# 具有跳跃的随机蚊子种群模型的不变测度Invariant Measure of Random Mosquito Population Model with Jumping

DOI: 10.12677/PM.2021.114075, PDF, HTML, XML, 下载: 14  浏览: 55  国家自然科学基金支持

Abstract: This paper mainly studies the invariant measures of the random mosquito population model with Markov chains. First, the Lyapunov function is cleverly constructed, and the Itô theorem and the comparison theorem are used to prove that the random mosquito population model has a unique global continuous positive solution. Second, if λ≤0, the sterile mosquito population will be extinct, and the distribution of the wild mosquito population weakly converges to the only constant probability measure; if λ > 0, then the system has an invariant probability measure, and the transition probability of the solution process converges to an invariant measure. Finally, it is proved that the transition probability of a stochastic process converges to the exponential convergence rate of its invariant measure.

1. 引言

$\left\{\begin{array}{l}\frac{\text{d}w}{\text{d}t}=\left[C\left(N\right)\frac{aw}{w+g}-\left({\mu }_{1}+{\xi }_{1}\left(w+g\right)\right)\right]w,\hfill \\ \frac{\text{d}g}{\text{d}t}=B\left(\cdot \right)-\left[{\mu }_{2}+{\xi }_{2}\left(w+g\right)\right]g.\hfill \end{array}$ (1.1)

$\left\{\begin{array}{l}\text{d}w\left(t\right)=\left[\frac{aw\left(t\right)}{1+w\left(t\right)+g\left(t\right)}-\left({\mu }_{1}+{\xi }_{1}\left(w\left(t\right)+g\left(t\right)\right)\right)\right]w\left(t\right)\text{d}t+{\sigma }_{1}w\left(t\right)\text{d}{B}_{1}\left(t\right)\hfill \\ \text{d}g\left(t\right)=\frac{bw\left(t\right)}{1+w\left(t\right)}-\left[{\mu }_{2}+{\xi }_{2}\left(w\left(t\right)+g\left(t\right)\right)\right]g\left(t\right)\text{d}t+{\sigma }_{2}g\left(t\right)\text{d}{B}_{2}\left(t\right),\hfill \end{array}$ (1.2)

$\left\{\begin{array}{l}\text{d}w\left(t\right)=\left[\frac{a\left(\alpha \left(t\right)\right)w\left(t\right)}{1+w\left(t\right)+g\left(t\right)}-{\mu }_{1}\left(w\left(t\right),\alpha \left(t\right)\right)-\left(w\left(t\right)+g\left(t\right)\right){\xi }_{1}\left(w\left(t\right),g\left(t\right),\alpha \left(t\right)\right)\right]w\left(t\right)\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\sigma }_{1}\left(\alpha \left(t\right)\right)w\left(t\right)\text{d}{B}_{1}\left(t\right)\\ \text{d}g\left(t\right)=\frac{b\left(\alpha \left(t\right)\right)w\left(t\right)}{1+w\left(t\right)}-\stackrel{˜}{k}\left(\alpha \left(t\right)\right)g\left(t\right)\text{d}t+{\sigma }_{2}\left(\alpha \left(t\right)\right)g\left(t\right)\text{d}{B}_{2}\left(t\right),\end{array}$ (1.3)

$P\left\{\alpha \left(t+\Delta \right)=j|\alpha \left(t\right)=i,{X}_{t}=x\right\}=\left\{\begin{array}{l}{\gamma }_{ij}\left(x\right)\Delta +ο\left(\Delta \right),i\ne j\\ 1+{\gamma }_{ij}\left(x\right)\Delta +ο\left(\Delta \right),i=j\end{array}$ (1.4)

2. 预备知识

$\begin{array}{l}{a}^{*}=\underset{i\in \mathcal{M}}{\mathrm{max}}a\left(i\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}^{*}=\underset{i\in \mathcal{M}}{\mathrm{max}}b\left(i\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{k}^{*}=\underset{i\in \mathcal{M}}{\mathrm{max}}\sigma \left(i\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{˜}{k}}^{*}=\underset{i\in \mathcal{M}}{\mathrm{max}}\stackrel{˜}{k}\left(i\right),\\ {a}_{*}=\underset{i\in \mathcal{M}}{\mathrm{min}}a\left(i\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{*}=\underset{i\in \mathcal{M}}{\mathrm{min}}b\left(i\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{k*}=\underset{i\in \mathcal{M}}{\mathrm{min}}\sigma \left(i\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{˜}{k}}_{*}=\underset{i\in \mathcal{M}}{\mathrm{min}}\stackrel{˜}{k}\left(i\right).\end{array}$

