﻿ 具有扩散项的互惠–寄生耦合模型图灵斑图研究 Analysis of Spatial Pattern in Mutualistic-Parasitic Coupled System with Diffusion

Advances in Applied Mathematics
Vol.06 No.07(2017), Article ID:22432,9 pages
10.12677/AAM.2017.67101

Analysis of Spatial Pattern in Mutualistic-Parasitic Coupled System with Diffusion

Lei Gao

School of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji Shaanxi

Received: Oct. 3rd, 2017; accepted: Oct. 17th, 2017; published: Oct. 24th, 2017

ABSTRACT

In this paper, we establish mutualistic-parasitic coupled spatial model incorporating diffusion, and study the dynamical behaviors of this spatial model. By spatial pattern theory, we obtain the conditions for Hopf bifurcation and Turing bifurcation, and give the exact parameter space for Turing domain. By utilizing difference approximation method, we find that the spatial model exhibits black eye spotted pattern, which shows that the diffusion can result in an isolated high density of parasite in the space.

Keywords:Mutualism, Parasitism, Bifurcation, Spatial Pattern, Diffusion

Copyright © 2017 by author and Hans Publishers Inc.

1. 引言

2. 主要结果

2.1. 建立空间模型

$\left\{\begin{array}{l}\frac{\text{d}H}{\text{d}t}=H\left[{r}_{1}-{\alpha }_{1}H+\left(\mu -\frac{P}{H}\right)\beta P\right]\equiv F\left(H,P\right)\\ \frac{\text{d}P}{\text{d}t}=P\left({r}_{2}-{\alpha }_{2}\frac{P}{H}\right)\equiv G\left(H,P\right)\end{array}$ (1)

$\frac{\partial H}{\partial t}={d}_{1}{\nabla }^{2}H$$\frac{\partial P}{\partial t}={d}_{2}{\nabla }^{2}P$

Table 1. Definitions and the symbols used in model (1)

$\left\{\begin{array}{l}\frac{\partial H}{\partial t}=F\left(H,P\right)+{d}_{1}{\nabla }^{2}H\\ \frac{\partial P}{\partial t}=G\left(H,P\right)+{d}_{2}{\nabla }^{2}P\end{array}$ (2)

2.2. 空间模型的分支分析

$\left\{\begin{array}{l}F\left({H}^{*},{P}^{*}\right)=0\\ G\left({H}^{*},{P}^{*}\right)=0\end{array}$ .

$\left\{\begin{array}{l}\frac{\partial h}{\partial t}={a}_{11}h+{a}_{12}p+{d}_{1}{\nabla }^{2}h\\ \frac{\partial p}{\partial t}={a}_{21}h+{a}_{22}p+{d}_{2}{\nabla }^{2}p\end{array}$ (3)

${a}_{11}={\frac{\partial F}{\partial H}|}_{\left({H}^{*},{P}^{*}\right)}=\frac{\left(\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}-{\alpha }_{1}\right)\cdot {r}_{1}}{{\alpha }_{1}-\left(\mu -{r}_{2}/{\alpha }_{2}\right)\beta \cdot {r}_{2}/{\alpha }_{2}}$ ,

${a}_{12}={\frac{\partial F}{\partial P}|}_{\left({H}^{*},{P}^{*}\right)}=\frac{\left(\mu -\text{2}{r}_{2}/{\alpha }_{2}\right)\cdot \beta \cdot {r}_{1}}{{\alpha }_{1}-\left(\mu -{r}_{2}/{\alpha }_{2}\right)\beta \cdot {r}_{2}/{\alpha }_{2}}$ ,

${a}_{21}={\frac{\partial G}{\partial H}|}_{\left({H}^{*},{P}^{*}\right)}={r}_{2}^{2}/{\alpha }_{2}$ , ${a}_{22}={\frac{\partial G}{\partial P}|}_{\left({H}^{*},{P}^{*}\right)}=-{r}_{2}$ .

