Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland.
Math Biosci Eng. 2013 Jun;10(3):861-72. doi: 10.3934/mbe.2013.10.861.
In the paper we consider a system of delay differential equations (DDEs) of Lotka-Volterra type with diffusion reflecting mutations from normal to malignant cells. The model essentially follows the idea of Ahangar and Lin (2003) where mutations in three different environmental conditions, namely favorable, competitive and unfavorable, were considered. We focus on the unfavorable conditions that can result from a given treatment, e.g. chemotherapy. Included delay stands for the interactions between benign and other cells. We compare the dynamics of ODEs system, the system with delay and the system with delay and diffusion. We mainly focus on the dynamics when a positive steady state exists. The system which is globally stable in the case without the delay and diffusion is destabilized by increasing delay, and therefore the underlying kinetic dynamics becomes oscillatory due to a Hopf bifurcation for appropriate values of the delay. This suggests the occurrence of spatially non-homogeneous periodic solutions for the system with the delay and diffusion.
在本文中，我们考虑了具有扩散的 Lotka-Volterra 型时滞微分方程组（DDEs），其中正常细胞向恶性细胞的突变会受到反射。该模型主要基于 Ahangar 和 Lin（2003）的思想，其中考虑了三种不同环境条件下的突变，即有利、竞争和不利。我们关注可能由特定治疗引起的不利条件，例如化疗。包含的延迟代表良性和其他细胞之间的相互作用。我们比较了 ODEs 系统、具有时滞和时滞与扩散的系统的动力学。我们主要关注存在正平衡点时的动力学。在没有时滞和扩散的情况下全局稳定的系统，由于时滞的适当值产生的 Hopf 分支，会被增加的时滞所破坏，因此由于动力学的动态变得振荡。这表明对于具有时滞和扩散的系统，会出现空间非均匀的周期性解。