${R}_{+}=\left[0,\infty \right)$${R}_{+}^{o}=\left(0,\infty \right)$${R}_{+}^{2}=\left[0,\infty \right)×\left[0,\infty \right)$${R}_{+}^{2,o}=\left(0,\infty \right)×\left(0,\infty \right)$，系统(1.3)与(1.4)的随机过程 $\left(w\left(t\right),g\left(t\right),\alpha \left(t\right)\right)$ 的相关算子：

$\mathcal{L}V\left(\varphi ,i\right)={V}_{\varphi }\left(\varphi ,i\right)\stackrel{˜}{f}\left(\varphi ,i\right)+\frac{1}{2}tr\left[\stackrel{˜}{g}\left(\varphi ,i\right){\stackrel{˜}{g}}^{\text{T}}\left(\varphi ,i\right){V}_{\varphi \varphi }\left(\varphi ,i\right)\right]+\underset{j\in M}{\sum }\text{ }\text{ }{\gamma }_{ij}V\left(\varphi ,i\right).$ (2.1)

$\stackrel{˜}{f}\left(\varphi ,i\right)={\left(\frac{a\left(i\right){w}^{2}}{1+w+g}-w{\mu }_{1}\left(w,i\right)-w\left(w+g\right){\xi }_{1}\left(w,g,i\right),\frac{wb\left(w,g,i\right)}{1+w}-g\stackrel{˜}{k}\left(i\right)\right)}^{\text{T}}$

$\stackrel{˜}{g}\left(\varphi ,i\right)=diag\left({\sigma }_{1}\left(i\right)w,{\sigma }_{2}\left(i\right)g\right)\in {R}^{2×2}$

1) ${B}_{1}\left(t\right)$${B}_{2}\left(t\right)$ 是独立于马尔科夫链 $\alpha \left(t\right)$ 的实值标准布朗运动。

2) 对任意 $i\in \mathcal{M}$$a\left(i\right),b\left(i\right),\stackrel{˜}{k}\left(i\right),{\sigma }_{1}\left(i\right),{\sigma }_{2}\left(i\right)$ 是非负的。

3) ${\mu }_{1}\left(w,i\right),{\xi }_{1}\left(w,g,i\right),b\left(w,g,i\right)$ 满足局部利普希茨条件； $\mu \left(0,i\right)={\xi }_{1}\left(0,g,i\right)=0$ 意味着对于 ${k}_{0}>1$$\underset{w\to \infty }{\mathrm{lim}}{\mu }_{1}\left(w,i\right)=\infty$$0\le b\left(w,g,i\right)\le {\kappa }_{0}\left({\mu }_{1}\left(w,i\right)\wedge {\xi }_{1}\left(w,g,i\right)\right)$${\xi }_{1}\left(w,g,i\right)\le {\kappa }_{0}\left(1+w\right)$。此外，对每个 $i\in \mathcal{M}$$b\left(w,g,i\right)$$g=0$ 时是一致连续的，即 $\underset{g\to 0}{\mathrm{lim}}\mathrm{sup}|b\left(w,g,i\right)-b\left(w,0,i\right)|=0$

4) 马尔科夫链或它的生成元 $\Gamma ={\left({\gamma }_{ij}\right)}_{N×N}$ 是不可约的，即对任意的 $i,j\in \mathcal{M}$，存在 $i={i}_{0},{i}_{1},\cdots ,{i}_{n}=j$ 使 ${\gamma }_{{i}_{k-1}},{i}_{k}>0,\text{\hspace{0.17em}}k=1,2,\cdots ,n$

1) 对于每个 $i\in \mathcal{M}$，任意的 $\left(w,g\right)\in \left[0,\infty \right)×\left(0,\infty \right)$$b\left(w,g,i\right)$ 在w上是非减的，在g上是非增的；

2) $\underset{i\in \mathcal{M}}{\sum }\left(\underset{s\to \infty }{\mathrm{lim}}\mathrm{sup}b\left(w,0,i\right)-\stackrel{˜}{k}\left(i\right)-\frac{{\sigma }_{2}^{2}\left(i\right)}{2}\right){\pi }_{\alpha }\left(i\right)<0;$ (2.2)

3) $\underset{i\in \mathcal{M}}{\sum }\left(\underset{s\to \infty }{\mathrm{lim}}\mathrm{inf}b\left(w,0,i\right)-\stackrel{˜}{k}\left(i\right)-\frac{{\sigma }_{2}^{2}\left(i\right)}{2}\right){\pi }_{\alpha }\left(i\right)>0.$ (2.3)

3. 存在性与不变测度

${P}_{w,g,i}\left\{w\left(t\right)>0,t>0\right\}=1$${P}_{w,g,i}\left\{g\left(t\right)=0,t>0\right\}=1$，如果 $g>0$，则 ${P}_{w,g,i}\left\{g\left(t\right)>0,t>0\right\}=1$

