Abstract: The Lotka-Volterra predator-prey biological model is mainly studied in this paper by introducing the new population which has no ability to prey the other population to form a simple competition between predator and prey when the number of predators is excessive. Based on above condition, differential algebraic biological model is established according to the conservation. Then, the stability of biological model is discussed when the parameters change in a certain range by applying the stability analysis method and the related criteria of differential algebraic system. Finally, the model is the simulated numerically by considering the results of the analysis and using the Matlab software, and the simulation results show that the system has a limit cycle when the parameters vary a certain value, which proved that the complex nonlinear phenomena exist in the differential algebraic biological model.

Key words: Differential algebraic model, Limit cycle, Stability, Lotka-Volterra predator-prey system

摘要： 研究了Lotka-Volterra食饵-捕食生物模型，考虑当捕食者数量过多时引入与捕食者形成一种简单竞争关系且不具有捕食食饵能力的物种来抑制捕食者的增长，根据守恒关系建立微分代数生物系统模型。然后，应用微分代数系统的稳定性分析方法和相关判据，讨论参数在一定范围内变化时生物模型稳定性问题。最后，结合分析结果应用Matlab软件对模型进行数值仿真。仿真结果表明,系统在参数取某一定值时出现极限环，所建立的微分代数生物系统模型产生复杂的非线性动力学现象。

关键词: 微分代数模型, 极限环, 稳定性, Lotka-Volterra食饵-捕食系统