一类强身型食饵-捕食者模型正平衡点稳定性注记 Note on the Global Asymptotic Stability of a Strengthening Type Predator-Prey Model with Stage Structure

doi:10.12677/AAM.2018.79132

Advances in Applied Mathematics
Vol. 07 No. 09 ( 2018 ), Article ID: 26706 , 6 pages
10.12677/AAM.2018.79132

Note on the Global Asymptotic Stability of a Strengthening Type Predator-Prey Model with Stage Structure

Xiaoyan Huang, Hang Deng, Fengde Chen

●How to Cite this Article

College of Mathematics and Computer Science, Fuzhou University, Fuzhou Fujian

Received: Aug. 16^th, 2018; accepted: Aug. 31^st, 2018; published: Sep. 6^th, 2018

ABSTRACT

A strengthening type predator-prey model with stage structure is revisited in this paper. We first show that the main results of the previous paper are incorrect. After that, by constructing some suitable Lyapunov functions, a set of sufficient conditions which ensure the globally asymptotically stable of the positive equilibrium is obtained. We show that the conditions which ensure the existence of the positive equilibrium are enough to ensure the globally asymptotically stable, and consequently, the system is permanent. Our results supplement and complement some known results.

Keywords:Stage Structure, Predator, Prey, Global Asymptotic Stability

一类强身型食饵-捕食者模型正平衡点稳定性注记

黄小燕，邓行，陈凤德

福州大学数学与计算机科学学院，福建福州

收稿日期：2018年8月16日；录用日期：2018年8月31日；发布日期：2018年9月6日

摘要

本文对一类强身型食饵-捕食者模型进行再探讨，首先指出前人的有关正平衡点全局稳定性的相关结果是错误的，之后通过构造适当的函数，我们证得了如果系统的正平衡点存在，则必是全局渐近稳定的，进而系统是一致持久的。本文所得结果推广和改进了前人的相关工作。

关键词 :阶段结构，捕食，食饵，全局渐近稳定

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

1. 引言

近年来，阶段结构种群模型的动力学行为研究引起了学者们的极大关注，见文 [1]- [18]以及所引文献。著名学者王稳地和他的学生于宇梅，张勇 [16]在构造阶段结构捕食者-食饵种群模型时，原创性的提出如下假设：捕食者种群通过捕食活动不增加生育能力，只是增加体质减少死亡，且幼年捕食者没有捕食能力，在这样的假设下，作者们提出了如下强身型的捕食者-食饵模型：

$\begin{array}{l} \dot{x} (t) = x (t) (r - b x (t) - a y_{2} (t)), \\ {\dot{y}}_{1} (t) = e y_{2} - (d_{1} + c) y_{1} (t), \\ {\dot{y}}_{2} (t) = e y_{1} (t) - d_{2} y_{2} (t) + k a x (t) y_{2} (t), \end{array}$ (1.1)

其中 $x (t)$ 为时刻t的食饵种群密度； $y_{1} (t)$ 为幼年捕食者在单位时间、单位面积内的生物量； $y_{2} (t)$ 为成年捕食者在单位时间、单位面积内的生物量，系统的各个参数的生态学含义见 [16]。

作变换 $\bar{x} = \frac{b x}{r}$ , ${\bar{y}}_{1} = \frac{a y_{1}}{e}$ , ${\bar{y}}_{2} = \frac{a y_{2}}{r}$ , $\bar{t} = r t$ ，并重新用 $x, y_{1}, y_{2}, t$ 表示 $\bar{x}, {\bar{y}}_{1}, {\bar{y}}_{2}, \bar{t}$ ，则系统(1.1)可以变换为

$\begin{array}{l} \dot{x} (t) = x (t) (1 - x (t) - y_{2} (t)), \\ {\dot{y}}_{1} (t) = y_{2} (t) - c_{1} y_{1} (t), \\ {\dot{y}}_{2} (t) = c_{2} y_{1} (t) - c_{3} y_{2} (t) + c_{4} x (t) y_{2} (t), \end{array}$ (1.2)

计算易知系统可能有三个非负平衡点， , $E_{0} (1, 0, 0)$ ，此外，进一步假设

$0 < c_{3} - \frac{c_{2}}{c_{1}} < c_{4}$ (1.3)

成立下，系统存在正平衡点 $(x^{*}, y_{1}^{*}, y_{2}^{*})$ ，其中

$x^{*} = \frac{c_{3} - \frac{c_{2}}{c_{1}}}{c_{4}}, y_{1}^{*} = \frac{y_{2}^{*}}{c_{1}}, y_{2}^{*} = 1 - x^{*} .$ (1.4)

有关系统(1.2)正平衡点的稳定性和系统的持久性等，作者得到了如下四个结果(见原文定理1，定理2，定理4和定理5)。

定理A

当(1.3)成立时，正平衡点 $E (x^{*}, y_{1}^{*}, y_{2}^{*})$ 是局部渐近稳定的。

定理B

若条件

(1.5)

