ABSTRACT

In this paper, we study the periodic solution of a delayed Leslie-Gower and Holling-Type III predator-prey model incorporating prey cannibalism to understand the dynamic relationship between prey and predator. Using the continuity theorem of coincidence degree theory and comparison theorem, if the condition $\overline{{\gamma }_1+c_1}>\overline{{\alpha }_1}{e}^{{\beta }_2}+\overline{\left(\frac{f}{d^2}\right)}{e}^{{\beta }_1}$ is met, then the system has a $\omega$ -positive periodic solution.

Keywords:Delayed Predator-Prey Model, Cannibalism, Continuity Theorem, Periodic Solution

具有Holling-III型食饵自食性的Leslie-Gower时滞捕食模型的周期解

牛璐^*，王晓云^#

太原理工大学数学学院，山西 太原

收稿日期：2020年7月18日；录用日期：2020年8月4日；发布日期：2020年8月11日

摘 要

本文研究具有Holling-III型食饵自食性的Leslie-Gower时滞捕食模型的周期解，了解食饵与捕食者之间的动态关系。利用重合度理论的连续性定理以及比较定理，如果满足条件 $\overline{{\gamma }_1+c_1}>\overline{{\alpha }_1}{e}^{{\beta }_2}+\overline{\left(\frac{f}{d^2}\right)}{e}^{{\beta }_1}$，则系统存在一个 $\omega$ -周期正解。

关键词 :时滞捕食模型，同类相食，连续性定理，周期解

Copyright © 2020 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

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

1. 引言

生态系统中存在多种捕食行为，捕食关系是生物数学界研究的一个重要课题。近几十年来，传统捕食模型由于其非常丰富的动态和在可再生资源管理中的重要作用受到了广泛的关注和研究 [1] - [6]。然而，同类相食(种内捕食)的行为在很多种群也会发生，使得简单的食物网变得更为复杂。同类相食是指同一类型的动物或者植物为了生存繁殖需要或者某种目的互相厮杀竞争的现象，也是一种种群内部的食饵物种–捕食物种之间的相互作用。它是一种消耗相同物种并有助于提供食物来源的行为，这种行为特征已经在各种各样的动物中被发现，包括鱼类，蜘蛛，浮游生物和昆虫等种群，因此研究具有食饵自食性的种群模型动力学行为尤为重要 [7]。近年来，具有Holling-II型的Leslie-Gower捕食模型已经得到了广泛的研究 [8] [9] [10]，因为Holling-III型可以清楚地描述食饵物种与捕食者之间的关系，且时滞微分方程比常微分方程更能展现出生物界复杂的动态关系，研究具有时滞的周期系统的周期解的存在性是有十分重要的意义的。所以本文考虑具有Holling-III型食饵自食性的Leslie-Gower时滞捕食模型：

$\{\begin{array}{l}{H}^{\prime }\left(t\right)=H\left(t\right)\left[{\gamma }_1\left(t\right)+c_1\left(t\right)-{\alpha }_1\left(t\right)P\left(t-{\tau }_1\left(t\right)\right)-b_1\left(t\right)H\left(t\right)\right]-\frac{f\left(t\right)H\left(t\right)H\left(t-\sigma \left(t\right)\right)}{d^2\left(t\right)+H^2\left(t-\sigma \left(t\right)\right)},\\ P^{\prime }\left(t\right)=P\left(t\right)\left[{\gamma }_2\left(t\right)-{\alpha }_2\left(t\right)\frac{P\left(t-{\tau }_2\left(t\right)\right)}{H\left(t-{\tau }_2\left(t\right)\right)}\right],\end{array}$ (1)

其中 $H\left(t\right)$，$P\left(t\right)$ 分别表示在时间t时食饵和捕食者的密度， ${\gamma }_1\left(t\right)$，$i=1,2$ 分别是食饵和捕食者的内在

增长率， $\frac{{\gamma }_1\left(t\right)}{b_1\left(t\right)}$ 是食饵的环境承载力， $f\left(t\right)$ 表示食饵的同类相食率， $C\left(H\right)\left(t\right)=f\left(t\right)\times H\left(t\right)\times \frac{H\left(t\right)}{H^2\left(t\right)+d^2(\; t\; )}$

