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Study of mathematical models of phenotype evolution and motion of cell populations

Abstract : In Chapter 1, we consider a cell population where the individuals live in the same environmental conditions for some fixed period of time where they compete for nutrients among themselves, considering that offspring has the same trait as their parents, we were defining a fitness function that is trait and density dependent, assuming there were a unique trait best adapted at fixed environmental conditions. We modeled this phenomenon using a Transport Equation. The main result have been obtaining a Dirac mass concentration like solutions for the asymptotic behavior, incorporating a parameter, which is biologically sustained. We applied the classical framework to obtain this result. First, we give the apriori estimates and existence result to the simplified problem, next we add terms to have a more realistic model, then we study an approximate problem given some regularity and properties at solutions, finally we obtain this limit. We used tools as BV convergence properties, Anzats, sub and super solutions, maximum principle, etc.Chapter 2 had been publishing in the following papers (see part II):- E. ESPEJO, K. VILCHES, C. CONCA (2012), Sharp conditon for blow-up and global existence in a two species chemotactic Keller-Segel system in R^2, European J. Appl. Math- C. CONCA, E. ESPEJO, K. VILCHES (2011), Remarks on the blow-up and global existence for a two species chemotactic Keller-Segel system in R^2. European J. Appl. Math.In this chapter, we give the main results obtained in these two publications. We have been studying the sharp condition to global existence and Blow-up in time to the parabolic PDE system in R^2, inspired by the studies were done in the one species case. We model the movement for two chemotactic populations produced by one chemical substance. The main result is to extend the result obtained to classical simplified Keller-Segel model in one species case to the multispecies case, using the adequately tools for PDE’s systems. We used the moment method to prove Blow-up and have been bounding the entropy to show global existence.
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Karina Vilches. Study of mathematical models of phenotype evolution and motion of cell populations. General Mathematics [math.GM]. Université Pierre et Marie Curie - Paris VI; Universidad de Chile, 2014. English. ⟨NNT : 2014PA066117⟩. ⟨tel-02103912⟩

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