}{\mathrm{max}}{E}_{h,t}\left[{W}_{t+1}\right]-\frac{a}{2}{V}_{h,t}\left[{W}_{t+1}\right],$

${z}_{h,t}=\frac{{E}_{h,t}\left[{p}_{t+1}+{y}_{t+1}-\left(1+r\right){p}_{t}\right]}{a{V}_{h,t}\left[{p}_{t+1}+{y}_{t+1}-\left(1+r\right){p}_{t}\right]}.$

${z}^{s}$ 表示投资者的风险外股票供应量， ${n}_{h,t}$ 为t时点交易者h的市场分数，市场供需均衡意味着

$\underset{h=1}{\overset{H}{\sum }}{n}_{h,t}\frac{{E}_{h,t}\left[{p}_{t+1}+{y}_{t+1}-\left(1+r\right){p}_{t}\right]}{a{V}_{h,t}\left[{p}_{t+1}+{y}_{t+1}-\left(1+r\right){p}_{t}\right]}={z}^{s}.$

$\left(1+r\right){p}_{t}=\underset{h=1}{\overset{H}{\sum }}{n}_{h,t}{E}_{h,t}\left[{p}_{t+1}+{y}_{t+1}\right]-a{\sigma }^{2}{z}^{s}.$

${p}_{t}^{*}=\underset{k=1}{\overset{\infty }{\sum }}\frac{{E}_{t}\left[{y}_{t+k}\right]}{{\left(1+r\right)}^{k}}=\frac{\stackrel{¯}{y}}{r}.$

${x}_{t}={p}_{t}-{p}^{*}.$

${E}_{h,t}\left({p}_{t+1}\right)={E}_{t}\left({p}_{t+1}^{*}\right)+{f}_{h}\left({x}_{t-1},\cdots ,{x}_{t-L}\right)={p}^{*}+{f}_{h}\left({x}_{t-1},\cdots ,{x}_{t-L}\right).$

${f}_{h}\left({x}_{t-1},\cdots ,{x}_{t-L}\right)$ 表示投资者对价格偏离基本价格的信念。则得到定价方程为：

$\left(1+r\right){x}_{t}=\underset{h=1}{\overset{H}{\sum }}{n}_{h,t}{f}_{h,t}.$

2.2. 交易者的预期信念函数

${f}_{1,t}={\lambda }_{1}{x}_{t-1}.$

${f}_{2,t}\left({x}_{t}\right)={\lambda }_{2}\phi \left({x}_{t-1}\right).$

$\begin{array}{c}\phi \left({x}_{t}\right)=\phi \left({x}_{0}\right)+{\phi }^{\prime }\left({x}_{0}\right)\left({x}_{t}-{x}_{0}\right)+\frac{{\phi }^{″}\left({x}_{0}\right)}{2!}\left({x}_{t}-{x}_{0}\right){}^{2}+\cdots \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{\phi }^{n}\left({x}_{0}\right)}{n!}{\left({x}_{t}-{x}_{0}\right)}^{n}+o{\left({x}_{t}-{x}_{0}\right)}^{m}.\end{array}$ (1)

${x}_{t}=\frac{1}{1+r}\left[{n}_{1,t}{f}_{1,t}+{n}_{2,t}{f}_{2,t}\right].$

${n}_{h,t}=\alpha {n}_{h,t-1}+\left(1-\alpha \right)\frac{\mathrm{exp}\left[\beta {U}_{h,t-1}\right]}{{\sum }_{i=1}^{H}\mathrm{exp}\left[\beta {U}_{i,t-1}\right]},h=1,\cdots ,H.$

${U}_{h,t+1}=\gamma {\pi }_{h,t}+\mu {U}_{h,t}.$

$R=1+r$，已实现的收益为

${\pi }_{h,t}={R}_{t}{z}_{h,t-1}-{C}_{h}=\left({x}_{t}-R{x}_{t-1}+{\delta }_{t}\right)\frac{{f}_{h,t-1}\left({x}_{t-2}\right)-R{x}_{t-1}}{a{\sigma }^{2}}-{C}_{h}.$

