## Fractal Feature and Control of Three-Species Predator-Prey Model

Shao Changxu, Liu Shutang,

 基金资助: 国家自然科学基金重点项目.  61533011

 Fund supported: the Key Program of National Natural Science Foundation.  61533011

Abstract

The law of population quantity change is one of the key problems in Animal Ecology and Resource Management. By studying the change of population quantity, we can effectively grasp the population dynamics and living habits, which is of great significance for the rational utilization of resources and the protection of ecology. In this paper, we discuss the three-species predator-prey model from the point of fractal theory. We construct the Julia set of 3D predatorprey model. By studying the property of Julia sets, we discuss the conditions for the model to be stable, and take feedback control terms to realize the transformation from instability to stability. In addition, the effects of single population changes on the other two populations and ecosystems were analyzed. Finally, the nonlinear coupling terms with different parameters are constructed, the response system is transformed into the target system, and the synchronization between different systems is realized. Simulation results show the effectiveness of the method.

Keywords： Predator-prey model ; Stability ; Julia set ; Feedback control ; Synchronization

Shao Changxu, Liu Shutang. Fractal Feature and Control of Three-Species Predator-Prey Model. Acta Mathematica Scientia[J], 2019, 39(4): 951-962 doi:

## 1 引言

Lotka-Volterra捕食者-食饵模型最初是由Lotka于1910年在自催化反应理论[1-4]中提出. 1926年数学家和物理学家Volterra提出了同样的方程组[5-6],并用它解释一战期间亚得里亚海掠食性鱼类数量与渔捞量之间的关系[7].由于种群竞争模型分别由Lotka和Volterra两人提出,因此后来称之为Lotka-Volterra模型. Lotka-Volterra模型被用来解释捕食者和被捕食者的自然种群的动态,例如哈德逊湾公司的山猫和雪兔的数量[8]、罗亚岛国家公园里的鹿群和狼群数量[9].该模型为研究种群竞争关系提供了理论基础,并对现代生态学理论的发展产生了重要影响.基于该模型的基本概念和理论框架,人们进行了一些改进和实践研究[10-13].徐瑞等人[14]研究了一类具有时滞和基于比率的三种群Lotka-Volterra模型.证明了该系统在适当条件下的一致持久性,并通过构造李亚普诺夫函数的方法,得到了使该系统正平衡点全局渐近稳定的充分条件.陆忠华等人[15]研究了既有捕食关系又有竟争关系的三种群Lotka-Volterra模型,从周期系数的观点出发,得到了全局渐近稳定周期解的唯一存在条件.伏升茂等人[16]研究了具有自扩散和互扩散的三种群Lotka-Volterra模型,从自扩散和交叉扩散的角度讨论了一致有界解的全局存在性,证明了正平衡点的全局渐近稳定性.在竞争模型中,物种的持久性和灭绝与物种的初始数量有关,而分形理论中的Julia集与初始点的轨迹相关,这为我们研究Lotka-Volterra模型提供了一种新思路.据我们所知,目前为止运用分形理论讨论Lotka-Volterra模型尚未少数,因此本文从分形的角度对竞争模型进行了研究.

$\begin{eqnarray} \left \{\begin{array}{l} 2x^*-(x^*)^2-a_1 x^*y^*-b_1 x^*z^* = x^*, \\ 2y^*-(y^*)^2-a_2 x^*y^*-b_2 y^*z^* = y^*, \\ (1-d_1)z^*-d_2(z^*)^{2}+c_1 x^*z^*+c_2 y^*z^* = z^*. \end{array} \right. \end{eqnarray}$

(1) $(x^*, y^*, z^*) = (0, 0, 0).$

(2) $x^* = 0, y^*\neq0, z^*\neq0$,代入(3.3)式得到不动点$(x^*, y^*, z^*) = (0, \frac{d_{2}+b_{2}d_{1}}{d_{2}+b_{2}c_{2}}, \frac{c_{2}-d_{1}}{d_{2}+b_{2}c_{2}}).$同理, $x^*\neq0, y^* = 0, z^*\neq0$时,不动点为$(x^*, y^*, z^*) = (\frac{d_{2}+b_{1}d_{1}}{d_{2}+b_{2}c_{1}}, 0, \frac{c_{1}-d_{1}}{d_{2}+b_{1}c_{1}}). $$x^*\neq0, y^*\neq0,$$ z^* = 0$时,不动点为$(x^*, y^*, z^*) = (\frac{1-a_{1}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, 0).$

(3) $x^* = 0, y^* = 0, z^*\neq0$,代入(3.3)式得到不动点$(x^*, y^*, z^*) = (0, 0, \frac{-d_{1}}{d_{2}})$同理,当$x^* = 0, y^*\neq0, z^* = 0 $$x^*\neq0, y^* = 0, z^* = 0 时,不动点分别为 (x^*, y^*, z^*) = (0, 1, 0).$$ (x^*, y^*, z^*) = (1, 0, 0).$

(4) $x^*\neq0, y^*\neq0, z^*\neq0$,考虑该离散系统的共存不动点$(x^*, y^*, z^*) = (\frac{D_x}{D}, \; \frac{D_y}{D}, \; \frac{D_z}{D})$,其中

