ИСТИНА

The talk is devoted to evolution of a particle population range. This problem goes back to the classical paper by A.N.Kolmogorov, I.G.Petrovski and N.S.Piskunov (1937). Important models were proposed and studied by K.B.Athreya, D.Bertacchi, J.D.Biggins, S.Foss, B.Mallein, S.A.Molchanov, Z.Shi, V.A.Topchij, V.A.Vatutin, E.B.Yarovaya, F.Zucca and other researchers. We provide a survey of the previous works and then concentrate on our new results. A catalytic branching random walk (CBRW) on $\mathbb{Z}^d$, $d\in\mathbb{N}$, is exploited as a mathematical model of the phenomena. The distinctive model trait is a finite number of catalysts present at fixed lattice points such that a particle in CBRW may produce offspring or die just at the location of a catalyst. Outside the catalysts a particle performs a random walk on $\mathbb{Z}^d$ until hitting a catalyst. A CBRW can be classified as supercritical, critical or subcritical (see \cite{B_TPA_15}). In a supercritical CBRW only, the particle population survives globally and locally with positive probability, whereas in subcritical and critical CBRWs the population degenerates locally with probability one although it can survive globally with positive probability. Whenever in a supercritical CBRW the population survives, it increases exponentially-fast in time. In this case the study of area settling in time by the particle population is reduced to analysis of the rate of the population spread. As shown in \cite{B_Arxiv_18} and \cite{B_Arxiv_19}, it depends on ``heaviness'' of the distribution tails of the random walk. Correspondingly, the growth of the particle population ``front'' covers the scale from linear to exponential in time mode. In critical and subcritical CBRWs the problem of inhabited territory variation is posed in another way. The main interest focuses on the maximal displacement of the particles in the CBRW during the whole history of particles existence (see \cite{B_SEMR_20}). We reveal new effects in the asymptotic behavior of the distribution tail of the maximal displacement which are not observed in the model of a non-catalytic branching random walk. Methods of investigation include renewal theory, ``many-to-one'' lemma, analysis of solution to a system of integral equations, Laplace transform, convex analysis, martingale change of measure, large deviation theory and the coupling method. {\bf Acknowledgement:} This work was supported by the Russian Science Foundation under grant 17-11-01173-Ext. \vskip0.3cm \begin{thebibliography}{1} \bibitem{B_TPA_15} E.Vl. Bulinskaya, {\it Complete classification of catalytic branching processes}, Theory Probab. Appl., {\bf 59}:4 (2015), 545--566. \bibitem{B_Arxiv_18} E.Vl. Bulinskaya, {\it Maximum of catalytic branching random walk with regularly varying tails}, J. Theor. Probab, DOI: 10.1007/s10959-020-01009-w (2020), 1--21. \bibitem{B_Arxiv_19} E.Vl. Bulinskaya, {\it Catalytic branching random walk with semiexponential increments}, Math. Popul. Stud., DOI: 10.1080/08898480.2020.1767424 (2020), 1--31. \bibitem{B_SEMR_20} E.Vl. Bulinskaya, {\it On the maximal displacement of catalytic branching random walk}, Sib. Elektron. Mat. Izv. (to appear) (2020), arXiv: 2007.05815, 1--11. \end{thebibliography}