${m}_{t}={n}_{1,t}-{n}_{2,t}$ 为市场分数差，则

${m}_{t}=\alpha {m}_{t-1}+\left(1-\alpha \right)\mathrm{tanh}\left\{\frac{\beta }{2}\left({U}_{1,t-1}-{U}_{2,t-1}\right)\right\}.$

${C}_{1}=C$${C}_{2}=0$，根据胡建梅(2013) [9] ，本文对逆风者引入逆风参数k，即逆风者在图表分析者中所占的比例为 $k\left(0\le k\le 1\right)$，则顺风者所占比例为 $1-k$，资产定价方程可写为

${x}_{t}=\frac{1}{R}\left[{n}_{1,t}{\lambda }_{1}{x}_{t-1}+{n}_{2,t}\left(1-2k\right){\lambda }_{2}\phi \left({x}_{t}\right)\right].$ (2)

$\left\{\begin{array}{l}{x}_{t}=\frac{1}{R}\left[\frac{1+{m}_{t}}{2}{\lambda }_{1}{x}_{t-1}+\frac{1-{m}_{t}}{2}\left(1-2k\right){\lambda }_{2}\phi \left({x}_{t-1}\right)\right],\\ {U}_{1,t}=\gamma \left[\left({x}_{t}-R{x}_{t-1}+{\delta }_{t}\right)\frac{{\lambda }_{1}{x}_{t-2}-R{x}_{t-1}}{a{\sigma }^{2}}-C\right]+\mu {U}_{1,t-1},\\ {U}_{2,t}=\gamma \left({x}_{t}-R{x}_{t-1}+{\delta }_{t}\right)\frac{\left(1-2k\right){\lambda }_{2}\phi \left({x}_{t-2}\right)-R{x}_{t-1}}{a{\sigma }^{2}}+\mu {U}_{2,t-1},\\ {m}_{t}=\alpha {m}_{t-1}+\left(1-\alpha \right)\mathrm{tanh}\left\{\frac{\beta }{2}\left({U}_{1,t-1}-{U}_{2,t-1}\right)\right\}.\end{array}$ (3)

${\delta }_{t}=0$，通过增维降阶动力系统变为：

$\left\{\begin{array}{l}{x}_{t}=\frac{1}{R}\left[\frac{1+{m}_{t}}{2}{\lambda }_{1}{x}_{t-1}+\frac{1-{m}_{t}}{2}\left(1-2k\right){\lambda }_{2}\phi \left({x}_{t-1}\right)\right],\\ {y}_{t}={x}_{t-1},\\ {z}_{t}={y}_{t-1},\\ {U}_{1,t}=\gamma \left[\left({x}_{t}-R{x}_{t-1}+{\delta }_{t}\right)\frac{{\lambda }_{1}{y}_{t-1}-R{x}_{t-1}}{a{\sigma }^{2}}-C\right]+\mu {U}_{1,t-1},\\ {U}_{2,t}=\gamma \left({x}_{t}-R{x}_{t-1}+{\delta }_{t}\right)\frac{\left(1-2k\right){\lambda }_{2}\phi \left({y}_{t-1}\right)-R{x}_{t-1}}{a{\sigma }^{2}}+\mu {U}_{2,t-1},\\ {u}_{1,t}={U}_{1,t-1},\\ {u}_{2,t}={U}_{2,t-1},\\ {m}_{t}=\alpha {m}_{t-1}+\left(1-\alpha \right)\mathrm{tanh}\left\{\frac{\beta \gamma }{2a{\sigma }^{2}}\left({x}_{t-1}-R{y}_{t-1}\right)\left[{\lambda }_{1}{z}_{t-1}-\left(1-2k\right){\lambda }_{2}\phi \left({z}_{t-1}\right)\right]+\frac{\beta \mu }{2}\left({u}_{1,t}-{u}_{2,t}\right)-\frac{\beta \gamma C}{2}\right\}.\end{array}$ (4)

