﻿ 动态价格下Gompertz系统的捕捞问题

# 动态价格下Gompertz系统的捕捞问题Harvesting in Gompertz System with Dynamic Price

Abstract: In this paper, a harvesting model with price changed by market inside control is established for open fishery. The only equilibrium is proved to be positive and stable. We analyze the dynamics of fishery and draw a conclusion that the population of biological species in equilibrium decreasing and fishing effort increasing when the constant price is greater than the equilibrium price. At last, we provide government departments with guidance in control price.

1. 引言

$\left\{\begin{array}{l}\frac{\text{d}s}{\text{d}t}=F\left(s\right)-Es,\\ \frac{\text{d}E}{\text{d}t}=v\left(pEs-wE\right),\end{array}$ (1)

$p\left(S\right)=\frac{n}{m+S},$

$\underset{S\to 0}{\mathrm{lim}}p\left(S\right)=\frac{n}{m}={p}_{\mathrm{max}},\text{\hspace{0.17em}}\underset{S\to \infty }{\mathrm{lim}}p\left(S\right)=0,$

$\left\{\begin{array}{l}\frac{\text{d}s}{\text{d}t}=F\left(s\right)-\alpha Es,\\ \frac{\text{d}E}{\text{d}t}=v\alpha Es\left(\frac{n}{m+\alpha Es}-\frac{w}{\alpha s}\right),\end{array}$ (2)

$\frac{\text{d}p}{\text{d}t}=\beta \left({q}^{d}-{q}^{s}\right),\beta >0.$ (3)

$\frac{\text{d}p}{\text{d}t}=\beta \left(a-c\right)+\beta \left(b-d\right)p.$ (4)

$\left\{\begin{array}{l}\frac{\text{d}s}{\text{d}t}=rs\mathrm{ln}\left(k/s\right)-\alpha Es,\\ \frac{\text{d}E}{\text{d}t}=v\pi =v\left(p\alpha Es-wE\right),\\ \frac{\text{d}p}{\text{d}t}=\beta \left(a-c\right)+\beta \left(b-d\right)p,\end{array}$ (5)

2. 平衡点及其稳定性分析

$\left\{\begin{array}{l}rs\mathrm{ln}\left(k/s\right)-\alpha Es=0,\\ p\alpha Es-wE=0,\\ \beta \left(a-c\right)+\beta \left(b-d\right)=0,\end{array}$

${E}_{*}=\left({s}^{*},{E}^{*},{p}^{*}\right).$

${s}^{*}=\frac{w}{{p}^{*}\alpha }$${E}^{*}=\frac{r}{\alpha }\mathrm{ln}\left(k/{s}^{*}\right)$${p}^{*}=\frac{a-c}{d-b}$

$J=\left(\begin{array}{ccc}r\mathrm{ln}\left(k/s\right)-r-\alpha E& -\alpha s& 0\\ vp\alpha E& v\left(p\alpha s-w\right)& v\alpha Es\\ 0& 0& \beta \left(b-d\right)\end{array}\right)$

${a}_{0}{\lambda }^{3}+{a}_{1}{\lambda }^{2}+{a}_{2}\lambda +{a}_{3}=0,$

${a}_{0}=1,{a}_{1}=r-\beta \left(b-d\right)$

${a}_{2}=\left[vwr\mathrm{ln}\left(k/{s}^{*}\right)-r\beta \left(b-d\right)\right]$

${a}_{3}=-\beta \left(b-d\right)vwr\mathrm{ln}\left(k/{s}^{*}\right)$

$\begin{array}{c}{\Delta }_{2}=\left[r-\beta \left(b-d\right)\right]\left[vwr\mathrm{ln}\left(k/{s}^{*}\right)-r\beta \left(b-d\right)\right]+\beta \left(b-d\right)vwr\mathrm{ln}\left(k/{s}^{*}\right)\\ =r\left[vwr\mathrm{ln}\left(k/{s}^{*}\right)-r\beta \left(b-d\right)\right]+r{\beta }^{2}{\left(b-d\right)}^{2}\end{array}$

3. 开放渔业的动力学行为

4. 小结

Table 1. Increasing or decreasing of population and effort in four quadrants

Figure 1. Equilibrium of population and equilibrium of effort divide the area into four quadrants

Figure 2. Derived from model (5), the trajectory of the system over time in equilibrium price

Figure 3. Derived from model (1), the trajectory of the system over time in constant price

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