Stability of the Darwinian Dynamics: Effect of Intraspecific Competition and Human Intervention

We analyze the stability of a game-theoretic model of a polymorphic eco-evolutionary system in the presence of human intervention. The goal is to understand how the intensity of this human intervention and competition within the system impact its stability, with cancer treatment as a case study. In this case study, the physician applies anti-cancer treatment, while cancer, consisting of treatment-sensitive and treatment-resistant cancer cells, responds by evolving more or less treatment-induced resistance, according to Darwinian evolution. We analyze how the existence and stability of the cancer eco-evolutionary equilibria depend on the treatment dose and rate of competition between cancer cells of the two different types. We also identify initial conditions for which the resistance grows unbounded. In addition, we adopt the level-set method to find viscosity solutions of the corresponding Hamilton-Jacobi equation to estimate the basins of attraction of the found eco-evolutionary equilibria and simulate typical eco-evolutionary dynamics of cancer within and outside these estimated basins. While we illustrate our results on the cancer treatment case study, they can be generalized to any situation where a human aims at containing, eradicating, or saving Darwinian systems, such as in managing antimicrobial resistance, fisheries management, and pest management. The obtained results help our understanding of the impact of human interventions and intraspecific competition on the possibility of containing, eradicating, or saving evolving species. This will help us with our ability to control such systems.