3. 系统平衡点，稳定性及分支的讨论

$x\left(k+1\right)=Ax\left(k\right),k\ge {k}_{0},A\left(k\right)={\left({a}_{ij}\right)}_{n×n}.$

1) 若矩阵A所有特征根的模都不大于1，且模等于1的模是单重根，则线性差分方程组的零解是稳定的；

2) 若矩阵A所有特征根的模小于1，则线性差分方程组的零解是渐进稳定的；

3) 若矩阵A有模大于1的特征根，则线性差分方程组的零解是不稳定的。

3.1. 基本面分析者主导市场时模型分析

$k=0.5$，即图表分析者中顺风者与逆风者占比相同时，模型退化为

${x}_{t}=\frac{1+{m}_{t}}{2R}{\lambda }_{1}{x}_{t-1}.$ (5)

${m}^{eq}=\mathrm{tanh}\left\{\frac{\beta }{2}\left[\mu \left({U}_{1}^{eq}-{U}_{2}^{eq}\right)-\gamma C\right]\right\}.$

${m}^{eq}=\mathrm{tanh}\left\{-\frac{\beta \gamma C}{2\left(1-\mu \right)}\right\}.$ (6)

${m}^{*}=\mathrm{tanh}\left\{\frac{\beta \gamma }{2a{\sigma }^{2}}\left(1-R\right)\left[{\lambda }_{1}{x}^{*}-\left(1-2k\right){\lambda }_{2}\phi \left({x}^{*}\right)\right]{x}^{*}+\frac{\beta \mu }{2}\left({u}_{1}^{*}-{u}_{2}^{*}\right)-\frac{\beta \gamma C}{2}\right\}$

${u}_{2}^{*}=\frac{\gamma }{1-\mu }\left[\frac{\left(1-R\right)\left[\left(1-2k\right){\lambda }_{2}\phi \left({x}^{*}\right)-R{x}^{*}\right]{x}^{*}}{a{\sigma }^{2}}\right].$

${u}_{1}^{*}-{u}_{2}^{*}=\frac{\gamma \left(1-R\right)}{a{\sigma }^{2}\left(1-\mu \right)}\left[{\lambda }_{1}{x}^{*}-\left(1-2k\right){\lambda }_{2}\phi \left({x}^{*}\right)\right]-\frac{\gamma C}{1-\mu }.$

${m}^{*}=\mathrm{tanh}\left\{-\frac{\beta \gamma \left(1-R\right)}{2a{\sigma }^{2}\left(1-\mu \right)}\left[{\lambda }_{1}{x}^{*}-\left(1-2k\right){\lambda }_{2}\phi \left({x}^{*}\right)\right]-\frac{\beta \gamma C}{2\left(1-\mu \right)}\right\}.$ (7)

3.2. 一般情况下模型分析

${X}_{t}=F\left({X}_{t-1}\right)$ (8)

$\begin{array}{l}{F}_{1}\left({X}_{t-1}\right)=\frac{1}{R}\left[\frac{1+{F}_{8}\left({X}_{t-1}\right)}{2}{\lambda }_{1}{x}_{t-1}+\frac{1-{F}_{8}\left({X}_{t-1}\right)}{2}\left(1-2k\right){\lambda }_{2}\phi \left({x}_{t-1}\right)\right];\\ {F}_{2}\left({X}_{t-1}\right)={x}_{t-1};\\ {F}_{3}\left({X}_{t-1}\right)={y}_{t-1};\\ {F}_{4}\left({X}_{t-1}\right)=\gamma \left[\left({F}_{1}\left({X}_{t-1}\right)-R{x}_{t-1}\right)\frac{{\lambda }_{1}{y}_{t-1}-R{x}_{t-1}}{a{\sigma }^{2}}-C\right]+\mu {U}_{1,t-1};\end{array}$