成立，则系统(1.2)是一致持续生存的。

定理C

若条件

$c_{3} - \frac{c_{1}}{c_{2}} > c_{4}$ (1.6)

成立，则捕食者种群绝灭。

定理D

若条件

$c_{3} > c_{4} + \frac{c_{1}}{c_{2}} + c_{4} y_{1}^{*}$ (1.7)

成立，则是 $E (x^{*}, y_{1}^{*}, y_{2}^{*})$ 全局渐近稳定的。

这里有两个问题。1)条件(1.6)成立时，必有条件(1.5)成立，由定理B此时系统(1.2)是一致持续生存的，而由定理C此时捕食者种群是绝灭的，这是两个相互矛盾的结论，所以，定理B和C必然有一个是

错误的；2)条件(1.7)隐含 $c_{3} - \frac{c_{1}}{c_{2}} > c_{4}$ ，也就是说，在(1.7)成立下，(1.3)不可能成立，从而系统(1.2)不可能

有正平衡点，更不要说其稳定性了。既然在条件(1.3)成立下，系统(1.2)存在正平衡点，而文 [16]所给条件是错误的，那么，到底什么条件才能保证系统(1.2)正平衡点的全局稳定性呢?此外，保证系统一致持久的条件是什么呢?定理B和定理C到底哪个会成立?哪个不成立呢?

本文的目的在于给出这些问题的肯定回答。我们首先给出保证系统的正平衡点全局渐近稳定性的充分性条件并给予严格证明，其后探讨了系统的持久性问题。本文所得结果补充和完善了文 [16]的主要结果之一。

2. 主要结果

下面叙述本文的主要结果。

定理2.1. 当(1.3)成立时，系统(1.2)的正平衡点 $E (x^{*}, y_{1}^{*}, y_{2}^{*})$ 是全局渐近稳定的。

证明：我们将通过构造适当的函数来证明这一结论。

今构造函数

$V (t) = K_{1} (x - x^{*} - x^{*} \ln \frac{x}{x^{*}}) + K_{2} (y_{1} - y_{1}^{*} - y_{1}^{*} \ln \frac{y_{1}}{y_{1}^{*}}) + K_{3} (y_{2} - y_{2}^{*} - y_{2}^{*} \ln \frac{y_{2}}{y_{2}^{*}})$ (2.1)

其中 $K_{i}, i = 1, 2, 3$ 是待定的正常数。注意到 $x^{*}, y_{1}^{*}, y_{2}^{*}$ 满足方程

$\begin{array}{l} 1 - x^{*} - y_{2}^{*} = 0, \\ y_{2}^{*} - c_{1} y_{1}^{*} = 0, \\ c_{2} y_{1}^{*} - c_{3} y_{2}^{*} + c_{4} x^{*} y_{2}^{*} = 0. \end{array}$ (2.2)

沿着系统(1.2)的正解计算导数，借助(2.2)，有

$\begin{matrix} D^{+} V (t) = K_{1} \frac{x - x^{*}}{x} x (1 - x - y_{2}) + K_{2} \frac{y_{1} - y_{1}^{*}}{y_{1}} (y_{2} - c_{1} y_{1}) \\ + K_{3} \frac{y_{2} - y_{2}^{*}}{y_{2}} (c_{2} y_{1} - c_{3} y_{2} + c_{4} x y_{2}) \\ = K_{1} (x - x^{*}) (x^{*} + y_{2}^{*} - x - y_{2}) \\ + K_{2} \frac{y_{1} - y_{1}^{*}}{y_{1}} (\frac{1}{y_{1}^{*}} (y_{2} y_{1}^{*} - y_{1} y_{2}^{*}) + \frac{y_{1} y_{2}^{*}}{y_{1}^{*}} - c_{1} y_{1}) \\ + K_{3} \frac{y_{2} - y_{2}^{*}}{y_{2}} (\frac{c_{2}}{y_{2}^{*}} (y_{1} y_{2}^{*} - y_{1}^{*} y_{2}) + \frac{c_{2} y_{1}^{*} y_{2}}{y_{2}^{*}} - c_{3} y_{2} + c_{4} x y_{2}) \end{matrix}$

$\begin{array}{l} = - K_{1} {(x - x^{*})}^{2} + K_{1} (x - x^{*}) (y_{2}^{*} - y_{2}) \\ + K_{2} \frac{y_{1} - y_{1}^{*}}{y_{1}} (\frac{1}{y_{1}^{*}} (y_{2} y_{1}^{*} - y_{2} y_{1} + y_{2} y_{1} - y_{1} y_{2}^{*})) \\ + K_{3} \frac{y_{2} - y_{2}^{*}}{y_{2}} (\frac{c_{2}}{y_{2}^{*}} (y_{1} y_{2}^{*} - y_{1} y_{2} + y_{1} y_{2} - y_{1}^{*} y_{2}) - c_{4} x^{*} y_{2} + c_{4} x y_{2}) \end{array}$