${\tau }_{e}=\mathrm{inf}\left\{t\ge 0:w\left(t\right)\vee g\left(t\right)=\infty \right\}$$\mathrm{inf}\varphi =\infty$，且解是一个强马尔科夫过程 [11]。定义

${\tau }_{k}=\mathrm{inf}\left\{t\ge 0:w\left(t\right)\vee g\left(t\right)>k\right\},$

${\tau }_{e}={\mathrm{lim}}_{k\to \infty }{\tau }_{k}$

$\begin{array}{c}\mathcal{L}{\stackrel{˜}{V}}_{1}\left(w,g,i\right)=\frac{{\kappa }_{0}a\left(i\right){w}^{2}}{1+w+g}-{\kappa }_{0}w{\mu }_{1}\left(w,i\right)-\stackrel{˜}{k}\left(i\right)g\\ \text{\hspace{0.17em}}+w\left[\frac{b\left(w,g,i\right)}{1+w}-{\kappa }_{0}\left(w+g\right){\xi }_{1}\left(w,g,i\right)\right]\\ \le \frac{{\kappa }_{0}a\left(i\right){w}^{2}}{1+w+g}\end{array}$

${P}_{w,g,i}\left\{{\tau }_{k}

${\mu }_{1}^{\left(n\right)}\left(w,i\right)={\mu }_{1}\left(w\wedge n,i\right)$${\xi }_{1}^{\left(n\right)}\left(w,g,i\right)={\xi }_{1}\left(w\wedge n,g\wedge n,i\right)$${b}^{\left(n\right)}\left(w,g,i\right)=b\left(w\wedge n,g\wedge n,i\right)$。特别地，让

${\mu }_{1}^{\left(n\right)},{\xi }_{1}^{\left(n\right)},{b}^{\left(n\right)}$ 替代 ${\mu }_{1},{\xi }_{1},b$$\left({w}^{\left(n\right)}\left(t\right),{g}^{\left(n\right)}\left(t\right)\right)$ 成为系统(1.3)与(1.4)的解。注意

$\begin{array}{r}\hfill {\eta }^{\left(n\right)}=\mathrm{inf}\left\{t\ge 0:{w}^{\left(n\right)}\left(t\right)\wedge {g}^{\left(n\right)}\left(t\right)\le 0\right\}\end{array}$

$\begin{array}{r}\hfill {\eta }_{k}^{\left(n\right)}=\mathrm{inf}\left\{t\ge 0:{w}^{\left(n\right)}\left(t\right)\wedge {g}^{\left(n\right)}\left(t\right)<\frac{1}{k}\right\}\end{array}$

$\begin{array}{r}\hfill {\stackrel{^}{V}}_{2}^{\left(n\right)}\left(w,g,i\right)=w-{c}_{1}^{\left(n\right)}-{c}_{1}^{\left(n\right)}\mathrm{ln}\frac{w}{{c}_{1}^{\left(n\right)}}+{c}_{2}\left(g-\mathrm{ln}g-1\right),\end{array}$

$\begin{array}{l}\mathcal{L}{\stackrel{^}{V}}_{2}^{\left(n\right)}\left(w,g,i\right)\\ =\left(1-\frac{{c}_{1}^{\left(n\right)}}{g}\right)\left(\frac{a\left(i\right){w}^{2}}{1+w+g}-w{\mu }_{1}^{\left(n\right)}\left(w,i\right)-w\left(w+g\right){\xi }_{1}^{\left(n\right)}\left(w,g,i\right)\right)+\frac{{c}_{1}^{\left(n\right)}{\sigma }_{1}^{2}\left(i\right)}{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({c}_{2}-\frac{{c}_{2}}{g}\right)\left(\frac{w{b}^{\left(n\right)}\left(w,g,i\right)}{1+w+g}-g\stackrel{˜}{k}\left(i\right)\right)+\frac{{c}_{2}{\sigma }_{2}^{2}\left(i\right)}{2}\end{array}$

$\begin{array}{l}\le \frac{a\left(i\right){w}^{2}}{1+w+g}+\frac{{c}_{1}^{\left(n\right)}w{\mu }_{1}^{\left(n\right)}\left(w,i\right)}{g}-w{\mu }_{1}^{\left(n\right)}\left(w,i\right)+\frac{{c}_{1}^{\left(n\right)}{\stackrel{^}{\sigma }}_{1}^{2}+{c}_{2}{\stackrel{^}{\sigma }}_{2}^{2}}{2}+{c}_{2}\stackrel{^}{k}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+w\left[\frac{{c}_{1}^{\left(n\right)}\left(w+g\right){\xi }_{1}^{\left(n\right)}\left(w,g,i\right)}{g}-\frac{{c}_{2}{b}^{\left(n\right)}\left(w,g,i\right)}{g\left(1+w\right)}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+w\left[\frac{{c}_{2}{b}^{\left(n\right)}\left(w,g,i\right)}{1+w}-\left(w+g\right){\xi }_{1}^{\left(n\right)}\left(w,g,i\right)\right]\\ \le \frac{a\left(i\right){w}^{2}}{1+w+g}+\frac{{c}_{1}^{\left(n\right)}{c}_{2}^{\left(n\right)}}{g}-{c}_{2}^{\left(2\right)}+\frac{{c}_{1}^{\left(n\right)}{\stackrel{^}{\sigma }}_{1}^{2}+{c}_{2}{\stackrel{^}{\sigma }}_{2}^{2}}{2}:={K}^{\left(n\right)};\text{\hspace{0.17em}}\text{\hspace{0.17em}}w,g>0,\end{array}$

