Tumor micro-ecology and competitive interactions

Abstract

Three nested models describing the growth of individual subpopulations in a heterogeneous environment are described. The models represent the dynamics of two populations which compete, to varying degrees, for common resources. The first model describes growth in a totally non-competitive micro-environment, the second model describes an ecology in which competition is proportional to competitor population size, and the third model ecology extends the model described by Jansson & Revesz (1974), which allows one population to emerge from the other. The critical points for each model are defined using the isoclines derived from the Ordinary Differential Equations (ODE's) describing competitive growth. The critical points for each model are characterized by the signs of the eigenvalues of the variational matrix at each point. The theoretical results of the analysis show that a competitive model ecology with Verhulstian logistics allows four critical points: the origin which is a repeller, two competitive exclusion points, and an equilibrium state (Waltman, 1983). The extended model ecology of Jansson & Revesz (1974), allows three critical points: the origin which is a repeller, competitive exclusion of the first population, and an equilibrium point. Data from a human adenocarcinoma of the colon and murine mammary tumors are used as qualitative measures of the dynamics of the three micro-ecologies. Issues such as stochastic extension to model small populations either for clonal extinction or heterogeneous emergence are discussed.