$\begin{array}{l} = - K_{1} {(x - x^{*})}^{2} + K_{1} (x - x^{*}) (y_{2}^{*} - y_{2}) + K_{2} \frac{y_{1} - y_{1}^{*}}{y_{1}} \frac{y_{2}}{y_{1}^{*}} (y_{1}^{*} - y_{1}) \\ + K_{2} \frac{y_{1} - y_{1}^{*}}{y_{1}} \frac{y_{1}}{y_{1}^{*}} (y_{2} - y_{2}^{*}) + K_{3} \frac{y_{2} - y_{2}^{*}}{y_{2}} \frac{c_{2}}{y_{2}^{*}} y_{1} (y_{2}^{*} - y_{2}) \\ + K_{3} \frac{y_{2} - y_{2}^{*}}{y_{2}} \frac{c_{2}}{y_{2}^{*}} y_{2} (y_{1} - y_{1}^{*}) + K_{3} \frac{y_{2} - y_{2}^{*}}{y_{2}} y_{2} (- c_{4} x^{*} + c_{4} x) \end{array}$

$\begin{array}{l} = - K_{1} {(x - x^{*})}^{2} - K_{1} (x - x^{*}) (y_{2} - y_{2}^{*}) - \frac{K_{2}}{y_{1}^{*}} \frac{y_{2}}{y_{1}} {(y_{1} - y_{1}^{*})}^{2} \\ + \frac{K_{2}}{y_{1}^{*}} (y_{1} - y_{1}^{*}) (y_{2} - y_{2}^{*}) - \frac{K_{3} c_{2}}{y_{2}^{*}} \frac{y_{1}}{y_{2}} {(y_{2} - y_{2}^{*})}^{2} \\ + \frac{K_{3} c_{2}}{y_{2}^{*}} (y_{2} - y_{2}^{*}) (y_{1} - y_{1}^{*}) + K_{3} c_{4} (y_{2} - y_{2}^{*}) (x - x^{*}) \\ = - K_{1} {(x - x^{*})}^{2} + (K_{3} c_{4} - K_{1}) (x - x^{*}) (y_{2} - y_{2}^{*}) - \frac{K_{2}}{y_{1}^{*}} \frac{y_{2}}{y_{1}} {(y_{1} - y_{1}^{*})}^{2} \\ + (\frac{K_{2}}{y_{1}^{*}} + \frac{K_{3} c_{2}}{y_{2}^{*}}) (y_{1} - y_{1}^{*}) (y_{2} - y_{2}^{*}) - \frac{K_{3} c_{2}}{y_{2}^{*}} \frac{y_{1}}{y_{2}} {(y_{2} - y_{2}^{*})}^{2} \end{array}$ (2.3)

今取

$K_{1} = c_{4}, K_{2} = \frac{c_{2} y_{1}^{*}}{y_{2}^{*}}, K_{3} = 1,$ (2.4)

则有

(2.5)

因此，对所有的 $x, y_{1}, y_{2} > 0$ ,且 $(x, y_{1}, y_{2}) \neq (x^{*}, y_{1}^{*}, y_{2}^{*})$ ，均有 $D^{+} V (t) < 0$ 。且仅在 $E (x^{*}, y_{1}^{*}, y_{2}^{*})$ 处 $D^{+} V (t) = 0$ 。由此可知 $E (x^{*}, y_{1}^{*}, y_{2}^{*})$ 是全局渐近稳定的。

定理2.1证毕。

作为定理2.1的直接推论，我们立即得到如下有关系统(1.2)持久性的结论。

推论2.1. 当(1.3)成立时，系统(1.2)是一致持久的。

3. 结论

于宇梅等学者 [16]提出了强身型的捕食者-食饵模型(模型(1.1))，在作适当变换后，变成了系统(1.2)，作者探讨了系统的各个平衡点的存在性，正平衡点的局部稳定性和全局稳定性，系统的持久性和捕食者种群的绝灭性。我们注意到作者的主要结果：定理B和C是互相矛盾的；定理D的条件跟正平衡点存在的条件也是矛盾的，也就是说，文 [16]的结果是否成立都需要进一步验证。本文中，我们证明了正平衡点如果存在，则必是全局吸引的，从而必有系统是一致持久的，我们的这一结果非常简洁完美，我们的结果补充和完善了文 [16]的主要结果。

文章引用

黄小燕,邓行,陈凤德. 一类强身型食饵-捕食者模型正平衡点稳定性注记
Note on the Global Asymptotic Stability of a Strengthening Type Predator-Prey Model with Stage Structure[J]. 应用数学进展, 2018, 07(09): 1141-1146. https://doi.org/10.12677/AAM.2018.79132

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