Institute of Mathematics, Academy of Mathematics and Systems Science (AMSS), Chinese Academy of Sciences, Beijing, China.
Adv Exp Med Biol. 2011;696:647-55. doi: 10.1007/978-1-4419-7046-6_66.
With the aid of Volterra multiplier, we study ecological equations for both tree system and cycle system. We obtain a set of sufficient conditions for the ultimate boundedness to nonautonomous n-dimensional Lotka-Volterra tree systems with continuous time delay. The criteria are applicable to cooperative model, competition model, and predator-prey model. As to cycle system, we consider a three-dimensional predator-prey Lotka-Volterra system. In order to get a condition under which the system is globally asymptotic stable, we obtain a Volterra multiplier, so that in a parameter region the system is with the Volterra multiplier it is globally stable. We have also proved that in regions in which the condition is not satisfied, the system is unstable or at least it is not globally stable. Therefore, we say that the three-dimensional cycle system is with global bifurcation.
借助 Volterra 乘数，我们研究了树系统和循环系统的生态方程。我们获得了一组非自治 n 维 Lotka-Volterra 树系统具有连续时滞的最终有界性的充分条件。这些准则适用于合作模型、竞争模型和捕食者-猎物模型。对于循环系统，我们考虑一个三维捕食者-猎物 Lotka-Volterra 系统。为了得到系统全局渐近稳定的条件，我们得到了一个 Volterra 乘数，使得在一个参数区域内，系统具有 Volterra 乘数时是全局稳定的。我们还证明了在条件不满足的区域内，系统是不稳定的，或者至少不是全局稳定的。因此，我们说这个三维循环系统具有全局分岔。