是同类相食行为， $c_1\left(t\right)\times H\left(t\right)$ 是由于自相残杀而产生的新后代。因为食饵会掠夺许多食饵以产生一个新后代，所以 $c_1\left(t\right)<f\left(t\right)$。 ${\alpha }_1\left(t\right)$，${\alpha }_2\left(t\right)$，$b_1\left(t\right)$，$d\left(t\right)$，$\sigma \left(t\right)$，${\tau }_i\left(t\right)$，$i=1,2$ 均为严格正的 $\omega$ -周期连

续函数 $\left(\omega >0\right)$，${\gamma }_1\left(t\right)$，${\gamma }_2\left(t\right)$，$c_1\left(t\right)$ 是 $\omega$ -周期连续函数，假设 ${\displaystyle {\int }_0^{\omega }{c}_1\left(t\right)\text{d}t}>0$，${\displaystyle {\int }_0^{\omega }{\gamma }_i\left(t\right)\text{d}t}>0$，$i=1,2$。

2. 预备知识

设 $X$，$Z$ 为实Banach空间， $L$ ： $\text{Dom}L\subset X\to Z$ 为线性映射， $N$ ： $X\to Z$ 为连续映射，且记 $K_P$ 为 $L_P$ 的倒数。如果满足条件 $\dim \text{Ker}L=\text{codimIm}L<+\infty$，$\mathrm{Im}L$ 在 $Z$ 中为闭，则称 $L$ 为指标为零的Fredholm映射。如果映射 $L$ 是指标为零的Fredholm映射，则存在连续的投影 $P$ ： $X\to X$ 、 $Q$ ： $Z\to Z$ 有 $\text{Ker}Q=\mathrm{Im}L=\text{Im}\left(I-Q\right)$，$\mathrm{Im}P=\text{Ker}L$。可见 $L$ 对 $\text{Dom}L\cap \text{Ker}P$ 的限制性 $L_P$ ： $\left(I-P\right)X\to \mathrm{Im}L$ 是可逆的。如果 $\text{Ω}$ 是 $X$ 的一个开放边界子集， $QN\left(\text{<mover accent="true"> <mo>Ω <mo> ¯ </mo> </mo></mover>}\right)$ 有界， $K_P\left(I-Q\right)N$ ： $\text{<mover accent="true"> <mo>Ω <mo> ¯ </mo> </mo></mover>}\to X$ 紧，那么可知 $Z$ 在 $\text{<mover accent="true"> <mo>Ω <mo> ¯ </mo> </mo></mover>}$ 上是L-紧。根据 $\mathrm{Im}Q$ 与 $\text{Ker}L$ 是同构性，可知存在同构映射 $J$ ： $\mathrm{Im}L\to \text{Ker}L$。

引理2.1 [11] (连续性定理) 若 $\text{Ω}\subset X$ 是一个有界开集， $L$ 是指标为零的Fredholm映射，且 $N$ 在 $\text{<mover accent="true"> <mo>Ω <mo> ¯ </mo> </mo></mover>}$ 上是L-紧的，假设

(i) 对于任意的 $\lambda \in \left(0,1\right)$，$x\in \partial \text{Ω}\cap \text{Dom}L$，$Lx\ne \lambda Nx$ ；

(ii) 对于任意的 $x\in \partial \text{Ω}\cap \text{Ker}L$，$QNx\ne 0$ ；

(iii) $\text{deg}\{JQN,\text{Ω}\cap \text{Ker}L,0\}\ne 0$。

那么 $Lx=Nx$ 在 $\text{<mover accent="true"> <mo>Ω <mo> ¯ </mo> </mo></mover>}\cap \text{Dom}L$ 内至少有一个解。

引理2.2 [12] 假设 $y\in P{C}_{\omega }^1=\{x:x\in C^1\left(R,R\right),x\left(t+\omega \right)\equiv x\left(t\right)\}$，则

