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种内竞争时滞对植被周期振荡模式的影响研究

李 静 孙桂全 靳 祯

李 静,孙桂全,靳 祯. 种内竞争时滞对植被周期振荡模式的影响研究 [J]. 应用数学和力学,2022,43(X):1-13 doi: 10.21656/1000-0887.420190
引用本文: 李 静,孙桂全,靳 祯. 种内竞争时滞对植被周期振荡模式的影响研究 [J]. 应用数学和力学,2022,43(X):1-13 doi: 10.21656/1000-0887.420190
Jing LI, Guiquan SUN, Zhen JIN. Effect of intraspecific competition delay on vegetation periodic oscillation pattern[J]. Applied Mathematics and Mechanics. doi: 10.21656/1000-0887.420190
Citation: Jing LI, Guiquan SUN, Zhen JIN. Effect of intraspecific competition delay on vegetation periodic oscillation pattern[J]. Applied Mathematics and Mechanics. doi: 10.21656/1000-0887.420190

种内竞争时滞对植被周期振荡模式的影响研究

doi: 10.21656/1000-0887.420190
基金项目: 重点研发计划政府间国际合作重点资助项目(2018YFE0109600);国家自然科学基金(61873154;11671241;12001340;11901363);山西省自然科学基金(201901D211411;201801D221002);山西省高等学校科技创新计划(2019L0472)
详细信息
    作者简介:

    李 静:李静(1989—),女,讲师,博士(E-mail:jingli_2016@126.com)

    孙桂全(1983—),男,教授,博士(通讯作者. E-mail:gquansun@126.com)

    靳 祯:靳祯(1965—),男,教授,博士(E-mail:jinzhn@263.net)

  • 中图分类号: O29;Q141

Effect of intraspecific competition delay on vegetation periodic oscillation pattern

  • 摘要: 考虑到干旱半干旱地区的幼年植被与成年植被之间存在竞争水资源的现象,该文构建一个具有种内竞争时滞的植被-土壤水动力学模型。分析出系统存在植被灭绝平衡点和植被生存平衡点并给出平衡点局部稳定的条件,分别给出非空间系统和空间系统产生Hopf分支周期解的条件。通过数值模拟展示出两种系统对应的植被随时间演化做周期振荡模式,并通过参数敏感性分析发现降雨量和植被的增长率对这种模式的产生和模式的振幅、周期有显著影响,但蒸发量的影响效果最不显著,表明降雨量和植被本身的特征对干旱半干旱地区植被的演化发展产生深刻的影响。同时发现空间扩散的引入会抑制这种模式的发生,但对振幅和周期没有任何影响。所获得的结果解释了自然界中广泛观察到的植被周期振荡现象,为植被系统的可持续发展提供了理论支撑。
  • 图  1  从2010年1月到2018年12月甘肃省植被覆盖度随年变化的柱状图,其中每一年的12个柱体代表从1月到12月的植被覆盖度

    Figure  1.  Time series of vegetation coverage in Gansu Province from 2010.1 to 2018.12, the 12 columns of each year represent the vegetation coverage from January to December

    图  2  方程(4)存在唯一正实数根的示意图:(a) 情形(I);(b) 情形(II)(i);(c) 情形(II)(ii),其中红色实心点代表正实数根,(a)中亮蓝色和蓝色曲线分别表示$\varDelta=0$$\varDelta < 0$的情形

    Figure  2.  Schematic diagrams of the existence of the unique positive real root for Eq. (4): (a) case1; (b)case2(i); (c)case2(ii), where red dot represents the positive real root, light blue and blue curves of (a) represent the cases of $\Delta=0$ and $\Delta<0$, respectively.