$\left(\begin{array}{l}h\\ p\end{array}\right)=\underset{\kappa }{\sum }\left(\begin{array}{l}{c}_{\kappa }^{1}\\ {c}_{\kappa }^{2}\end{array}\right){\text{e}}^{{\lambda }_{\kappa }t+i\kappa h}$ (4)

${\lambda }_{\kappa }\left(\begin{array}{l}{c}_{\kappa }^{1}\\ {c}_{\kappa }^{2}\end{array}\right)=\left(\begin{array}{cc}{a}_{11}-{\kappa }^{2}{d}_{1}& {a}_{12}\\ {a}_{21}& {a}_{22}-{\kappa }^{2}{d}_{2}\end{array}\right)\left(\begin{array}{l}{c}_{\kappa }^{1}\\ {c}_{\kappa }^{2}\end{array}\right)$ (5)

${\lambda }_{\kappa }^{2}-t{r}_{\kappa }{\lambda }_{\kappa }+{\Delta }_{\kappa }=0$

$t{r}_{\kappa }={a}_{11}+{a}_{22}-{\kappa }^{2}\left({d}_{1}+{d}_{2}\right)=t{r}_{0}-{\kappa }^{2}\left({d}_{1}+{d}_{2}\right)$

$\begin{array}{c}{\Delta }_{\kappa }={a}_{11}{a}_{22}-{a}_{21}{a}_{12}-{\kappa }^{2}\left({a}_{11}{d}_{2}+{a}_{22}{d}_{1}\right)+{\kappa }^{4}{d}_{1}{d}_{2}\\ ={\Delta }_{0}-{\kappa }^{2}\left({a}_{11}{d}_{2}+{a}_{22}{d}_{1}\right)+{\kappa }^{4}{d}_{1}{d}_{2}\end{array}$ ,

${\lambda }_{\kappa }=\frac{t{r}_{\kappa }±\sqrt{t{r}_{\kappa }^{2}-4{\Delta }_{\kappa }}}{2}$ .

1) 在跨临界分支中，非扩散模型的一个特征值消失。因此，解 ${\lambda }_{1}\cdot {\lambda }_{2}={\Delta }_{\kappa }=0$$\kappa =0$ ，可得跨临界分支参数 $\mu$ 的临界值表达式：

$\mu ={\alpha }_{1}{\alpha }_{2}/\left({r}_{2}\beta \right)+{r}_{2}/{\alpha }_{2}$ .

2) Hopf分支不稳定的出现对应于下面情况：对于非扩散模型来说，当根从负到正跨越实轴时产生了一对纯虚根。从数学角度看，当满足下列条件：

$Im\left({\lambda }_{\kappa }\right)\ne 0$ , $Re\left({\lambda }_{\kappa }\right)=0$ .

$\kappa =0$ 系统产生了Hopf分支，即

$t{r}_{0}={a}_{11}+{a}_{22}=0$ ,

$\left(\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}-{\alpha }_{1}\right)\cdot {r}_{1}/\left({\alpha }_{1}-\left(\mu -{r}_{2}/{\alpha }_{2}\right)\beta \cdot {r}_{2}/{\alpha }_{2}\right)-{r}_{2}=0$ .

${r}_{1}\left(\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}-{\alpha }_{1}\right)-{r}_{2}\left({\alpha }_{1}-\left(\mu -{r}_{2}/{\alpha }_{2}\right)\beta \cdot {r}_{2}/{\alpha }_{2}\right)=0$ ,

${r}_{1}\left(\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}-{\alpha }_{1}\right)-{r}_{2}\left({\alpha }_{1}+\beta \cdot {\left({r}_{2}/{\alpha }_{2}\right)}^{2}\right)+\left({r}_{2}\cdot \beta \cdot {r}_{2}/{\alpha }_{2}\right)\mu =0$ ,

$\mu ={r}_{1}/{r}_{2}\cdot {\alpha }_{1}{\alpha }_{2}/\left({r}_{2}\beta \right)+{\alpha }_{1}{\alpha }_{2}/\left({r}_{2}\beta \right)+{r}_{2}/{\alpha }_{2}-{r}_{1}/{\alpha }_{2}$ .

3) 令

$B\left({\kappa }^{2}\right)=\left(\begin{array}{cc}{a}_{11}-{\kappa }^{2}{d}_{1}& {a}_{12}\\ {a}_{21}& {a}_{22}-{\kappa }^{2}{d}_{2}\end{array}\right)$ ,

