数学物理学报 ›› 2022, Vol. 42 ›› Issue (2): 570-582.
带有经济效益的时滞分数阶微分-代数捕食-被捕食系统的Hopf分岔
张道祥,李奔,陈丹丹,林雅婷,王鑫梅
- 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:2022-04-26发布日期:2022-04-18 -
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)
Hopf Bifurcation for a Fractional Differential-Algebraic Predator-Prey System with Time Delay and Economic Profit
Daoxiang Zhang,Ben Li,Dandan Chen,Yating Lin,Xinmei Wang
- School of Mathematics and Statistics, Anhui Normal University, Wuhu Anhui 241002
-
Received:2020-11-30Online:2022-04-26Published:2022-04-18 -
Supported by:the NSFC(11671013);the NSFC(11302002);the NSF of Anhui Province(2008085MA13)
摘要：
该文首先提出了一类带有经济效益的时滞分数阶微分-代数捕食-被捕食系统. 利用稳定性理论, 得到了在零经济收益条件下, 系统的正平衡点是局部渐近稳定的; 在正经济收益条件下, 时滞产生Hopf分岔的充分条件. 最后借助于数值模拟验证了理论的正确性, 并进一步讨论了分数阶阶数、经济收益和时滞对系统稳定性的影响.
中图分类号:
- O175.1
引用本文
张道祥,李奔,陈丹丹,林雅婷,王鑫梅. 带有经济效益的时滞分数阶微分-代数捕食-被捕食系统的Hopf分岔[J]. 数学物理学报, 2022, 42(2): 570-582.
Daoxiang Zhang,Ben Li,Dandan Chen,Yating Lin,Xinmei Wang. Hopf Bifurcation for a Fractional Differential-Algebraic Predator-Prey System with Time Delay and Economic Profit[J]. Acta mathematica scientia,Series A, 2022, 42(2): 570-582.
表 1
系统(1.1)相关变量和参数的意义"
| 变量(参数) | 意义 |
| 浮游植物的密度 | |
| 浮游动物的密度 | |
| 浮游植物种内干扰系数 | |
| 成熟捕食者的捕食率 | |
| 反应时间 | |
| 捕食者(浮游动物)之间的干扰程度 | |
| 喂食率 | |
| 转换效率 | |
| 浮游动物的死亡率 | |
| 时滞 | |
| Crowley-Martin[ |
表 1
图 1
$ \tau=0 $时系统(1.2)的积分曲线"
图 1
图 2
$ \tau = 2 $和$ \tau = 10 $时系统(1.2)的积分曲线"
图 2
图 3
系统(1.2)的积分曲线和相图. $ a = 1, b = 8, c = 3.9, d = 1.2, e = 0.0005, h = 1.1, $ $ g = 8, {p_1} = 10, {p_2} = 0.1, Q = 0.3, m = 0.26, q_1 = 0.8, q_2 = 0.9, \tau = 2.4 < {\tau _0} $"
图 3
图 4
系统(1.2)的积分曲线和相图. $ a = 1, b = 8, c = 3.9, d = 1.2, e = 0.0005, h = 1.1, $ $ g = 8, {p_1} = 10, {p_2} = 0.1, Q = 0.3, m = 0.26, q_1 = 0.8, q_2 = 0.9, \tau = 2.68 > {\tau _0} $"
图 4
图 5
$ m $与正平衡点$ {X^*} $的关系图"
图 5
图 6
稳定区域和不稳定区域随$ m $的增加而变化"
图 6
图 7
$ \tau=1, m = 0.5 $时, 系统(1.2)的积分曲线"
图 7
图 8
$ \tau=1, m = 0.65 $时, 系统(1.2)产生Hopf分岔"
图 8
图 9
$ q_1 $和$ {\tau _0} $的关系图"
图 9
图 10
$ q_2 $和$ {\tau _0} $的关系图"
图 10
图 11
$ {q_2} = 0.9, \tau = 2, q_1 = 0.7, 0.75, 0.8 $时系统(1.2)的积分曲线"
图 11
图 12
$ {q_2} = 0.9, \tau = 2, q_1 = 0.7, 0.75, 0.8 $时系统(1.2)的积分曲线"
图 12
图 13
$ {q_2} = 0.9, \tau = 2, q_1 = 0.7, 0.75, 0.8 $时系统(1.2)的相图"
图 13
图 14
$ {q_1} = 0.8, \tau = 2.2, q_2 = 0.825, 0.875, 0.925 $时系统(1.2)的积分曲线"
图 14
图 15
$ {q_1} = 0.8, \tau = 2.2, q_2 = 0.825, 0.875, 0.925 $时系统(1.2)的积分曲线"
图 15
图 16
$ {q_1} = 0.8, \tau = 2.2, q_2 = 0.825, 0.875, 0.925 $时系统(1.2)的相图"
图 16
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