TWO MODELS OF ADAPTIVE DYNAMICS OF STRUCTURED POPULATIONS: ANALYSIS AND SIMULATION - XINRAN RUAN - LJLL, SORBONNE UNIVERSITé
Structured population models offer us a powerful way of studying the adaptive dynamics of some hidden characteristics, or “traits”, which are different from individual to individual. In this talk, Dr. Ruan will introduce two types of recently studied models, namely the age-structured models and the structured cell growth model with heterogeneous mobility and proliferation rate.
In the age-structured models, Dirac concentrations on particular phenotypical traits appear in the case without mutation, which makes the numerical resolution of the problem challenging. Dr. Ruan will briefly review the asymptotic results of the model and further design an asymptotic preserving (A-P) scheme based on the WKB ansatz of the solutions. With the method, we can accurately capture the concentrations on a coarse, parameter-independent mesh. Important properties, including the A-P property, are rigorously proved. The scheme can be generalized to the case with mutation, where a nonlinear Hamilton-Jacobi equation will be involved in the limiting model. It can be formally shown that the generalized scheme is A-P as well.
In the second model, we study the dynamics of a growing population of cells with heterogeneous mobility and proliferation rate. Specifically, we consider the case where the more mobile cells are less proliferative and viceversa. An implicit finite volume scheme is designed and proved with several nice properties. In particular, we found via simulation that, in the case where mobility is bounded, compactly supported traveling fronts emerge, while the stretching fronts may occur in the case where mobility is unbounded. Formal asymptotic analysis will be presented to explain the numerical results.
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