$\mathrm{det}\left(B\left({\kappa }^{2}\right)\right)$ 是关于 ${\kappa }^{2}$ 的二次多项式，它在某些 ${\kappa }^{2}$ 值处取得极小值。首先假设非扩散模型在平衡点处是稳定的(特别的 $\mathrm{det}\left(B\left(0\right)\right)>0$ )，且图灵不稳定的发生需满足下面两个条件：(a) ${\kappa }^{2}>0$ (如果这个条件不成立，对于所有的 ${\kappa }^{2}=0$ ，由 $\mathrm{det}\left(B\left(0\right)\right)>0$ 可推出 $\mathrm{det}\left(B\left({\kappa }^{2}\right)\right)>0$ )是图灵不稳定的必要条件；(b) 图灵不稳定的充分条件是 $\mathrm{det}\left(B\left({\kappa }^{2}\right)\right)<0$ ，线性分析给出了图灵斑图产生的条件如下：

$t{r}_{0}={a}_{11}+{a}_{22}<0$ , ${\Delta }_{0}={a}_{11}{a}_{22}-{a}_{21}{a}_{12}>0$

${a}_{11}{d}_{2}+{a}_{22}{d}_{1}>0$${\left({a}_{11}{d}_{2}+{a}_{22}{d}_{1}\right)}^{2}>4{d}_{2}{d}_{1}\left({a}_{11}{a}_{22}-{a}_{21}{a}_{12}\right)$

$Im\left({\lambda }_{\kappa }\right)=0$ , $Re\left({\lambda }_{\kappa }\right)=0$ , $\kappa ={\kappa }_{T}\ne 0$ ,

${\kappa }_{T}^{2}=\sqrt{\frac{{a}_{11}{a}_{22}-{a}_{21}{a}_{12}}{{d}_{1}{d}_{2}}}$ .

${\lambda }_{{\kappa }_{T}}=\frac{t{r}_{{\kappa }_{T}}±\sqrt{t{r}_{{\kappa }_{T}}^{2}-4{\Delta }_{{\kappa }_{T}}}}{2}$

$\left\{\begin{array}{l}t{r}_{{\kappa }_{T}}=0\\ t{r}_{{\kappa }_{T}}^{2}-4{\Delta }_{{\kappa }_{T}}=0\end{array}$

$t{r}_{\kappa }={a}_{11}+{a}_{22}-{\kappa }_{T}^{2}\left({d}_{1}+{d}_{2}\right)$ ,

${\Delta }_{\kappa }={a}_{11}{a}_{22}-{a}_{21}{a}_{12}-{\kappa }_{T}^{2}\left({a}_{11}{d}_{2}+{a}_{22}{d}_{1}\right)+{\kappa }_{T}^{4}{d}_{1}{d}_{2}$ ,

$\left\{\begin{array}{l}{a}_{11}+{a}_{22}-{\kappa }_{T}^{2}\left({d}_{1}+{d}_{2}\right)=0\\ {a}_{11}{a}_{22}-{a}_{21}{a}_{12}-{\kappa }_{T}^{2}\left({a}_{11}{d}_{2}+{a}_{22}{d}_{1}\right)+{\kappa }_{T}^{4}{d}_{1}{d}_{2}=0\end{array}$ ,

$\left\{\begin{array}{l}{a}_{11}+{a}_{22}-{\kappa }_{T}^{2}\left({d}_{1}+{d}_{2}\right)=0\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}}\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}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{ }\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}}\left(6\right)\\ 2\left({a}_{11}{a}_{22}-{a}_{21}{a}_{12}\right)-{\kappa }_{T}^{2}\left({a}_{11}{d}_{2}+{a}_{22}{d}_{1}\right)=0\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}}\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}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(7\right)\end{array}$

$\left(7\right)×\left({d}_{1}+{d}_{2}\right)-\left(6\right)×\left({a}_{11}{d}_{2}+{a}_{22}{d}_{1}\right)$

$2\left({a}_{11}{a}_{22}-{a}_{21}{a}_{12}\right)\left({d}_{1}+{d}_{2}\right)-\left({a}_{11}+{a}_{22}\right)\left({a}_{11}{d}_{2}+{a}_{22}{d}_{1}\right)=0$ ,

$\left({a}_{11}-{a}_{22}\right)\left({a}_{22}{d}_{1}-{a}_{11}{d}_{2}\right)-2{a}_{12}{a}_{21}\left({d}_{1}+{d}_{2}\right)=0$ ,

