# American Institute of Mathematical Sciences

August  2014, 7(4): 807-821. doi: 10.3934/dcdss.2014.7.807

## On an initial value problem modeling evolution and selection in living systems

 1 Department of Mathematics and Informatics, University of Perugia, Via Vanvitelli 1, 06123 Perugia, Italy, Italy

Received  September 2013 Revised  November 2013 Published  February 2014

This paper is devoted to the qualitative analysis of a new broad class of nonlinear initial value problems that model evolution and selection in living systems derived by the mathematical tools of the kinetic theory of active particles. The paper is divided into two parts. The first shows how to obtain the nonlinear equations with proliferative/distructive nonlinear terms. The latter presents a detailed analysis of the related initial value problem. In particular, it is proved that the corresponding initial value problem admits a unique non--negative maximal solution. However, the solution cannot be in general global in time, due to the possibility of blow--up. The blow--up occurs when the biological life system is globally proliferative, see Theorem 3.3.
Citation: Patrizia Pucci, Maria Cesarina Salvatori. On an initial value problem modeling evolution and selection in living systems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 807-821. doi: 10.3934/dcdss.2014.7.807
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