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| Evolution and mutation phenomena have an important role in developmental biology and life sciences in general. Recently the study and modeling of these phenomena became one of the most challenging new frontiers of applied mathematics. We obtain and study a new broad class of nonlinear integro-differential equations, derived by the mathematical tools
of the kinetic theory for active particles, suitable to model the dynamics of large interacting populations. These equations, which offer the basis for the derivation of specific examples, can model mutations, namely generation of new populations with a phenotype different form the origin genotype that could be more (or less) fitted to an environment that evolves in time with known dynamics, and subsequently proliferative/distructive events that can bring to growth of new populations and, in some cases, of extinction events.
We present a detailed qualitative analysis of the related initial value problem associated to applications.In particular, we prove that the 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. |
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