$\begin{array}{l}{F}_{5}\left({X}_{t-1}\right)=\gamma \left({F}_{1}\left({X}_{t-1}\right)-R{x}_{t-1}\right)\frac{\left(1-2k\right){\lambda }_{2}\phi \left({y}_{t-1}\right)-R{x}_{t-1}}{a{\sigma }^{2}}+\mu {U}_{2,t-1};\\ {F}_{6}\left({X}_{t-1}\right)={U}_{1,t-1};\\ {F}_{7}\left({X}_{t-1}\right)={U}_{2,t-1};\\ {F}_{8}\left({X}_{t-1}\right)=\alpha {m}_{t-1}+\left(1-\alpha \right)\mathrm{tanh}\left\{\frac{\beta \gamma }{2a{\sigma }^{2}}\left({x}_{t-1}-R{y}_{t-1}\right)\left[{\lambda }_{1}{z}_{t-1}-\left(1-2k\right){\lambda }_{2}\phi \left({z}_{t-1}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\beta \mu }{2}\left({u}_{1,t-1}-{u}_{2,t-1}\right)-\frac{\beta \gamma C}{2}\right\}.\end{array}$

${X}^{*}={\left({x}^{*},{y}^{*},{z}^{*},{U}_{1}^{*},{U}_{2}^{*},{u}_{1}^{*},{u}_{2}^{*},{m}^{*}\right)}^{\text{T}}$ 为系统的平衡解，它满足 ${X}^{*}=F\left({X}^{*}\right)$，即满足下列方程组：

$\left\{\begin{array}{l}{x}^{*}=\frac{1}{R}\left[\frac{1+{m}^{2}}{2}{\lambda }_{1}{x}^{*}+\frac{1-{m}^{*}}{2}\left(1-2k\right){\lambda }_{2}\phi \left({x}^{*}\right)\right],\\ {y}^{*}={x}^{*},\\ {z}^{*}={y}^{*}\\ {U}_{1}^{*}=\gamma \left[\left({x}^{*}-R{x}^{*}+{\delta }_{t}\right)\frac{{\lambda }_{1}{y}^{*}-R{x}^{*}}{a{\sigma }^{2}}-C\right]+\mu {U}_{1}^{*},\\ {U}_{2}^{*}=\gamma \left({x}^{*}-R{x}^{*}+{\delta }_{t}\right)\frac{\left(1-2k\right){\lambda }_{2}\phi \left({y}^{*}\right)-R{x}^{*}}{a{\sigma }^{2}}+\mu {U}_{2}^{*},\\ {u}_{1}^{*}={U}_{1}^{*},\\ {u}_{2}^{*}={U}_{2}^{*},\\ {m}^{*}=\mathrm{tanh}\left\{\frac{\beta \gamma }{2a{\sigma }^{2}}\left({x}^{*}-R{y}^{*}\right)\left[{\lambda }_{1}{z}^{*}-\left(1-2k\right){\lambda }_{2}\phi \left({z}^{*}\right)\right]+\frac{\beta \mu }{2}\left({u}_{1}^{*}-{u}_{2}^{*}\right)-\frac{\beta \gamma C}{2}\right\}.\end{array}$ (9)

1) 当 $k\in \left[0,0.5\right)$ 时，系统有基本平衡解 ${E}_{0}$ 和非零平衡解 ${E}_{1}$，其中

${E}_{1}=\left(+{x}^{*},+{x}^{*},+{x}^{*},{U}_{1}^{*},{U}_{2}^{*},{u}_{1}^{*},{u}_{2}^{*},{m}^{*}\right).$

2) 当 $k\in \left(0.5,1\right]$ 时，系统有基本平衡解 ${E}_{0}$ 和非零平衡解 ${E}_{2}$，其中

${E}_{2}=\left(-{x}^{*},-{x}^{*},-{x}^{*},{U}_{1}^{*},{U}_{2}^{*},{u}_{1}^{*},{u}_{2}^{*},{m}^{*}\right).$

3) 当 $k=0.5$ 时，系统只有基本平衡解 ${E}_{0}$

${x}^{*}=\frac{1}{R}\left[\frac{1+{m}^{2}}{2}{\lambda }_{1}{x}^{*}+\frac{1-{m}^{*}}{2}\left(1-2k\right){\lambda }_{2}\phi \left({x}^{*}\right)\right]$

