`a`
Discrete and Continuous Dynamical Systems - Series S (DCDS-S)
 

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

Pages: 807 - 821, Volume 7, Issue 4, August 2014      doi:10.3934/dcdss.2014.7.807

 
       Abstract        References        Full Text (407.0K)       Related Articles       

Patrizia Pucci - Department of Mathematics and Informatics, University of Perugia, Via Vanvitelli 1, 06123 Perugia, Italy (email)
Maria Cesarina Salvatori - Department of Mathematics and Informatics, University of Perugia, Via Vanvitelli 1, 06123 Perugia, Italy (email)

Abstract: 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.

Keywords:  Population dynamics, multi--scale modeling, active particles, living systems, kinetic theory, complexity in biology, nonlinearity, nonlinear interactions.
Mathematics Subject Classification:  Primary: 47J35, 45K05, 92D25; Secondary: 74A25, 92D15.

Received: September 2013;      Revised: November 2013;      Available Online: February 2014.

 References