Page 183 - Special Topic Session (STS) - Volume 4

STS579 Anastasiia Rytova et al.

Survival analysis of particle populations in
branching random walks
Anastasiia Rytova, Elena Yarovaya
Lomonosov Moscow State University, Moscow, Russia

Abstract
Application of the branching random walk models in the population studies is
discussed. The main results obtained for the models of symmetric continuous-
time branching random walks on a multidimensional lattice with a few sources
of particle birth and death at lattice points. We will be mainly interested in
studying the problems related to the limiting behavior of branching random
walks such as existence of phase transitions under change of various
parameters, the properties of the limiting distribution and the survival ability
of the particle population. The survival analysis of such particle system is
related with the notions of local extension probability of the branching
random walk at every lattice point and of the survival probability of the particle
population. Emphasis is made on the survival analysis and study of branching
random walk properties depending on the configuration of the sources and
their intensities. The answer to these and other questions heavily depend on
numerous factors which affect the properties of a branching random walk.
Therefore, we will try to describe how the properties of a branching random
walk depend on such characteristics of an underlying branching walk as
finiteness or infiniteness of the variance of jumps.

Keywords
population dynamics; asymptotic behavior; heavy-tailed distribution; survival
probability

1. Introduction
The evolution of systems with several elements that able to move, produce
descendants and die can be described by a random walk and branching
process. The corresponding mathematical models were used to explore the
genetic patterns in Haldane (1927), axon growing in Zhizhina, Komech, &
Descombes (2015), reliability for servers system in Vatutin, Topchii & Yarovaya
(2004), biological populations in Bolker, Pacala, & Neuhauser (2003), human
population in Molchanov & Whitmeyer (2017), epidemic spread in Ermakova,
Makhmutova, & Yarovaya (2019). It can be interesting to determine the
conditions that lead to a special state of the system, such as the degeneration
or exponential growth of the population, the form and stability of the spatial
distribution. For example, if there is an area in space in which the particle is
rather die than give descendants, the strategy increases the population

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