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The two-type Richardson model describes the growth of two competing infections on $\mathbb{Z}^d$ and the main question is whether both infection types can simultaneously grow to occupy infinite parts of $\mathbb{Z}^d$. For bounded initial…
We consider the model of Deijfen et al. for competing growth of two infection types in R^d, based on the Richardson model on Z^d. Stochastic ball-shaped infection outbursts transmit the infection type of the center to all points of the ball…
The two-type Richardson model describes the growth of two competing infection types on the two or higher dimensional integer lattice. For types that spread with the same intensity, it is known that there is a positive probability for…
We consider a stochastic model, describing the growth of two competing infections on $\mathbb{R}^d$. The growth takes place by way of spherical outbursts in the infected region, an outburst in the type 1 (2) infected region causing all…
This paper provides a survey of known results and open problems for the two-type Richardson model, which is a stochastic model for competition on $\mathbb{Z}^d$. In its simplest formulation, the Richardson model describes the evolution of a…
A stochastic model, describing the growth of two competing infections on $\mathbb{R}^d$, is introduced. The growth is driven by outbursts in the infected region, an outburst in the type 1 (2) infected region transmitting the type 1 (2)…
In the two-type Richardson model on a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, each vertex is at a given time in state $0$, $1$ or $2$. A $0$ flips to a $1$ (resp.\ $2$) at rate $\lambda_1$ ($\lambda_2$) times the number of…
We study a large family of competing spatial growth models. In these the vertices in Z^d can take on three possible states {0,1,2}. Vertices in states 1 and 2 remain in their states forever, while vertices in state 0 which are adjacent to a…
We study two competing growth models. Each of these models describes the spread of a finite number of infections on a graph. Each infection evolves like an (oriented or unoriented) first passage percolation process except that once a vertex…
We construct an edge-weight distribution for i.i.d. first-passage percolation on $\mathbb{Z}^2$ whose limit shape is not a polygon and whose extreme points are arbitrarily dense in the boundary. Consequently, the associated Richardson-type…
We study the growth of two competing infection types on graphs generated by the configuration model with a given degree sequence. Starting from two vertices chosen uniformly at random, the infection types spread via the edges in the graph…
We study competing first passage percolation on graphs generated by the configuration model. At time 0, vertex 1 and vertex 2 are infected with the type 1 and the type 2 infection, respectively, and an uninfected vertex then becomes type 1…
We study competing first passage percolation on graphs generated by the configuration model with infinite-mean degrees. Initially, two uniformly chosen vertices are infected with type 1 and type 2 infection, respectively, and the infection…
We study two simple mathematical models of the epidemic. At first, we study the repetitive infection spreading in a simplified SIRS model including the effect of the decay of the acquired immune. The model is an intermediate model of the…
The irreversible growth of a binary mixture under far-from-equilibrium conditions is studied in three-dimensional confined geometries of size $L_x \times L_y \times L_z$, where $L_z \gg L_x = L_y$ is the growing direction. A competing…
We consider a symmetric finite-range contact process on $\mathbb{Z}$ with two types of particles (or infections), which propagate according to the same supercritical rate and die (or heal) at rate $1$. Particles of type 1 can occupy any…
The focus of this article is on the dynamics of a new susceptible-infected model which consists of a susceptible group ($S$) and two different infectious groups ($I_1$ and $I_2$). Once infected, an individual becomes a member of one of…
We study two famous interacting particle systems, the so-called Richardson's model and the contact process, when we add a stirring dynamics to them. We prove that they both satisfy an asymptotic shape theorem, as their analogues without…
We study models of spatial growth processes where initially there are sources of growth (indicated by the colour green) and sources of a growth-stopping (paralyzing) substance (indicated by red). The green sources expand and may merge with…
Simple growth mechanisms have been proposed to explain the emergence of seemingly universal network structures. The widely-studied model of preferential attachment assumes that new nodes are more likely to connect to highly connected nodes.…