Spatio-temporal Structure of Branching Random Walks in Non Homogeneous Environments

Last modified: 2014-03-28

#### Abstract

In this work we discuss a spatio-temporal evolutionary model of a cell population. In modelling a cell population evolution, the key characteristics are the existence or the absence of sources where cells can die, after having

produced or not produced offspring and having migrated or not migrated to different compartments. Based on such characteristics, we can apply continuous-time branching random walks on multidimensional lattices to study

the evolution of a cell population with migration and division of cells. Consider particles living independently of each other and of their history. Each particle walks on the lattice until it reaches the source where its behavior changed. The spatio-temporal modeling is implemented as a stochastic

evolutionary system on a multidimensional lattices. At first, we present an approach to investigate the number of cells in the system and in every point of the lattice under fixation of space coordinates in these spatial models. Secondly, we examine the effect of one-point potential on the spatial

dynamics on the lattice in case when the spatial and temporal variables jointly tend to infinity. In particular, we construct a scale for measuring the transition probability as a function of time $t$ assuming that the

spatial variable is of order $t^{\alpha}$ for various values of $\alpha\geq0$. Further, we examine the effect of phase transitions on behavior of a cell population. Based on the obtained results, we discuss possible strategies

that may delay a cell population progression to some extent.

produced or not produced offspring and having migrated or not migrated to different compartments. Based on such characteristics, we can apply continuous-time branching random walks on multidimensional lattices to study

the evolution of a cell population with migration and division of cells. Consider particles living independently of each other and of their history. Each particle walks on the lattice until it reaches the source where its behavior changed. The spatio-temporal modeling is implemented as a stochastic

evolutionary system on a multidimensional lattices. At first, we present an approach to investigate the number of cells in the system and in every point of the lattice under fixation of space coordinates in these spatial models. Secondly, we examine the effect of one-point potential on the spatial

dynamics on the lattice in case when the spatial and temporal variables jointly tend to infinity. In particular, we construct a scale for measuring the transition probability as a function of time $t$ assuming that the

spatial variable is of order $t^{\alpha}$ for various values of $\alpha\geq0$. Further, we examine the effect of phase transitions on behavior of a cell population. Based on the obtained results, we discuss possible strategies

that may delay a cell population progression to some extent.

#### Keywords

Branching random walks, Evolution operators, Large Deviations