Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvesting
Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvesting
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DOI：
10.1016/j.nonrwa.2016.05.010
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发表时间：
2017-02-01
影响因子：
2
通讯作者：
Cao, Hongjun
中科院分区：
文献类型：
--
作者：
Hu, Dongpo;Cao, Hongjun
考虑了具有非线性 Michaelis-Menten 型捕食者收获的捕食者猎物系统的稳定性和分岔分析。研究了可能平衡的存在性和稳定性。特别地，由于某些正平衡点处雅可比矩阵的行列式和迹的表达式非常复杂，因此需要采用数值模拟的方法来确定这些平衡点的稳定性。借助索托马约尔定理，给出了鞍结分岔和跨临界分岔存在性的严格数学证明。此外，为了确定Hopf分岔极限环的稳定性，计算了第一李亚普诺夫数，并给出数值例子进行了图形化说明。选择系统的两个参数作为分岔参数，通过计算尖点附近的通用展开，证明系统表现出余维2的Bogdanov-Takens分岔。进行数值模拟以证明理论结果的有效性。当捕食者种群存在非线性米氏收获效应时，我们的研究将有助于理解生态系统或物理系统的动态复杂性。这种非线性收获比恒产收获、恒力收获的模型更加现实合理。它可以被认为是对该系统动力学现有文献的补充，因为迄今为止很少有文献涉及该系统的非线性类型收获。 (C) 2016 Elsevier Ltd. 保留所有权利。
The stability and bifurcation analysis for a predator prey system with the nonlinear Michaelis-Menten type predator harvesting are taken into account. The existence and stability of possible equilibria are investigated. Specially, the stability of some positive equilibria is determined by using numerical simulation method due to the fact that the expressions of determinant and trace of the Jacobian matrix at these equilibria are very complex. The rigorous mathematical proofs of the existence of saddle node bifurcation and transcritical bifurcation are derived with the help of Sotomayor's theorem. Furthermore, in order to determine the stability of limit cycle of Hopf bifurcation, the first Lyapunov number is calculated and a numerical example is given to illustrate graphically. Choosing two parameters of the system as bifurcation parameters, we prove that the system exhibits Bogdanov-Takens bifurcation of codimension 2 by calculating a universal unfolding near the cusp. Numerical simulations are carried out to demonstrate the validity of theoretical results. Our research will be useful for understanding the dynamic complexity of ecosystems or physical systems when there is the nonlinear Michaelis-Menten type harvesting effect on predator population. This kind of nonlinear harvesting is more realistic and reasonable than the model with constant-yield harvesting and constant effort harvesting. It can be thought as a supplement to existing literature on the dynamics of this system, since there is little literature involved in nonlinear type harvesting for the system up to now. (C) 2016 Elsevier Ltd. All rights reserved.