${x}^{*}=\frac{\left(1-{m}^{*}\right)\left(1-2k\right){\lambda }_{2}}{2R-\left(1+{m}^{*}\right){\lambda }_{1}}\phi \left(x*\right)$

1) 当 $k\in \left[0,0.5\right)$ 时， $1-2k>0$，则 $\left(1-{m}^{*}\right)\left(1-2k\right){\lambda }_{2}>0$，故 ${x}^{*}>0$，记为 $+{x}^{*}$ 。此时系统有基本平衡解 ${E}_{0}$ 和非零平衡解 ${E}_{1}$，其中 ${E}_{1}=\left(+{x}^{*},+{x}^{*},+{x}^{*},{U}_{1}^{*},{U}_{2}^{*},{u}_{1}^{*},{u}_{2}^{*},{m}^{*}\right)$

2) 当 $k\in \left(0.5,1\right]$ 时， $\text{1}-\text{2}k<0$，则 $\left(1-{m}^{*}\right)\left(1-2k\right){\lambda }_{2}<0$，故 ${x}^{*}<0$，记为 $-{x}^{*}$ 。此时系统有基本平衡解 ${E}_{0}$ 和非零平衡解 ${E}_{2}$，其中 ${E}_{1}=\left(-{x}^{*},-{x}^{*},-{x}^{*},{U}_{1}^{*},{U}_{2}^{*},{u}_{1}^{*},{u}_{2}^{*},{m}^{*}\right)$

3) 当 $k=0.5$ 时，系统只有基本平衡解 ${E}_{0}$，证明过程由定理1可得。

1) 当 $k\in \left[0,0.5\right)$，即市场中顺风者在图表分析者中所占的比例大于逆风者时，

(i) 当 $1<{\lambda }_{2}<\frac{R}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}$ 时， ${E}_{0}$ 是局部渐进稳定的。

(ii) 当 $\frac{R}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}<{\lambda }_{2}<\frac{2R-{\lambda }_{1}}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}$ 时，

1o、当 $0<\beta <\stackrel{¯}{\beta }$ 时， ${E}_{0}$ 是渐进稳定的。

2o、当 $\beta >\stackrel{¯}{\beta }$ 时， ${E}_{0}$ 是不稳定的。

3o、当 $\beta =\stackrel{¯}{\beta }$ 时，系统出现分支。

(iii) 当 ${\lambda }_{2}>\frac{2R-{\lambda }_{1}}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}$ 时， ${E}_{0}$ 是全局不稳定的。

2) 当 $k\in \left(0.5,1\right]$，即市场中逆风者在图表分析者中所占的比例大于顺风者时，

(i) 当 $1<{\lambda }_{2}<-\frac{R}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}$ 时， ${E}_{0}$ 是全局渐进稳定的。

(ii) 当 $-\frac{R}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}<{\lambda }_{2}<-\frac{2R+{\lambda }_{1}}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}$ 时，

1o、当 $0<\beta <\stackrel{˜}{\beta }$ 时， ${E}_{0}$ 是渐进稳定的。

2o、当 $\beta >\stackrel{˜}{\beta }$ 时， ${E}_{0}$ 是不稳定的。

3o、当 $\beta =\stackrel{˜}{\beta }$ 时，系统出现分支。

(iii) 当 ${\lambda }_{2}>-\frac{2R+{\lambda }_{1}}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}$ 时， ${E}_{0}$ 是全局不稳定的。

3) 当 $k=0.5$ 时，此时平衡解 ${E}_{0}$ 是系统唯一的平衡解，且是全局渐进稳定的。

$J=\left\{\begin{array}{cccccccc}A& 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& \mu & 0& 0& 0& 0\\ 0& 0& 0& 0& \mu & 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& B& -B& \alpha \end{array}\right\}.$