$0\le \underset{s\in \left[0,\omega \right]}{\max }y\left(s\right)-\underset{s\in \left[0,\omega \right]}{\min }y\left(s\right)\le \frac{1}{2}{\displaystyle {\int }_0^{\omega }\left|y^{\prime }\left(s\right)\right|\text{d}s}.$

3. 周期解的存在性

方便起见，记 $\bar{f}=\frac{1}{\omega }{\displaystyle {\int }_0^{\omega }f\left(t\right)\text{d}t}$，其中 $f\left(t\right)$ 是 $\omega$ -周期连续函数。

定理3.1 如果条件 $\overline{{\gamma }_1+c_1}>\overline{{\alpha }_1}{\text{e}}^{{\beta }_2}+\overline{\left(\frac{f}{d^2}\right)}{\text{e}}^{{\beta }_1}$ 成立，那么系统(1)至少存在一个 $\omega$ -周期正解。

证明 令 $H\left(t\right)={\text{e}}^{x_1\left(t\right)}$，$P\left(t\right)={\text{e}}^{x_2\left(t\right)}$，则系统(1)变为

$\{\begin{array}{l}{x^{\prime }}_1\left(t\right)={\gamma }_1\left(t\right)+c_1\left(t\right)-{\alpha }_1\left(t\right){\text{e}}^{x_2\left(t-{\tau }_1\left(t\right)\right)}-b_1\left(t\right){\text{e}}^{x_1\left(t\right)}-\frac{f\left(t\right){\text{e}}^{x_1\left(t-\sigma \left(t\right)\right)}}{d^2\left(t\right)+{\text{e}}^{2x_1\left(t-\sigma \left(t\right)\right)}},\\ {x^{\prime }}_2\left(t\right)={\gamma }_2\left(t\right)-{\alpha }_2\left(t\right)\frac{{\text{e}}^{x_2\left(t-{\tau }_2\left(t\right)\right)}}{{\text{e}}^{x_1\left(t-{\tau }_2\left(t\right)\right)}},\end{array}$ (2)

从上可知系统(2)具有一个以 $\omega$ 为周期的解 ${\left(x_1^{\ast }\left(t\right),x_2^{\ast }\left(t\right)\right)}^{\text{T}}$，则 ${\left(H^{\ast }\left(t\right),P^{\ast }\left(t\right)\right)}^{\text{T}}={\left({\text{e}}^{x_1^{\ast }\left(t\right)},{\text{e}}^{x_2^{\ast }\left(t\right)}\right)}^{\text{T}}$

是系统(1)的一个以 $\omega$ 为周期的周期正解。所以，在这里只要证明系统(2)至少有一个 $\omega$ -周期解即可。

令 $X=Z=\left\{x\left(t\right)={\left(x_1\left(t\right),x_2\left(t\right)\right)}^{\text{T}}\in C\left(R,R^2\right):x\left(t+\omega \right)=x\left(t\right)\right\}$，且 $\Vert x\Vert =\underset{s\in \left[0,\omega \right]}{\max }\left\{\left|x_1\left(t\right)+x_2\left(t\right)\right|\right\}$，那么在 $\Vert \cdot \Vert$ 范数下， $X$ 和 $Z$ 为Banach空间。

定义映射 $L$，$P$，$Q$ 分别为

$L:\text{Dom}L\cap X\to Z,Lx=x^{\prime };$

$P\left(x\right)=\frac{1}{\omega }{\displaystyle {\int }_0^{\omega }x\left(t\right)\text{d}t},x\in X;$

$Q\left(x\right)=\frac{1}{\omega }{\displaystyle {\int }_0^{\omega }z\left(t\right)\text{d}t},z\in Z.$