$\begin{eqnarray} &&\lambda^{3}+(x^*+y^*+d_{2}z^*-3)\lambda^{2}+[(d_{2}+b_{1}c_{1})x^*z^*+(d_{2}+b_{2}c_{2})y^*z^*\\ &&+(1-a_{1}a_{2})x^*y^*-2x^*-2y^*-2d_{2}z^*+3]\lambda+(d_{2}-a_{1}b_{2}c_{1}\\ &&-a_{2}b_{1}c_{2}+b_{1}c_{1}+b_{2}c_{2}-a_{1}a_{2}d_{2})x^*y^*z^*+(a_{1}a_{2}-1)x^*y^*\\ &&-(b_{1}c_{1}+d_{2})x^*z^*-(b_{2}c_{2}+d_{2})y^*z^*+x^*+y^*+d_{2}z^* = 0. \end{eqnarray}$

## 4 竞争模型Julia集的控制

$\begin{eqnarray} \left \{\begin{array}{l} x_{n+1} = 2x_{n}-x_{n}^2-a_1 x_{n}y_{n}-b_1 x_{n}z_{n}+k_{1}(x_{n}-x_{ref}), \\ y_{n+1} = 2y_{n}-y_{n}^2-a_2 x_{n}y_{n}-b_2 y_{n}z_{n}+k_{2}(y_{n}-y_{ref}), \\ z_{n+1} = (1-d_1)z_{n}-d_2z_{n}^2+c_1 x_{n}z_{n}+c_2y_{n}z_{n}+k_{3}(z_{n}-z_{ref}). \end{array} \right. \end{eqnarray}$

$\begin{eqnarray} \left \{\begin{array}{l} x_{n+1} = 2x_{n}-x_{n}^2-a_1 x_{n}y_{n}-b_1 x_{n}z_{n}+k_{1}(x_{n}-x^*), \\ y_{n+1} = 2y_{n}-y_{n}^2-a_2 x_{n}y_{n}-b_2 y_{n}z_{n}+k_{2}(y_{n}-y^*), \\ z_{n+1} = (1-d_1)z_{n}-d_2z_{n}^2+c_1 x_{n}z_{n}+c_2y_{n}z_{n}+k_{3}(z_{n}-z^*). \end{array} \right. \end{eqnarray}$

$\begin{eqnarray} &&\lambda^3+(x^*+y^*+d_2z^*-k_1-k_2-k_3-3)\lambda^2+[(1-a_1a_2)x^*y^*+(b_1c_1+d_2)x^*z^*\\ &&+(b_2c_2+d_2)y^*z^*-(k_2+k_3+2)x^*-(k_1+k_3+2)y^*-(k_1+k_2+2)d_2z^*+ 2(k_1+k_2+k_3)\\ &&+k_1k_2+k_1k_3+k_2k_3+3]\lambda+(b_1c_1+b_2c_2-a_1a_2d_2-a_1b_2c_1-a_2b_1c_2+d_2)x^*y^*z^*\\ &&-(1-a_1a_2)(k_3+1)x^*y^*-(b_1c_1+d_2)(k_2+1)x^*z^*-(b_2c_2+d_2)(k_1+1)y^*z^*\\ &&+(1+k_2+k_3+k_2k_3)x^*+(1+k_1+k_3+k_1k_3)y^*+(1+k_1+k_2+k_1k_2)d_2z^*\\ &&-(k_1+k_2+k_3+k_1k_2+k_1k_3+k_2k_3+k_1k_2k_3+1) = 0. \end{eqnarray}$

 行数 $z^0$ $z^1$ $z^2$ $z^3$ 1 $w'$ $n'$ $m'$ 1 2 1 $m'$ $n'$ $w'$ 3 $b_0$ $b_1$ $b_2$

## 5 竞争模型Julia集的同步

$\begin{eqnarray} \left \{\begin{array}{l} p_{n+1} = 2p_{n}-p_{n}^2-\alpha_1 p_{n}q_{n}-\beta_1 p_{n}w_{n}, \\ q_{n+1} = 2q_{n}-q_{n}^2-\alpha_2 p_{n}q_{n}-\beta_2 q_{n}w_{n}, \\ w_{n+1} = (1-\xi_1)w_{n}-\xi_2w_{n}^2+\gamma_1 p_{n}w_{n}+\gamma_2 q_{n}w_{n}. \end{array} \right. \end{eqnarray}$

由于Julia集以及系统参数的有界性,存在$M_1 > 0$满足$|p^2_n|+|x^2_n| < M_1$, $|a_1||x_ny_n|+|\alpha_1p_nq_n| < M_1$, $|b_1||x_nz_n|+|\beta_1||p_nw_n| < M_1$.

$m_1\rightarrow2, \; h_1\rightarrow1$时, $|p_{n+1}-x_{n+1}|\rightarrow0$.同理,当$n\rightarrow+\infty$, $|q_{n+1}-y_{n+1}|\rightarrow0$.

由于Julia集以及系统参数的有界性,存在$M_2 > 0$满足$|d_2||z^2_n|+|\xi_2||w^2_n| < M_3$, $|\gamma_1||p_nw_n|+|c_1||x_nz_n| < M_3$, $|\gamma_2||q_nw_n|+|c_2||y_nz_n| < M_3$.

### 图 6

(a)响应系统. (b) $m_1 = 1.7, \; m_2 = 1.7.\; h_1 = 0.7, \; h_2 = 0.7, \; h_3 = 0.7.$ (c) $m_1 = 1.95, \; m_2 = 1.95, \; h_1 = 0.95, \; h_2 = 0.95, \; h_3 = 0.95.$ (d)目标系统

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