Resilience Analysis for Competing Populations.

Ecological resilience refers to the ability of a system to retain its state when subject to state variables perturbations or parameter changes. While understanding and quantifying resilience is crucial to anticipate the possible regime shifts, characterizing the influence of the system parameters on resilience is the first step toward controlling the system to avoid undesirable critical transitions. In this paper, we apply tools of qualitative theory of differential equations to study the resilience of competing populations as modeled by the classical Lotka-Volterra system. Within the high interspecific competition regime, such model exhibits bistability, and the boundary between the basins of attraction corresponding to exclusive survival of each population is the stable manifold of a saddle point. Studying such manifold and its behavior in terms of the model parameters, we characterized the populations resilience: While increasing competitiveness leads to higher resilience, it is not always the case with respect to reproduction. Within a pioneering context where both populations initiate with few individuals, increasing reproduction of one population leads to an increase in its resilience; however, within an environment previously dominated by one population and then invaded by the other, an increase in the resilience of a population is obtained by decreasing its reproduction rate. Besides providing interesting insights for the dynamics of competing populations, this work brings near to each other the concepts of ecological resilience and the methods of differential equations and stimulates the development and application of new tools for ecological resilience.

To Cure or Not to Cure: Consequences of Immunological Interactions in CML Treatment.

Recent clinical findings in chronic myeloid leukemia (CML) patients suggest that the number and function of immune effector cells are modulated by tyrosine kinase inhibitors (TKI) treatment. There is further evidence that the success or failure of treatment cessation at least partly depends on the patients immunological constitution. Here, we propose a general ODE model to functionally describe the interactions between immune effector cells with leukemic cells during the TKI treatment of CML. In total, we consider 20 different sub-models, which assume different functional interactions between immune effector and leukemic cells. We show that quantitative criteria, which are purely based on the quality of model fitting, are not able to identify optimal models. On the other hand, the application of qualitative criteria based on a dynamical system framework allowed us to identify nine of those models as more suitable than the others to describe clinically observed patterns and, thereby, to derive conclusion about the underlying mechanisms. Additionally, including aspects of early CML onset, we can demonstrate that certain critical parameters, such as the strength of immune response or leukemia proliferation rate, need to change during CML growth prior to diagnosis, leading to bifurcations that alter the attractor landscape. Finally, we show that the crucial parameters determining the outcome of treatment cessation are not identifiable with tumor load data only, thereby highlighting the need to measure immune cell number and function to properly derive mathematical models with predictive power.