其中 $DomL=\left\{x\in X:x\left(t\right)\in c^1\left(R,R^2\right)\right\}$。

定义变换 $N:X\to Z$

$Nx=\left[\begin{array}{c}{\gamma }_1\left(t\right)+c_1\left(t\right)-{\alpha }_1\left(t\right){\text{e}}^{x_2\left(t-{\tau }_1\left(t\right)\right)}-b_1\left(t\right){\text{e}}^{x_1\left(t\right)}-\frac{f\left(t\right){\text{e}}^{x_1\left(t-\sigma \left(t\right)\right)}}{d^2\left(t\right)+{\text{e}}^{2x_1\left(t-\sigma \left(t\right)\right)}}\\ {\gamma }_2\left(t\right)-{\alpha }_2\left(t\right)\frac{{\text{e}}^{x_2\left(t-{\tau }_2\left(t\right)\right)}}{{\text{e}}^{x_1\left(t-{\tau }_2\left(t\right)\right)}}\end{array}\right],$

显然， $\text{Ker}L=R^2$，$\mathrm{Im}L=\left\{z|z\in Z,{\displaystyle {\int }_0^{\omega }z\left(t\right)\text{d}t=0}\right\}$ 为 $Z$ 中的闭集， $\text{Ker}L=\text{codimIm}I=2$，$P$，$Q$ 是满足 $\mathrm{Im}P=\text{Ker}L$，$\text{Ker}Q=\mathrm{Im}L=\mathrm{Im}\left(I-Q\right)$ 的连续映射，故L是指标为零的Fredholm算子，而且 $K_P$ 为 $K_P\left(z\right)={\displaystyle {\int }_0^{t}z\left(s\right)\text{d}s}-\frac{1}{\omega }{\displaystyle {\int }_0^{\omega }{\displaystyle {\int }_0^{t}z\left(s\right)\text{d}s\text{d}t}}$，于是

$QNx=\left[\begin{array}{c}\frac{1}{\omega }{\displaystyle {\int }_0^{\omega }\left[{\gamma }_1\left(t\right)+c_1\left(t\right)-{\alpha }_1\left(t\right){\text{e}}^{x_2\left(t-{\tau }_1\left(t\right)\right)}-b_1\left(t\right){\text{e}}^{x_1\left(t\right)}-\frac{f\left(t\right){\text{e}}^{x_1\left(t-\sigma \left(t\right)\right)}}{d^2\left(t\right)+{\text{e}}^{2x_1\left(t-\sigma \left(t\right)\right)}}\right]}\\ \frac{1}{\omega }{\displaystyle {\int }_0^{\omega }\left[{\gamma }_2\left(t\right)-{\alpha }_2\left(t\right)\frac{{\text{e}}^{x_2\left(t-{\tau }_2\left(t\right)\right)}}{{\text{e}}^{x_1\left(t-{\tau }_2\left(t\right)\right)}}\right]}\end{array}\right],$

