[Mathematical modeling of life history evolution: a brief history and main trends].

A brief history is recounted of the mathematical theory of life history evolution which originates from works by L. Euler, T. Malthus, A. Lotka, and R. Fisher, but has been developed in its final form in 1970s. Basic approaches to modeling life history evolutionary ecology--the explicit, optimization, and adaptive dynamics ones--are evaluated with an emphasis on their methodological advantages and shortcomings. The explicit approach deals with direct modeling of coupled changes in size of populations--which are treated as assemblages of individuals with different life history strategies--and takes into account the interactions between populations themselves and between populations and their environment. The approach is methodologically transparent, though requires a great deal of computing resources for resolving problems complex in formulation. Besides, since it deals with specific modeling situations, it is not well suited for drawing general conclusions. The optimization approach is grounded on a search for life history strategies ensuring the maximization of the given measure of evolutionary fitness--most often, lifetime reproductive success or Malthusian parameter. Owing to the possibility for applying efficient analytical or numerical methods from the mathematical theory of optimal control, it allows to reveal optimal evolutionary strategies in rather complicated situations. Still, its usage is associated with significant methodological obstacles originated from the necessity of optimization criterion substantiation. The adaptive dynamics approach addresses to methods of the qualitative theory of differential equations while investigating evolutionary changes in phenotypic traits and life history strategies in particular. Within its framework, interactions between the main population (resident) and an alien population (invader), which is very small in size, are modeled. This approach allows to explore subtle aspects of evolutionary dynamics and to formulate and analyze mathematically some major problems of the theory of evolution--for example, the problem of sympatric speciation. However, application of this approach is also restricted due to specificity of objectives formulation as well as complexity of respective mathematical tools.