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Bifurcation patterns in generalized models for the dynamics of normal and leukemic stem cells with signaling. (English) Zbl 1336.34068
Summary: The dynamics of normal and leukemic stem cell clones competing for resources in the same bone marrow niche can be modeled in a systems biology approach. We derive three related models describing the time evolution of normal and chronic myelogenous leukemia (CML) stem cells. The approach is based on the idea that stem cell proliferation is regulated by feedback mechanisms. We assume here that the growth of stem cell clones can be approximated by Tsallis functions, which depend on population numbers to ensure the communication between different stem cell strains. Of particular interest is the influence of medication, for example, by the CML drug imatinib, on the stem cell dynamics, which can be simulated by a variation of the parameterization of the differential equation systems. The basic 2D model represents the contest between cycling normal and wild-type CML stem cells. In addition, extension of the basic model (i) by coupled reservoirs of quiescent stem cells and (ii) by a third stem cell species, which could be interpreted as an imatinib-resistant mutant, is investigated. We analyze the global dynamics of the corresponding 2D, 3D, and 4D equation systems by analytic means and provide a complete map of the bifurcation landscape that occurs for changing values of the respective signaling strengths and growth or death parameters. This includes a complete classification of equilibria and their linear and non-linear Lyapunov stability.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
34C23 Bifurcation theory for ordinary differential equations
37B25 Stability of topological dynamical systems
92C37 Cell biology
34D20 Stability of solutions to ordinary differential equations
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