数学物理学报 ›› 2024, Vol. 44 ›› Issue (6): 1577-1594.
一个扩散模型中恐惧效应与集群行为协同诱导的时空动力学研究
肖江龙1(
),宋永利2(),夏永辉3,*()
- 1浙江师范大学数学科学学院 浙江金华 321004
2杭州师范大学数学学院 杭州 311121
3佛山大学数学学院 广东佛山 528000
-
收稿日期:2023-08-02修回日期:2024-04-16出版日期:2024-12-26发布日期:2024-11-22 -
通讯作者:*夏永辉,Email: yhxia@zjnu.cn -
作者简介:肖江龙,Email:jianglongxiao@zjnu.edu.cn ;|宋永利,Email:songyl@hznu.edu.cn -
基金资助:国家自然科学基金(11931016);浙江省自然科学基金(LZ23A010001);浙江省自然科学基金(LZ24A010006)
Spatiotemporal Dynamics Induced by the Interaction Between Fear and Schooling Behavior in a Diffusive Model
Xiao Jianglong1(),Song Yongli2(),Xia Yonghui3,*()
- 1School of Mathematical Sciences, Zhejiang Normal University, Zhejiang Jinhua 321004
2School of Mathematics, Hangzhou Normal University, Hangzhou 311121
3School of Mathematics, Foshan University, Guangdong Foshan 528000
-
Received:2023-08-02Revised:2024-04-16Online:2024-12-26Published:2024-11-22 -
Supported by:NSFC(11931016);Natural Science Foundation of Zhejiang Province(LZ23A010001);Natural Science Foundation of Zhejiang Province(LZ24A010006)
摘要：
该文研究了在齐次 Neumann 边界条件下扩散的捕食者--猎物模型的时空动力学. 该文研究表明, 恐惧和集群行为之间的相互作用产生了非常丰富和有趣的时空动力学. 详细讨论了 Turing 不稳定性和 Turing-Hopf 分支产生的条件. 利用规范型理论对 Turing-Hopf 分支点附近的时空动力学进行了分类, 并用丰富的数值模拟验证了理论分析. 最后, 总结了恐惧效应对集群行为的巨大影响. 发现猎物的集群行为不能抵消高水平的恐惧, 但可以抵消低水平的恐惧. 此外, 恐惧效应与集群行为共同诱发了系统的 Turing 不稳定性. 然而, 对于没有集群行为的系统, 恐惧效应既不会改变共存平衡点的稳定性, 也不会引起系统产生周期解或 Turing 不稳定性.
中图分类号:
- 0175.23
引用本文
肖江龙, 宋永利, 夏永辉. 一个扩散模型中恐惧效应与集群行为协同诱导的时空动力学研究[J]. 数学物理学报, 2024, 44(6): 1577-1594.
Xiao Jianglong, Song Yongli, Xia Yonghui. Spatiotemporal Dynamics Induced by the Interaction Between Fear and Schooling Behavior in a Diffusive Model[J]. Acta mathematica scientia,Series A, 2024, 44(6): 1577-1594.
图9
(A) 恐惧对猎物和捕食者密度的负面影响图. 这里取 $c=1.5 ＜ \sigma=20$. (B)-(C) 系统 (2.1) 在 $\sigma-\theta$ 平面上的分支图. 红线和绿线分别表示正平衡点存在的分界线和 Hopf 分支曲线."
图1
系统 （1.5） 在平面 $\sigma-d_{1}$ 上的稳定性区域和分支图. 取 $d_{2}=0.5,\theta=3,c=1.5$. $H_{0}$ 和虚线分别表示 Hopf 分支曲线和 $E_{*}$ 存在的分界线. $l_{s}(s=2,3,4)$ 是 Turing 分支曲线. $H_{0}$ 和 $l_{2}$ 相交于点 $P(\sigma^{*}, d^{*}_{1})=(3.0625, 0.05)$."
图2
系统 （5.2） 在 Turing-Hopf 分支点 $(3.0625,0.05)$ 处的分支图"
图3
当 $(\epsilon_{1},\epsilon_{2})=(0.015,0.5)$ 位于区域 $S_{1}$ 内时, $E_{*}(0.3194,0.3194)$ 稳定. 初值是 $(0.3194+0.2\cos 2x, 0.3194-0.24 \cos 2x)$."
图4
当 $(\epsilon_{1},\epsilon_{2})=(0.001,0.002)$ 位于区域 $S_{2}$ 内时, $E_{*}(0.2503,0.2503)$ 不稳定且形如 $\cos 2x$ 个稳定的非常数稳态出现. 图(A)-(B) 的初值取 $(0.2503+0.1\cos 2x, 0.2503+0.01 \cos 2x)$, 图(C)-(D) 的初值取 $(0.2503-0.1\cos 2x, 0.2503-0.01 \cos 2x)$."
图5
当 $(\epsilon_{1},\epsilon_{2})=(0.0002,-0.0001)$ 位于区域 $S_{3}$ 内时, $E_{*}(0.25,0.25)$ 不稳定. 一个异宿轨将不稳定的空间齐次周期解与稳定的空间非齐次稳态连接起来. 初始值为 $(0.25+0.00001 \cos 2x, 0.25+0.00001 \cos 2x)$."
图6
当 $(\epsilon_{1},\epsilon_{2})=(-0.02,-0.3)$ 位于区域 $S_{4}$ 内时, $E_{*}(0.2063,0.2063)$ 不稳定且稳定的非常数稳态出现. 初值取 $(0.2063-0.006 \cos 2x, 0.2063+0.001 \cos 2x)$."
图7
当 $(\epsilon_{1},\epsilon_{2})=(-0.001,-0.0032)$ 位于区域 $S_{5}$ 内时, $E_{*}(0.2454,0.2454)$ 不稳定且稳定的周期解出现. 初值取 $(0.2454-0.008 \cos 2x, 0.28-0.008 \cos 2x)$."
图8
当 $(\epsilon_{1},\epsilon_{2})=(-0.005,-0.013)$ 位于区域 $S_{6}$ 内时, $E_{*}(0.2486,0.2486)$ 不稳定且稳定的空间齐次周期解出现. 初值取 $(0.249, 0.249+0.00001\cos 2x)$."
图10
(A) 取 $\sigma=3.5625,\theta=1000,c=0.1$. (B)--(C) 取 $d_{1}=0.065,d_{2}=0.5$ 且初值为 $(0.1+0.1\cos 2x,0.1+0.1 \cos 2x)$. 高水平的恐惧使猎物和捕食者都死亡."
图11
(A) 取 $\sigma=100,\theta=0.01,c=1.5$. (B)--(C) 取 $d_{1}=0.065,d_{2}=0.5$ 且初值为 $(30+30\cos 2x,30+30\cos 2x)$. 低水平的恐惧能使捕食者和猎物稳定到一定数量."
图12
(A) 取 $\sigma=3.0622,\theta=3,c=1.5$. (B)--(C) 取 $d_{1}=0.065,d_{2}=0.5$ 且初值为 $(0.26+0.001\cos 2x,0.26+0.001 \cos 2x)$. 恐惧效应引起系统产生周期解."
图13
恐惧效应对捕食者密度的负面影响图. 这里取 $c=1.5 ＜ \sigma=20$. 恐惧效应不影响猎物密度."
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