## Complex Dynamics of an Intraguild Predation Model

Yang Xiaomin,, Qiu Zhipeng, Ding Ling

 基金资助: 国家自然科学基金.  11671206

 Fund supported: the NSFC.  11671206

Abstract

The complex dynamics of an intraguild predation (IGP) model is investigated in this paper, and the model incorporates the Holling-Ⅱ functional response functions. The sufficient conditions are obtained for the existence and local stability of boundary equilibria. Then, the numerical simulations are applied to the model under the given values of parameters. The numerical results show that the system may have an attracting invariant torus but no positive equilibrium. Furthermore, the Poincaré map and Fourier transform spectrum analysis are performed to study the complex dynamics of the system on the invariant torus. The results suggest that the dynamics on the invariant torus is almost periodic.

Keywords： IGP model ; Invariant torus ; Numerical simulation ; Poincaré map ; Fourier spectrum analysis

Yang Xiaomin, Qiu Zhipeng, Ding Ling. Complex Dynamics of an Intraguild Predation Model. Acta Mathematica Scientia[J], 2019, 39(4): 963-970 doi:

## 1 引言

1997年, Holt和Polis[3-4]根据上述种间关系建立了三种群动力学模型,简称IGP模型.一个典型的IGP系统包括三个种群:共享资源, IG食饵, IG捕食者,分别记为$R$, $C$, $P$.在IGP系统中, IG捕食者$P$既捕食IG食饵$C$,又以共享资源$R$为食; IG食饵$C$与共享资源$R$之间为捕食与被捕食关系; IG捕食者$P$与IG食饵$C$在对共享资源$R$的捕食过程中又存在竞争关系. IGP系统的能量传递关系见图 1所示.

### 图 1

$$$\left\{ \begin{array}{l} \frac{{{\rm d}R}}{{{\rm d}\tau}} = R \bigg[r(1 - \frac{R}{K}) - \frac{{{m_1}C}}{{1 + {n_1}R}} - \frac{{{m_2}P}}{{1 + {n_2}R}} \bigg], \\ \frac{{{\rm d}C}}{{{\rm d}\tau}} = C \bigg({b_1}\frac{{{m_1}R}}{{1 + {n_1}R}} - \frac{{{m_3}P}}{{1 + {n_3}C}} - {c_1} \bigg), \\ \frac{{{\rm d}P}}{{{\rm d}\tau}} = P \bigg({b_2}\frac{{{m_2}R}}{{1 + {n_2}R}} + \beta \frac{{{m_3}C}}{{1 + {n_3}C}} - {c_2} \bigg), \end{array} \right.$$$

## 2 边界平衡点的存在性及稳定性

$$$\left\{ \begin{array}{l} \frac{{{\rm d}x}}{{{\rm d}t}} = x \bigg(1 - x - \frac{{y}}{{1 + {\alpha _1}x}} - \frac{{z}}{{1 + {\alpha _2}x}} \bigg), \\ \frac{{{\rm d}y}}{{{\rm d}t}} = {\gamma _1}y \bigg(\frac{{x}}{{1 + {\alpha _1}x}} - \frac{{e_1}z}{{1 + hy}} - {d_1} \bigg), \\ \frac{{{\rm d}z}}{{{\rm d}t}} = {\gamma _2}z \bigg(\frac{{x}}{{1 + {\alpha _2}x}} + \frac{{e_2}y}{{1 + hy}} - {d_2} \bigg). \end{array} \right.$$$

(1)当$\frac{1}{1+{\alpha_1}} > d_1$时,平衡点$E_{xy}$存在;

(2)当$\frac{1}{1+{\alpha_2}} > d_2$时,平衡点$E_{xz}$存在.