$\begin{array}{l}\left(\frac{\left(\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}-{\alpha }_{1}\right)\cdot {r}_{1}}{{\alpha }_{1}-\left(\mu -{r}_{2}/{\alpha }_{2}\right)\beta \cdot {r}_{2}/{\alpha }_{2}}+{r}_{2}\right)\left(-{r}_{2}\cdot {d}_{1}-\frac{\left(\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}-{\alpha }_{1}\right)\cdot {r}_{1}}{{\alpha }_{1}-\left(\mu -{r}_{2}/{\alpha }_{2}\right)\beta \cdot {r}_{2}/{\alpha }_{2}}\cdot {d}_{2}\right)\\ -2\frac{\left(\mu -\text{2}{r}_{2}/{\alpha }_{2}\right)\cdot \beta \cdot {r}_{1}}{{\alpha }_{1}-\left(\mu -{r}_{2}/{\alpha }_{2}\right)\beta \cdot {r}_{2}/{\alpha }_{2}}\cdot {r}_{2}^{2}/{\alpha }_{2}\cdot \left({d}_{1}+{d}_{2}\right)=0\end{array}$ ,

$\begin{array}{l}\left(\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}-{\alpha }_{1}\right)\cdot {r}_{2}{d}_{1}+{\left(\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}-{\alpha }_{1}\right)}^{2}\cdot {r}_{1}/\left({\alpha }_{1}-\left(\mu -{r}_{2}/{\alpha }_{2}\right)\beta \cdot {r}_{2}/{\alpha }_{2}\right)\cdot {d}_{2}\\ +{r}_{2}^{2}\cdot {d}_{1}\left({\alpha }_{1}-\left(\mu -{r}_{2}/{\alpha }_{2}\right)\beta \cdot {r}_{2}/{\alpha }_{2}\right)/{r}_{1}+\left(\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}-{\alpha }_{1}\right)\cdot {r}_{2}{d}_{2}\\ +2\beta \left(\mu -2{r}_{2}/{\alpha }_{2}\right)\cdot {r}_{2}^{2}/{\alpha }_{2}\left({d}_{1}+{d}_{2}\right)=0\end{array}$ ,

$\begin{array}{l}\left(\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}-{\alpha }_{1}\right)\cdot \left({\alpha }_{1}-\left(\mu -{r}_{2}/{\alpha }_{2}\right)\beta \cdot {r}_{2}/{\alpha }_{2}\right)\cdot {d}_{1}/{d}_{2}\\ +{\left(\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}-{\alpha }_{1}\right)}^{2}\cdot {r}_{1}/{r}_{2}+{r}_{2}/{r}_{1}\cdot {d}_{1}/{d}_{2}\cdot {\left({\alpha }_{1}-\left(\mu -{r}_{2}/{\alpha }_{2}\right)\beta \cdot {r}_{2}/{\alpha }_{2}\right)}^{2}\\ +\left(\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}-{\alpha }_{1}\right)\cdot \left({\alpha }_{1}-\left(\mu -{r}_{2}/{\alpha }_{2}\right)\beta \cdot {r}_{2}/{\alpha }_{2}\right)\\ +2\beta \left(\mu -2{r}_{2}/{\alpha }_{2}\right)\cdot {r}_{2}/{\alpha }_{2}\cdot \left(1+{d}_{1}/{d}_{2}\right)\left({\alpha }_{1}-\left(\mu -{r}_{2}/{\alpha }_{2}\right)\beta \cdot {r}_{2}/{\alpha }_{2}\right)=0\end{array}$

$A=\beta \cdot {\left({r}_{2}/{\alpha }_{2}\right)}^{2}-{\alpha }_{1}$ , $D={d}_{1}/{d}_{2}$ , $r={r}_{1}/{r}_{2}$ ,

$\begin{array}{l}\left(1+D\right){\alpha }_{1}-\mu \beta {r}_{2}/{\alpha }_{2}\cdot \left(1+D\right)+\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}\left(1+D\right)+Ar\\ +D/r\cdot \left({\alpha }_{1}^{2}-2{\alpha }_{1}\left(\mu -{r}_{2}/{\alpha }_{2}\right)\beta {r}_{2}/{\alpha }_{2}+{\left(\mu -{r}_{2}/{\alpha }_{2}\right)}^{2}{\beta }^{2}{\left({r}_{2}/{\alpha }_{2}\right)}^{2}\right)\\ +2\mu \beta {r}_{2}/{\alpha }_{2}\cdot \left(\left(1+D\right)/A\right)\left({\alpha }_{1}-\left(\mu -{r}_{2}/{\alpha }_{2}\right)\beta {r}_{2}/{\alpha }_{2}\right)\\ -4\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}\left(\left(1+D\right)/A\right)\left({\alpha }_{1}-\left(\mu -{r}_{2}/{\alpha }_{2}\right)\beta {r}_{2}/{\alpha }_{2}\right)=0\end{array}$ .