$B={\frac{\partial {F}_{8}}{\partial {u}_{1}}|}_{{E}_{0}}=\frac{\left(1-\alpha \right)\beta \mu }{2}\mathrm{tan}{h}^{\prime }\left(\theta \right).$

$\Gamma \left(\lambda \right)=\left(\lambda -A\right)\left(\lambda -\alpha \right){\left(\lambda -\mu \right)}^{2}{\lambda }^{4}.$

$\alpha \in \left[0,1\right)$$\mu \in \left[0,1\right)$，则当 $|A|<1$ 时基本平衡点 ${E}_{0}$ 是稳定的。由于

$A=\frac{1}{2R}\left[\left(1+{m}^{eq}\right){\lambda }_{1}+\left(1-{m}^{eq}\right)\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)\right]$ ,

1) 当 $k\in \left[0,0.5\right)$ 时， $A>0$，即 $A>-1$ 成立，下面讨论 $A<1$ 的情况。

$\left(1+{m}^{eq}\right){\lambda }_{1}+\left(1-{m}^{eq}\right)\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)<2R$

${m}^{eq}\left[{\lambda }_{1}-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)\right]<2R-{\lambda }_{1}-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)$

(i) 当 $0<{\lambda }_{1}<\left(1-2k\right){\phi }^{\prime }\left(0\right)$ 时，有 ${\lambda }_{1}-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)<0$，则

${m}^{eq}>1+\frac{2\left(R-{\lambda }_{1}\right)}{{\lambda }_{1}-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)},$

$\stackrel{¯}{m}=1+\frac{2\left(R-{\lambda }_{1}\right)}{{\lambda }_{1}-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)}$

1o、当 $1<{\lambda }_{2}<\frac{R}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}$ 时， $\stackrel{¯}{m}\le -1$，则恒有 ${m}^{eq}>\stackrel{¯}{m}$ 成立，则 ${E}_{0}$ 是渐进稳定的。

2o、当 ${\lambda }_{2}>\frac{R}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}$ 时，要使得 ${m}^{eq}>\stackrel{¯}{m}$ 成立，则

$\mathrm{tanh}\left\{-\frac{\beta \gamma C}{2\left(1-\mu \right)}\right\}>1+\frac{2\left(R-{\lambda }_{1}\right)}{{\lambda }_{1}-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)}$

$\stackrel{¯}{\beta }=\frac{1-\mu }{\gamma C}\mathrm{ln}\left(\frac{R-{\lambda }_{1}}{\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)-R}\right)$

$0<\beta <\stackrel{¯}{\beta }$ 时， ${E}_{0}$ 是渐进稳定的。

$\beta >\stackrel{¯}{\beta }$ 时， ${E}_{0}$ 是不稳定的。

(ii) 当 $\left(1-2k\right){\phi }^{\prime }\left(0\right)<{\lambda }_{1}<1$ 时，

1o、当 $1<{\lambda }_{2}<\frac{{\lambda }_{1}}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}$ 时，有 $1-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)>0$，则 $\stackrel{¯}{m}>1$ 成立。

2o、当 $\frac{{\lambda }_{1}}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}<{\lambda }_{2}<\frac{R}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}$ 时，有 ${\lambda }_{1}-R<{\lambda }_{1}-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)<0$，则可以得到 $\stackrel{¯}{m}=1+\frac{2\left(R-{\lambda }_{1}\right)}{{\lambda }_{1}-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)}<-1$ 成立。此时 ${m}^{eq}>\stackrel{¯}{m}$，则基本平衡解 ${E}_{0}$ 是全局渐进稳定的。

3o、当 $\frac{R}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}<{\lambda }_{2}<\frac{2R-{\lambda }_{1}}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}$ 时，有 $-2<\frac{2\left(R-{\lambda }_{1}\right)}{{\lambda }_{1}-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)}<-1$，则 $\stackrel{¯}{m}\in \left(-1,1\right)$