$\begin{array}{c}{K}_P\left(I-Q\right)Nx=\left(\begin{array}{c}{\displaystyle {\int }_0^t\left[{\gamma }_1\left(t\right)+c_1\left(t\right)-{\alpha }_1\left(t\right){\text{e}}^{x_2\left(t-{\tau }_1\left(t\right)\right)}-b_1\left(t\right){\text{e}}^{x_1\left(t\right)}-\frac{f\left(t\right){\text{e}}^{x_1\left(t-\sigma \left(t\right)\right)}}{d^2\left(t\right)+{\text{e}}^{2x_1\left(t-\sigma \left(t\right)\right)}}\right]}\\ {\displaystyle {\int }_0^t\left[{\gamma }_2\left(t\right)-{\alpha }_2\left(t\right)\frac{{\text{e}}^{x_2\left(t-{\tau }_2\left(t\right)\right)}}{{\text{e}}^{x_1\left(t-{\tau }_2\left(t\right)\right)}}\right]}\end{array}\right)\\ -\left(\begin{array}{c}\frac{1}{\omega }{\displaystyle {\int }_0^{\omega }{\displaystyle {\int }_0^t\left[{\gamma }_1\left(t\right)+c_1\left(t\right)-{\alpha }_1\left(t\right){\text{e}}^{x_2\left(t-{\tau }_1\left(t\right)\right)}-b_1\left(t\right){\text{e}}^{x_1\left(t\right)}-\frac{f\left(t\right){\text{e}}^{x_1\left(t-\sigma \left(t\right)\right)}}{d^2\left(t\right)+{\text{e}}^{2x_1\left(t-\sigma \left(t\right)\right)}}\right]}}\\ \frac{1}{\omega }{\displaystyle {\int }_0^{\omega }{\displaystyle {\int }_0^t\left[{\gamma }_2\left(t\right)-{\alpha }_2\left(t\right)\frac{{\text{e}}^{x_2\left(t-{\tau }_2\left(t\right)\right)}}{{\text{e}}^{x_1\left(t-{\tau }_2\left(t\right)\right)}}\right]}}\end{array}\right)\\ -\left(\begin{array}{c}\left(\frac{t}{\omega }-\frac{1}{2}\right){\displaystyle {\int }_0^{\omega }\left[{\gamma }_1\left(t\right)+c_1\left(t\right)-{\alpha }_1\left(t\right){\text{e}}^{x_2\left(t-{\tau }_1\left(t\right)\right)}-b_1\left(t\right){\text{e}}^{x_1\left(t\right)}-\frac{f\left(t\right){\text{e}}^{x_1\left(t-\sigma \left(t\right)\right)}}{d^2\left(t\right)+{\text{e}}^{2x_1\left(t-\sigma \left(t\right)\right)}}\right]}\\ \left(\frac{t}{\omega }-\frac{1}{2}\right){\displaystyle {\int }_0^{\omega }\left[{\gamma }_2\left(t\right)-{\alpha }_2\left(t\right)\frac{{\text{e}}^{x_2\left(t-{\tau }_2\left(t\right)\right)}}{{\text{e}}^{x_1\left(t-{\tau }_2\left(t\right)\right)}}\right]}\end{array}\right).\end{array}$

易证， $QN$ 和 $K_P\left(I-Q\right)N$ 都是连续的。对于任意的有界开集 $\text{Ω}\subset X$，$QN\left(\text{<mover accent="true"> <mo>Ω <mo> ¯ </mo> </mo></mover>}\right)$，$K_P\left(I-Q\right)N\left(\text{<mover accent="true"> <mo>Ω <mo> ¯ </mo> </mo></mover>}\right)$ 紧，所以有 $N$ 在 $\text{<mover accent="true"> <mo>Ω <mo> ¯ </mo> </mo></mover>}$ 上是L-紧的。

对应方程 $Lx=\lambda Nx$，$\lambda \in \left(0,1\right)$，有

$\{\begin{array}{l}{x^{\prime }}_1\left(t\right)=\lambda \left[{\gamma }_1\left(t\right)+c_1\left(t\right)-{\alpha }_1\left(t\right){\text{e}}^{x_2\left(t-{\tau }_1\left(t\right)\right)}-b_1\left(t\right){\text{e}}^{x_1\left(t\right)}-\frac{f\left(t\right){\text{e}}^{x_1\left(t-\sigma \left(t\right)\right)}}{d^2\left(t\right)+{\text{e}}^{2x_1\left(t-\sigma \left(t\right)\right)}}\right],\\ {x^{\prime }}_2\left(t\right)=\lambda \left[{\gamma }_2\left(t\right)-{\alpha }_2\left(t\right)\frac{{\text{e}}^{x_2\left(t-{\tau }_2\left(t\right)\right)}}{{\text{e}}^{x_1\left(t-{\tau }_2\left(t\right)\right)}}\right],\end{array}$ (3)

将(3)式从 $\left[0,\omega \right]$ 积分有

$\{\begin{array}{l}{\displaystyle {\int }_0^{\omega }\left[{\gamma }_1\left(t\right)+c_1\left(t\right)-{\alpha }_1\left(t\right){\text{e}}^{x_2\left(t-{\tau }_1\left(t\right)\right)}-b_1\left(t\right){\text{e}}^{x_1\left(t\right)}-\frac{f\left(t\right){\text{e}}^{x_1\left(t-\sigma \left(t\right)\right)}}{d^2\left(t\right)+{\text{e}}^{2x_1\left(t-\sigma \left(t\right)\right)}}\right]\text{d}t}=0,\\ {\displaystyle {\int }_0^{\omega }\left[{\gamma }_2\left(t\right)-{\alpha }_2\left(t\right)\frac{{\text{e}}^{x_2\left(t-{\tau }_2\left(t\right)\right)}}{{\text{e}}^{x_1\left(t-{\tau }_2\left(t\right)\right)}}\right]\text{d}t}=0,\end{array}$ (4)

