Dynamics Analysis of Cannibalistic Model With Density Dependence in Mature Stage
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摘要:
在考虑成熟阶段具有密度制约的基础上，建立了一类具有卵-成熟阶段的同类相食模型。该文从两个方面讨论了模型的动力学性态：当种群不存在同类相食时，构造Lyapunov函数证明平衡点的全局渐近稳定性；当种群存在同类相食时，利用中心流形定理证明同类相食使模型产生鞍结点分支，通过构造Dulac函数说明在二维自治系统中不存在极限环，得到了平衡点的全局稳定性。最后，利用数值模拟验证了所得相应结果的正确性。
Abstract:In view of the density dependence of mature individuals, a two-stage cannibalistic model with the egg-to-maturity stage was established. The dynamic behaviors of the model were discussed from two aspects. In the case without cannibalism, the global asymptotic stability of the equilibrium points was proved through construction of the Lyapunov function, while in the case with cannibalism, the existence of saddle-node bifurcation due to cannibalism was proved with the center manifold theorem. Through construction of the Dulac function, nonexistence of the limit cycle in the two-dimensional autonomous system was elucidated, and therefore, the global stability of the equilibrium points was obtained. Finally, the theoretical results were verified through numerical simulation.
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Key words:
- cannibalism /
- saddle-node bifurcation /
- density dependence /
- Dulac function
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图 1 模型(1)存在种群灭绝平衡点时的动力学性态
Figure 1. The model (1) dynamics with a population extinction equilibrium point
图 2 模型(1)在
$ d \lt r $ 时存在唯一种群存活平衡点的动力学性态Figure 2. The model (1) dynamics with a unique population survival equilibrium point for
$ d \lt r $ 
图 3 模型(1)存在两个种群存活平衡点的动力学性态
Figure 3. The model (1) dynamics with 2 population survival equilibrium points
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