    图  3  (a) 植被生物量$n$随着参数$p$的变化;(b) 在w-n平面上零斜率线$f_{1}(n,w)=0$(紫色/洋红色)和$f_{2}(n,w)=0$(橘色)形成的交点(黑色实心点),其中$E_{0}$$E_{{\rm{u}}}$表示植被灭绝平衡点和植被生存平衡点,参数$\gamma=1.6$$\sigma=0.8$$\mu=0.2$$b=0.2$

    Figure  3.  (a) The vegetation biomass $n$ varies with $p$. (b) The intersection(black solid point) formed by nullclines of $f_{1}(n,w)=0$(purple/magenta) and $f_{2}(n,w)=0$(orange) on $w-n$ plane, where $E_{0}$ and $E_{u}$ are the vegetation extinction equilibrium and the vegetation existence equilibrium. Parameters are $\gamma=1.6$$\sigma=0.8$$\mu=0.2$$b=0.2$.

    图  4  (a) $\tau_{0}^{0}$随参数$p$的变化情况;(b) $(p,n)$平面上的单参数分支图,参数$\sigma=0.8$$\mu=0.2$$b=0.2$$\gamma=1.6$ and $\tau=3$

    Figure  4.  (a) the variation of $\tau_{0}^{0}$ with $p$; (b) bifurcation diagram in $(p,n)$ plane with $\sigma=0.8$, $\mu=0.2$, $b=0.2$, $\gamma=1.6$ and $\tau=3$.

    图  5  非空间系统(2)解的数值模拟,(a) $\tau=1$;(b)(c) $\tau=3$

    Figure  5.  Numerical simulation of the solution of non-spatial system (2), (a) $\tau=1$; (b)(c) $\tau=3$

    图  6  对于不同的$p=1.0, 1.3, 1.6$$\tau_{0}^{0}$随着$\gamma$,$b$$\sigma$的变化情况,其他参数为$\sigma=0.8$$\mu=0.2$$b=0.2$$\gamma=1.6$

    Figure  6.  The variation of $\tau_{0}^{0}$ with $\gamma$, $b$, $\sigma$ for $p=1.0, 1.3, 1.6$. Other parameters are $\sigma=0.8$, $\mu=0.2$, $b=0.2$, $\gamma=1.6$

    图  7  $n(x,t)$的时空演化图:(a) $\tau=1$;(b) $\tau=4.8$;(c) 截取图(b)形成的部分图;(d) 图(c)的二维时空图

    Figure  7.  The time series of $n(x,t)$: (a) $\tau=1$; (b) $\tau=4.8$; (c) partial graph formed by intercepting (b); (d) two dimensional spatial-temporal graph of (c)

    图  8  对于不同的$k$$\beta$$\tau_{k_{*}}^{0}$随着参数$p$的变化

    Figure  8.  The variation of $\tau_{k_{*}}^{0}$ with $p$ for different $k$ and $\beta$

    图  9  对于不同的参数$p$$\tau_{k_{*}}^{0}$随着$d_{2}$$\beta$$\gamma$$b$$\sigma$的变化情况,这里$k=2$

    Figure  9.  The variation of $\tau_{k_{*}}^{0}$ with (a) $d_{2}$, (b) $\beta$, (c) $\gamma$, (d) $b$, (e) $\sigma$ for different $p$, here $k=2$

    图  10  周期振荡模式的敏感性分析,参数$\sigma=0.8$$\mu=0.2$$b=0.2$$\gamma=1.6$$d_{1}=0.02$$d_{2}=0.2$$\beta=2$$k=2$

    Figure  10.  Sensitivity analysis of periodic oscillation pattern, parameters are $\sigma=0.8$, $\mu=0.2$, $b=0.2$, $\gamma=1.6$, $d_{1}=0.02$, $d_{2}=0.2$, $\beta=2$, $k=2$

    表  1  系统(3)的正平衡点的存在情况

    Table  1.   Existence properties of equilibria for system (3)

    existence conditionsequilibria
    $\gamma-\mu\sigma \leqslant 0$$E_{0}$
    $\gamma-\mu\sigma> 0$$\varDelta\leq0$$E_{0}$,$E_{*}$
    $\varDelta > 0$,$\alpha_{1}>0$$E_{0}$,$E^{*}$
    $\varDelta > 0$,$\alpha_{1}<0$,$\alpha_{2}<0$$E_{0}$,$E_{1}$
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出版历程
  • 收稿日期:  2021-07-06
  • 修回日期:  2021-10-05
  • 网络出版日期:  2022-04-24

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