由(4)得

${\displaystyle {\int }_0^{\omega }\left[{\alpha }_1\left(t\right){\text{e}}^{x_2\left(t-{\tau }_1\left(t\right)\right)}+b_1\left(t\right){\text{e}}^{x_1\left(t\right)}+\frac{f\left(t\right){\text{e}}^{x_1\left(t-\sigma \left(t\right)\right)}}{d^2\left(t\right)+{\text{e}}^{2x_1\left(t-\sigma \left(t\right)\right)}}\right]\text{d}t}=\overline{{\gamma }_1+c_1}\omega ,$ (5)

${\displaystyle {\int }_0^{\omega }\left[{\alpha }_2\left(t\right)\frac{{\text{e}}^{x_2\left(t-{\tau }_2\left(t\right)\right)}}{{\text{e}}^{x_1\left(t-{\tau }_2\left(t\right)\right)}}\right]\text{d}t}=\overline{{\gamma }_2}\omega ,$ (6)

(3)结合(5)有

$\begin{array}{c}{\displaystyle {\int }_0^{\omega }\left|x_1\left(t\right)\right|\text{d}t}=\lambda {\displaystyle {\int }_0^{\omega }\left|{\gamma }_1\left(t\right)+c_1\left(t\right)-{\alpha }_1\left(t\right){\text{e}}^{x_2\left(t-{\tau }_1\left(t\right)\right)}-b_1\left(t\right){\text{e}}^{x_1\left(t\right)}-\frac{f\left(t\right){\text{e}}^{x_1\left(t-\sigma \left(t\right)\right)}}{d^2\left(t\right)+{\text{e}}^{2x_1\left(t-\sigma \left(t\right)\right)}}\right|\text{d}t}\\ \le \left(\overline{\left|{\gamma }_{１}+c_1\right|}+\overline{{\gamma }_{１}+c_1}\right)\omega .\end{array}$ (7)

存在 ${\xi }_i,{\eta }_i,i=1,2$ 满足

$x_i\left({\xi }_i\right)=\underset{s\in \left[0,\omega \right]}{\text{sup}}{x}_i\left(t\right),x_i\left({\eta }_i\right)=\underset{s\in \left[0,\omega \right]}{\inf }{x}_i\left(t\right),$ (8)

由(5)和(8)得

$\overline{{\gamma }_1+c_1}\omega \ge {\displaystyle {\int }_0^{\omega }{b}_1\left(t\right){\text{e}}^{x_1\left(t\right)}}\text{d}t\ge \omega \overline{b_1}{\text{e}}^{x_1\left({\eta }_1\right)},$ (9)

化简有

$x_1\left({\eta }_1\right)\le \ln \frac{\overline{{\gamma }_1+c_1}}{\overline{b_1}}.$ (10)

结合(7)，(10)和引理2.2得

$x_1\left(t\right)\le x_1\left({\eta }_1\right)+\frac{1}{2}{\displaystyle {\int }_0^{\omega }\left|x_1\left(t\right)\right|\text{d}t}\le \ln \frac{\overline{{\gamma }_1+c_1}}{\overline{b_1}}+\frac{1}{2}\left(\overline{\left|{\gamma }_{１}+c_1\right|}+\overline{{\gamma }_{１}+c_1}\right)\omega =:{\beta }_1.\text{}$ (11)

