This book examines common issues for particular situations and provides examples of several asymptotic and analytical techniques used in the study of random evolutionary systems. Here, the trajectories of diffusion-type processes—rather than those of the Wiener process—are more accurately described by constructive mathematical models of natural phenomena.

We study models in which there is some free gap between two successive collisions of particles. Simultaneously, we study two scenarios: the semi-Markov evolutionary system, with an arbitrary distribution of the switching process, and the Markov evolutionary system, where the time spent by the particle moving towards a direction is distributed exponentially with intensity parameter λ. As a result, the models examined here explain both the finite speed motion of particles and the suggested random evolutionary process, which has free run and finite propagation speed, in common with a natural physical process. There are either infinite or finitely many alternative evolutionary paths in the models that have been suggested.