$\begin{array}{l}{E}_{w,g,i}{\stackrel{^}{V}}_{2}^{\left(n\right)}\left({w}^{\left(n\right)}\left({\eta }_{k}^{\left(n\right)}\wedge t\right),{g}^{\left(n\right)}\left({\eta }_{k}^{\left(n\right)}\wedge t\right),{\alpha }^{\left(n\right)}\left({\eta }_{k}^{\left(n\right)}\wedge t\right)\right)\\ ={\stackrel{^}{V}}_{2}^{\left(n\right)}\left(w,g,i\right)+{E}_{w,g,i}{\int }_{0}^{{\eta }_{k}^{\left(n\right)}\wedge t}\mathcal{L}{\stackrel{^}{V}}_{2}^{\left(n\right)}\left({w}^{\left(n\right)}\left(u\right),{g}^{\left(n\right)}\left(u\right),{\alpha }^{\left(n\right)}\left(u\right)\right)\text{d}u\\ \le {\stackrel{^}{V}}_{2}^{\left(n\right)}\left(w,g,i\right)+{K}^{\left(n\right)}t.\end{array}$

${\stackrel{^}{V}}_{2}^{\left(n\right)}\left({w}^{\left(n\right)}\left({\eta }_{k}^{\left(n\right)}\wedge t\right),{g}^{\left(n\right)}\left({\eta }_{k}^{\left(n\right)}\wedge t\right),{\alpha }^{\left(n\right)}\left({\eta }_{k}^{\left(n\right)}\wedge t\right)\right)\ge \left({c}_{1}^{\left(n\right)}\mathrm{ln}k{c}_{1}^{\left(n\right)}-{c}_{1}^{\left(n\right)}\right)\wedge \left({c}_{2}\mathrm{ln}k-{c}_{2}\right).$

${P}_{w,g,i}\left\{{\eta }_{k}^{\left(n\right)}

$\begin{array}{r}\hfill {P}_{w,g,i}\left\{{w}^{\left(n\right)}\left(t\right),{g}^{\left(n\right)}\left(t\right)>0:\forall t>0\right\}=1.\end{array}$

$w\left(y\right)\left(\omega \right)\vee g\left(y\right)\left(\omega \right)<{n}_{0};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall 0\le y\le t.$

$\begin{array}{l}w\left(t\right)\left(\omega \right)={w}^{\left({n}_{0}\right)}\left(t\right)\left(\omega \right)>0,\\ g\left(t\right)\left(\omega \right)={g}^{\left({n}_{0}\right)}\left(t\right)\left(\omega \right)>0.\end{array}$

${P}_{w,g,i}\left\{w\left(t\right)>0:t>0\right\}={P}_{w,g,i}\left\{g\left(t\right)>0:t>0\right\}=1;\forall w,g>0.$ (3.1)

$\frac{a\left(i\right){\stackrel{˜}{w}}^{2}}{1+\stackrel{˜}{w}+g}-\stackrel{˜}{w}{\mu }_{1}\left(\stackrel{˜}{w},i\right)-\stackrel{˜}{w}\left(\stackrel{˜}{w}+g\right){\xi }_{1}\left(\stackrel{˜}{w},g,i\right)\ge 0.$ (3.2)

${\stackrel{˜}{\tau }}_{1}=\mathrm{inf}\left\{t>0:w\left(t\right)+|g\left(t\right)-g|\ge \epsilon \right\}.$

(3.3)

(3.4)

i) 对于的情况，对任意

(3.5)

ii) 对于的情况，此时存在一个，有

(3.6)

(3.7)

(3.8)

1)的情况。

(3.9)

(3.10)

(3.11)

(3.12)

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)

(3.20)

(3.21)

(3.22)

(3.23)

2）的情况。

3)的情况。

(3.24)

(3.25)

(3.26)

(3.27)

(3.28)

(3.29)

，根据( [13]，命题4.1)，存在一些，满足

(3.30)

(3.31)

(3.32)

(3.33)

(3.34)

(3.35)

，有

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