根据(3)和(6)有

${\displaystyle {\int }_0^{\omega }\left|x_2\left(t\right)\right|\text{d}t}=\lambda {\displaystyle {\int }_0^{\omega }\left|{\gamma }_2\left(t\right)-{\alpha }_2\left(t\right)\frac{{\text{e}}^{x_2\left(t-{\tau }_2\left(t\right)\right)}}{{\text{e}}^{x_1\left(t-{\tau }_2\left(t\right)\right)}}\right|\text{d}t}\le \left(\overline{\left|{\gamma }_2\right|}+\overline{{\gamma }_2}\right)\omega ,$ (12)

由(6)，(8)和(11)得

$\frac{\omega \overline{{\alpha }_2}{\text{e}}^{x_2\left({\eta }_2\right)}}{{\text{e}}^{{\beta }_1}}\le {\displaystyle {\int }_0^{\omega }\left[{\alpha }_2\left(t\right)\frac{{\text{e}}^{x_2\left(t-{\tau }_2\left(t\right)\right)}}{{\text{e}}^{x_1\left(t-{\tau }_2\left(t\right)\right)}}\right]\text{d}t}=\overline{{\gamma }_2}\omega ,$ (13)

化简得

$x_2\left({\eta }_2\right)\le \ln \frac{\overline{{\gamma }_2}{\text{e}}^{{\beta }_1}}{\overline{{\alpha }_2}}=\ln \frac{\overline{{\gamma }_2}}{\overline{{\alpha }_2}}+{\beta }_1,$ (14)

结合(12)，(14)和引理2.2得

$x_2\left(t\right)\le x_2\left({\eta }_2\right)+\frac{1}{2}{\displaystyle {\int }_0^{\omega }\left|x_2\left(t\right)\right|\text{d}t}\le \ln \frac{\overline{{\gamma }_2}}{\overline{{\alpha }_2}}+{\beta }_1+\frac{1}{2}\left(\overline{\left|{\gamma }_2\right|}+\overline{{\gamma }_2}\right)\omega =:{\beta }_2.$ (15)

由(4)和(8)得

$\begin{array}{c}\omega \overline{b_1}{\text{e}}^{x_1\left({\xi }_1\right)}\ge \overline{{\gamma }_1+c_1}\omega -{\displaystyle {\int }_0^{\omega }{\alpha }_1\left(t\right){\text{e}}^{x_2\left(t-{\tau }_1\left(t\right)\right)}}\text{d}t-{\displaystyle {\int }_0^{\omega }\frac{f\left(t\right){\text{e}}^{x_1\left(t-\sigma \left(t\right)\right)}}{d^2\left(t\right)+{\text{e}}^{2x_1\left(t-\sigma \left(t\right)\right)}}\text{d}t}\\ \ge \overline{{\gamma }_1+c_1}\omega -\overline{{\alpha }_1}\omega {\text{e}}^{{\beta }_2}-\overline{\left(\frac{f}{d^2}\right)}\omega {\text{e}}^{{\beta }_{１}},\end{array}$ (16)

即

$x_{１}\left({\xi }_{１}\right)\ge \ln \frac{\overline{{\gamma }_1+c_1}-\overline{{\alpha }_1}{\text{e}}^{{\beta }_2}-\overline{\left(\frac{f}{d^2}\right)}{\text{e}}^{{\beta }_1}}{\overline{b_1}}.$ (17)

由(7)，(17)和引理2.2得

$\begin{array}{c}{x}_1\left(t\right)\ge x_1\left({\xi }_1\right)-\frac{1}{2}{\displaystyle {\int }_0^{\omega }\left|x_1\left(t\right)\right|\text{d}t}\\ \ge \ln \frac{\overline{{\gamma }_1+c_1}-\overline{{\alpha }_1}{\text{e}}^{{\beta }_2}-\overline{\left(\frac{f}{d^2}\right)}{\text{e}}^{{\beta }_1}}{\overline{b_1}}-\frac{1}{2}\left(\overline{\left|{\gamma }_{１}+c_1\right|}+\overline{{\gamma }_{１}+c_1}\right)\omega =:{\beta }_3.\end{array}$ (18)