(ⅱ)系统(2.1)始终存在平衡点${E_x}$.如果$\frac{1}{1+{\alpha _1}} < {d_1} $$\frac{1}{1+{\alpha _2}} < {d_2} , {E_x} 为稳定的结点;如果 \frac{1}{1+{\alpha _1}} > {d_1}$$ \frac{1}{1+{\alpha _2}} > {d_2}$, ${E_x}$为鞍点;如果$\frac{1}{1+{\alpha _1}} = {d_1} $$\frac{1}{1+{\alpha _2}}\neq{d_2} ,或 \frac{1}{1+{\alpha _1}}\neq{d_1}$$ \frac{1}{1+{\alpha _2}} = {d_2}$, ${E_x}$为鞍-结点;如果$\frac{1}{1+{\alpha _1}} = {d_1} $$\frac{1}{1+{\alpha _2}} = {d_2} ,则平衡点 {E_x} 是余维 2 的退化点. (ⅲ)当 \frac{1}{1+{\alpha _1}} > {d_1} 时,平衡点 {E_{xy}} 存在.进一步,如果 以及 {E_{xy}} 是局部渐近稳定的;如果 {E_{xy}} 不稳定. (ⅳ)当 \frac{1}{1+{\alpha _2}} > {d_2} 时,平衡点 {E_{xz}} 存在.进一步,如果 以及 {E_{xz}} 是局部渐近稳定的;如果 {E_{xz}} 不稳定. 对于系统(2.1),当 x = 0 时,有 \frac{{\rm d}x}{{\rm d}t} = 0 ,因此坐标平面 yz 是不变面.同理可知,坐标平面 xy, xz 都是不变面,从而系统(2.1)在 {\Bbb R}_+^3 中是正向不变的. (ⅰ)在平衡点 E_0 = (0, 0, 0) 处,系统(2.1)的线性化矩阵为 直接计算可得,矩阵 J({E_0}) 的特征值分别为 又因为坐标平面 yz 是不变面,且容易知道在坐标面内的轨线均收敛于 E_0 .同理可得,坐标轴 x -轴也是不变轴,且在坐标轴内的轨线当 t\rightarrow-\infty 时均趋向于 E_0 .因此由Hartman-Grobman定理容易知道,平衡点 E_0 的稳定流形是 yz 平面,不稳定流形是 \{(x, 0, 0)\mid 0\leq x < 1\} . (ⅱ)在平衡点 {E_x} = (1, 0, 0) 处,系统(2.1)的线性化矩阵为 由计算可得 J({E_x}) 的特征值分别为 因此,如果 \frac{1}{{\alpha _1} + 1} < {d_1}$$ \frac{1}{{\alpha _2} + 1} < {d_2}$,则有${\lambda_2} < 0, {\lambda_3} < 0$,从而${E_x}$是稳定的结点;如果$\frac{1}{{\alpha _1} + 1} > {d_1} $$\frac{1}{{\alpha _2} + 1} > {d_2} ,则有 {\lambda_2} > 0$$ {\lambda_3} > 0$,从而${E_x}$是鞍点;如果$\frac{1}{1+{\alpha _1}} = {d_1} $$\frac{1}{1+{\alpha _2}}\neq{d_2} ,或 \frac{1}{1+{\alpha _1}}\neq{d_1}$$ \frac{1}{1+{\alpha _2}} = {d_2}$, ${E_x}$是鞍-结点;如果$\frac{1}{1+{\alpha _1}} = {d_1} $$\frac{1}{1+{\alpha _2}} = {d_2} ,特征值 {\lambda_2} = {\lambda_3} = 0 ,则平衡点 {E_x} 是余维 2 的退化点. (ⅲ)当 \frac{1}{{\alpha _1} + 1} > {d_1} ,则平衡点 存在,其对应的系统(2.1)的线性化矩阵为 计算可得 J({E_{xy}}) 的特征值分别为 其中 b = \frac{2d_1}{1-{\alpha_1}d_1} - d_1(1+{\alpha_1}). 因此,当 以及 \lambda_1 < 0, \lambda_2 < 0, \lambda_3 < 0 时, {E_{xy}} 是局部渐近稳定的;当 \lambda_{2, 3} > 0$$ \lambda_1 > 0$,因此平衡点${E_{xy}}$不稳定.

(ⅳ)证明思路及过程与情形(ⅲ)相同,故在此省略.

## 3 数值仿真

### 图 2

(a) $x, y, z-t$状态曲线; (b) $x, y, z$三维相图; (c) $x, y$二维相图; (d) $x, z$二维相图; (e) $y, z$二维相图

### 图 3

($c'$)平行于平面$xy$的截面图; ($d'$)平行于平面$xz$的截面图; ($e'$)平行于平面$yz$的截面图

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