RESUMO
Evolution is a stochastic yet inevitable process that lies at the heart of biology yet in the multi-cellular environments within patients, ecological complexities arise via heterogeneity and microenvironments. The interplay of ecology and mutation is thus fundamental to predicting the evolution of complex diseases and engineering optimal treatment solutions. As experimental evidence of ecological interactions between disease agents continues to grow, so does the need for evolutionary theory and modeling that incorporates these interaction effects. Inspired by experimental cell biology, we transform the variables in the interaction payoff matrix to encode cell-cell interactions in our mathematical approach as growth-rate modifying, frequency-dependent interactions. In this way, we can show the extent to which the presence of these cell-extrinsic ecological interactions can modify the evolutionary trajectories that would be predicted from cell-intrinsic properties alone. To do this we form a Fokker-Planck equation for a genetic population undergoing diffusion, drift, and interactions and generate a novel, analytic solution for the stationary distribution. We use this solution to determine when these interactions can modify evolution in such ways as to maintain, mask, or mimic mono-culture fitness differences. This work has implications for the interpretation and understanding of experimental and patient evolution and is a result that may help to explain the abundance of apparently neutral evolution in cancer systems and heterogeneous populations in general. In addition, the derivation of an analytical result for stochastic, ecologically dependent evolution paves the way for treatment approaches requiring knowledge of a stationary solution for the development of control protocols.