结合(6)，(8)和(18)得

$\frac{\overline{{\alpha }_2}\omega {\text{e}}^{x_2\left({\xi }_2\right)}}{{\text{e}}^{{\beta }_3}}\ge {\displaystyle {\int }_0^{\omega }\left[{\alpha }_2\left(t\right)\frac{{\text{e}}^{x_2\left(t-{\tau }_2\left(t\right)\right)}}{{\text{e}}^{x_1\left(t-{\tau }_2\left(t\right)\right)}}\right]\text{d}t}=\overline{{\gamma }_2}\omega ,$ (19)

化简有

$x_2\left({\xi }_2\right)\ge \ln \frac{\overline{{\gamma }_2}{\text{e}}^{{\beta }_3}}{\overline{{\alpha }_2}}=\ln \frac{\overline{{\gamma }_2}}{\overline{{\alpha }_2}}+{\beta }_3.$ (20)

根据(12)，(20)和引理2.2得

$\begin{array}{c}{x}_2\left(t\right)\ge x_2\left({\xi }_2\right)-\frac{1}{2}{\displaystyle {\int }_0^{\omega }\left|x_2\left(t\right)\right|\text{d}t}\\ \ge \ln \frac{\overline{{\gamma }_2}}{\overline{{\alpha }_2}}+{\beta }_3-\frac{1}{2}\left(\overline{\left|{\gamma }_2\right|}+\overline{{\gamma }_2}\right)\omega =:{\beta }_4.\end{array}$ (21)

结合(11)，(15)，(18)，(21)有 $\Vert x\Vert \le \left|{\beta }_1\right|+\left|{\beta }_2\right|+\left|{\beta }_3\right|+\left|{\beta }_4\right|=:{\beta }_0$ ( ${\beta }_0$ 与 $\lambda$ 无关)。

考虑下面方程组

$\{\begin{array}{l}\overline{{\gamma }_1+c_1}-\overline{{\alpha }_1}{\text{e}}^{x_2}-\overline{b_1}{\text{e}}^{x_1}-\frac{1}{\omega }{\displaystyle {\int }_0^{\omega }\frac{f\left(t\right){\text{e}}^{x_1}}{d^2\left(t\right)+{\text{e}}^{2x_1}}\text{d}t=0,}\\ \overline{{\gamma }_2}-\frac{1}{\omega }{\displaystyle {\int }_0^{\omega }{\alpha }_2\frac{{\text{e}}^{x_2}}{{\text{e}}^{x_1}}\text{d}t=0,}\end{array}$ (22)

若(22)有解 $x^{\ast }={\left(x_1^{\ast },x_2^{\ast }\right)}^{\text{T}}$，则满足 $\Vert {\left(x_1^{\ast },x_2^{\ast }\right)}^{\text{T}}\Vert =\max \left\{\left|x_1^{\ast }\right|+\left|x_2^{\ast }\right|\right\}<{\beta }_0$。当 $x\in \partial \text{Ω}$，且 $\lambda \in \left(0,1\right)$，满足引理2.1 (i)。假设当 $x\in \partial \text{Ω}\cap \text{Ker}L$ 且 $\Vert x\Vert ={\beta }_0$，$QNx\ne 0$，满足引理2.1 (ii)。定义 $J:\mathrm{Im}Q\to \ker L$，${\left(x_1,x_2\right)}^{\text{T}}={\left(x_1,x_2\right)}^{\text{T}}$，可直接算得 $\text{deg}\{JQN,\text{Ω}\cap \text{Ker}L,0\}\ne 0$，满足引理2.1 (iii)。因此，(2)至少一个 $\omega$ -周期解。所以，系统(1)至少有一个 $\omega$ -周期正解。即可证定理3.1。

基金项目

国家自然科学基金青年项目(No. 11801398)。

文章引用

牛 璐,王晓云. 具有Holling-III型食饵自食性的Leslie-Gower时滞捕食模型的周期解
Periodic Solution for a Delayed Leslie-Gower and Holling-Type III Predator-Prey Model Incorporating Prey Cannibalism[J]. 应用数学进展, 2020, 09(08): 1170-1176. https://doi.org/10.12677/AAM.2020.98136

参考文献