$a={\beta }^{2}\cdot {\left({r}_{2}/{\alpha }_{2}\right)}^{2}\cdot \left(D/r-2\left(1+D\right)/A\right)$ ,

$b=-\beta \left({r}_{2}/{\alpha }_{2}\right)\cdot \left(1+D\right)+2{\alpha }_{1}\beta \cdot {r}_{2}/{\alpha }_{2}\cdot \left(\left(1+D\right)/A-D/r\right)+2{\beta }^{2}{\left({r}_{2}/{\alpha }_{2}\right)}^{3}\cdot \left(3\left(1+D\right)/A-D/r\right)$$c=\left(1+D\right)\cdot \left({\alpha }_{1}+\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}\right)+Ar+D/r\cdot {\left(\left({\alpha }_{1}+\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}\right)\right)}^{2}+4\beta {\left({r}_{2}/{\alpha }_{2}\right)}^{2}\cdot \left(1+D\right)$ .

$a{\mu }^{2}+b\mu +c=0$ ,

$\mu =\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$ .

3. 数值模拟

$\left\{\begin{array}{l}{H}^{k+1}\left(i,j\right)={H}^{k}\left(i,j\right)+\Delta t\cdot {F}_{1}\left(H,P\right)+\frac{\Delta t\cdot {d}_{1}}{{\left(\Delta h\right)}^{2}}\left[{H}^{k}\left(i+1,j\right)+{H}^{k}\left(i-1,j\right)+{H}^{k}\left(i,j+1\right)+{H}^{k}\left(i,j-1\right)-4{H}^{k}\left(i,j\right)\right]\\ {P}^{k+1}\left(i,j\right)={P}^{k}\left(i,j\right)+\Delta t\cdot {F}_{2}\left(H,P\right)+\frac{\Delta t\cdot {d}_{2}}{{\left(\Delta h\right)}^{2}}\left[{P}^{k}\left(i+1,j\right)+{P}^{k}\left(i-1,j\right)+{P}^{k}\left(i,j+1\right)+{P}^{k}\left(i,j-1\right)-4{P}^{k}\left(i,j\right)\right]\end{array}$

Figure 1. The parameter space of bifurcation

Figure 2. The time evolution of spatial pattern for parasitism

${F}_{1}\left(H,P\right)={H}^{k}\left(i,j\right)\cdot \left[{r}_{1}-{\alpha }_{1}{H}^{k}\left(i,j\right)+\left[\mu -\frac{{P}^{k}\left(i,j\right)}{{H}^{k}\left(i,j\right)}\right]\beta {P}^{k}\left(i,j\right)\right]$ . ${F}_{2}\left(H,P\right)={P}^{k}\left(i,j\right)\cdot \left[{r}_{2}-{\alpha }_{2}\frac{{P}^{k}\left(i,j\right)}{{H}^{k}\left(i,j\right)}\right]$

${H}_{n}\left({x}_{0},{y}_{j}\right)={H}_{n}\left({x}_{1},{y}_{j}\right),{H}_{n}\left({x}_{N+1},{y}_{j}\right)={H}_{n}\left({x}_{N},{y}_{j}\right)$ ,

${H}_{n}\left({x}_{i},{y}_{0}\right)={H}_{n}\left({x}_{i},{y}_{1}\right),{H}_{n}\left({x}_{i},{y}_{N+1}\right)={H}_{n}\left({x}_{i},{y}_{N}\right)$ ,

${P}_{n}\left({x}_{0},{y}_{j}\right)={P}_{n}\left({x}_{1},{y}_{j}\right),{P}_{n}\left({x}_{N+1},{y}_{j}\right)={P}_{n}\left({x}_{N},{y}_{j}\right)$ ,

${P}_{n}\left({x}_{i},{y}_{0}\right)={P}_{n}\left({x}_{i},{y}_{1}\right),{P}_{n}\left({x}_{i},{y}_{N+1}\right)={P}_{n}\left({x}_{i},{y}_{N}\right)$ .

Analysis of Spatial Pattern in Mutualistic-Parasitic Coupled System with Diffusion[J]. 应用数学进展, 2017, 06(07): 841-849. http://dx.doi.org/10.12677/AAM.2017.67101

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