$\beta >\stackrel{¯}{\beta }$ 时， ${E}_{0}$ 是不稳定的。

4o、当 ${\lambda }_{2}>\frac{2R-{\lambda }_{1}}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}$ 时，有 $\frac{R-{\lambda }_{1}}{\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)-R}<1$ 。则 $\stackrel{¯}{\beta }=\frac{1-\mu }{\gamma C}\mathrm{ln}\left(\frac{R-{\lambda }_{1}}{\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)-R}\right)<0$

2) 当 $k\in \left(0.5,1\right]$$A<1$ 成立，下面讨论 $A>-1$ 的情况，即

$\left(1+{m}^{eq}\right){\lambda }_{1}+\left(1-{m}^{eq}\right)\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)>-2R$

${m}^{eq}\left[{\lambda }_{1}-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)\right]>-2R-{\lambda }_{1}-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)$

$\stackrel{˜}{m}=1-\frac{2\left(R+{\lambda }_{1}\right)}{{\lambda }_{1}-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)}$ .

1o、当 $1<{\lambda }_{2}<-\frac{R}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}$ 时，有 $\frac{2\left(R+{\lambda }_{1}\right)}{{\lambda }_{1}-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)}>2$，则 $\stackrel{˜}{m}<-1$ 。此时， ${m}^{eq}>\stackrel{˜}{m}$ 恒成立，则 ${E}_{0}$ 是渐进稳定的。

2o、当 $-\frac{R}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}<{\lambda }_{2}<-\frac{2R+{\lambda }_{1}}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}$ 时，有 $1<\frac{2\left(R+{\lambda }_{1}\right)}{{\lambda }_{1}-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)}<2$，则 $\stackrel{˜}{m}\in \left(-1,1\right)$ 。要使得 ${m}^{eq}>\stackrel{˜}{m}$ 成立，则

$\mathrm{tanh}\left\{-\frac{\beta \gamma C}{2\left(1-\mu \right)}\right\}>1-\frac{2\left(R+{\lambda }_{1}\right)}{{\lambda }_{1}-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)}$

$\beta <\frac{1-\mu }{\gamma C}\mathrm{ln}\left(\frac{R+{\lambda }_{1}}{-R-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)}\right)$

$\stackrel{˜}{\beta }=\frac{1-\mu }{\gamma C}\mathrm{ln}\left(\frac{R+{\lambda }_{1}}{-R-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)}\right)$

$\beta >\stackrel{˜}{\beta }$ 时， ${E}_{0}$ 是不稳定的。

3o、当 ${\lambda }_{2}>-\frac{2R+{\lambda }_{1}}{\left(1-2k\right){\phi }^{\prime }\left(0\right)}$ 时，有 $\frac{R+{\lambda }_{1}}{-R-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)}<1$，则 $\stackrel{˜}{\beta }=\frac{1-\mu }{\gamma C}\mathrm{ln}\left(\frac{R+{\lambda }_{1}}{-R-\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)}\right)<0$ 成立。

$k=0.5$ 时， $A=\frac{1}{2R}\left[\left(1+{m}^{eq}\right){\lambda }_{1}\right]$ 成立。由于 ${\lambda }_{1}\in \left[0,1\right)$$R>1$，则 $|A|<1$ 恒成立，即此时平衡解 ${E}_{0}$

Figure 1. Phase plots

3.3. 主要参数对系统分支的影响

Figure 2. Bifurcation diagrams of the intensity of choice (β)

Figure 3. Bifurcation diagrams of ${\lambda }_{2}$

$\stackrel{¯}{\beta }=\frac{1-\mu }{\gamma C}\mathrm{ln}\left(\frac{R-{\lambda }_{1}}{\left(1-2k\right){\lambda }_{2}{\phi }^{\prime }\left(0\right)-R}\right)$ 。这里 $\stackrel{¯}{\beta }$ 关于记忆参数 $\mu$ 是递减的，即当 $\mu$ 增大时，过去效用对适应度

Figure 4. The effect of parameters on $\stackrel{¯}{\beta }$ when $k\in \left[0,0.5\right)$

Figure 5. The effect of parameters on $\stackrel{¯}{\beta }$ when $k\in \left(0.5,1\right]$

4. 总结

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