数学物理学报 ›› 2021, Vol. 41 ›› Issue (6): 1980-1992.
• 论文 • 上一篇
恐惧效应对带时滞的反应扩散捕食系统的稳定区间的影响
孙悦(
),张道祥*(),周文
- 安徽师范大学数学与统计学院 安徽芜湖 241002
-
收稿日期:2020-11-30出版日期:2021-12-26发布日期:2021-12-02 -
通讯作者:张道祥 E-mail:1921011863@ahnu.edu.cn;18955302433@163.com -
作者简介:孙悦, E-mail:1921011863@ahnu.edu.cn -
基金资助:国家自然科学基金(11671013);国家自然科学基金(11302002);安徽省自然科学基金(2008085MA13)
The Influence of Fear Effect on Stability Interval of Reaction-Diffusion Predator-Prey System with Time Delay
Yue Sun(),Daoxiang Zhang*(),Wen Zhou
- School of Mathematics and Statistics, Anhui Normal University, Anhui Wuhu 241002
-
Received:2020-11-30Online:2021-12-26Published:2021-12-02 -
Contact:Daoxiang Zhang E-mail:1921011863@ahnu.edu.cn;18955302433@163.com -
Supported by:the NSFC(11671013);the NSFC(11302002);the NSF of Anhui Province(2008085MA13)
摘要：
该文结合理论推导和数值模拟两个方面研究了带有恐惧效应和时滞效应的反应扩散捕食-食饵模型的动力学.首先研究了系统的正平衡点的存在性和稳定性.其次，通过线性稳定性分析研究了系统的Hopf分支问题，结果表明恐惧效应影响Hopf分支点，继而影响着系统的稳定区间.最后，通过数值模拟验证了理论结果，并发现恐惧效应与稳定区间的非线性关系，即随着恐惧效应的持续增加，系统将会由稳定状态变为不稳定状态，再变为稳定状态.
中图分类号:
- O175.2
引用本文
孙悦,张道祥,周文. 恐惧效应对带时滞的反应扩散捕食系统的稳定区间的影响[J]. 数学物理学报, 2021, 41(6): 1980-1992.
Yue Sun,Daoxiang Zhang,Wen Zhou. The Influence of Fear Effect on Stability Interval of Reaction-Diffusion Predator-Prey System with Time Delay[J]. Acta mathematica scientia,Series A, 2021, 41(6): 1980-1992.
图 1
系统(1.3)的参数: $k_{0} = 3, t = 3000, r = 6, d = 0.2, a = 1, $ $ b = 5, q = 2, c = 2.5, m_{1} = 0.4, m_{2} = 1, d_{1} = 0.01, d_{2} = 0.1. $ (a) $\tau = 1.4$, 食饵的密度演化图; (b) $\tau = 1.4$, 捕食者的密度演化图; (c) $\tau = 2$, 食饵的密度演化图; (d) $\tau = 2$, 捕食者的密度演化图"
图 2
系统(1.3)的参数: $k_{0} = 3, t = 20000, r = 6, d = 0.2, a = 1, b = 5, $ $ q = 2, c = 2.5, m_{1} = 0.4, m_{2} = 1, d_{1} = 0.01, d_{2} = 0.1. $ (a) $\tau = 1.4$, (b) $\tau = 2$. 相图"
图 3
系统(1.3)的参数: $k_{0} = 3, \tau = 1.4, r = 6, d = 0.2, a = 1, $ $ b = 5, q = 2, c = 2.5, m_{1} = 0.4, m_{2} = 1, d_{1} = 0.01, d_{2} = 0.1. $ (a) $t = 200$, (b) $t = 400$, (c) $t = 600$, (d) $t = 20000. $ 捕食者的空间斑图"
图 4
系统(1.3)的参数: $k_{0} = 3, \tau = 2, r = 6, d = 0.2, a = 1, $ $ b = 5, q = 2, c = 2.5, m_{1} = 0.4, m_{2} = 1, d_{1} = 0.01, d_{2} = 0.1. $ (a) $t = 400$, (b) $t = 2000$, (c) $t = 4000$, (d) $t = 20000. $ 捕食者的空间斑图"
表 1
系统(1.3)的参数: $r = 6, d = 0.2, a = 1, b = 5, q = 2, c = 2.5, m_{1} = 0.4, $ $ m_{2} = 1, d_{1} = 0.01, d_{2} = 0.1. $恐惧效应与Hopf分支和平衡点的关系表"
| 恐惧交应 | Hopf分支临界值 | 平衡点 |
| 2 | 6.0441 | (1.5819, 0.6418) |
| 3 | 1.9491 | (1.2469, 0.5458) |
| 6 | 1.2727 | (0.9279, 0.3908) |
| 8 | 1.2269 | (0.8453, 0.3354) |
| 9.7 | 1.2199 | (0.7997, 0.3015) |
| 10 | 1.2201 | (0.7931, 0.2964) |
| 10.3 | 1.2205 | (0.7868, 0.2915) |
| 11 | 1.2222 | (0.7733, 0.2808) |
| 13 | 1.2308 | (0.7416, 0.2548) |
| 20 | 1.2732 | (0.6748, 0.1958) |
| 30 | 1.3265 | (0.6269, 0.1501) |
| 40 | 1.3665 | (0.5997, 0.1229) |
| 60 | 1.4222 | (0.5690, 0.0913) |
| 80 | 1.4572 | (0.5517, 0.0730) |
| 100 | 1.4822 | (0.5405, 0.0610) |
图 5
系统(1.3)的参数: $r = 6, d = 0.2, a = 1, b = 5, q = 2, c = 2. 5, $ $m_{1} = 0.4, m_{2} = 1, d_{1} = 0. 01, d_{2} = 0.1.$ 恐惧效应与Hopf分支临界点的关系图"
图 6
系统(1.3)的参数: $\tau = 1.4, t = 20000, r = 6, d = 0.2, a = 1, b = 5, q = 2, c = 2. 5, $ $m_{1} = 0.4, m_{2} = 1, d_{1} = 0. 01, d_{2} = 0.1. $ (a) $k_{0} = 10$, (b) $k_{0} = 80$. 捕食者的空间斑图"
图 7
系统(1.3)的参数: $r = 6, d = 0.2, a = 1, b = 5, q = 2, c = 2. 5, $ $m_{1} = 0.4, m_{2} = 1, d_{1} = 0. 01, d_{2} = 0.1 $. 恐惧效应与平衡点的关系图"
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