1,721,012 research outputs found
Modeling and simulating animal grouping Individual-based models
No full textThis paper contains a review of recent results about two interacting particle models describing aggregation of social individuals;
a cellular automaton model and a system of of Itô type stochastic differential equations are described
Cellular automata and many-particle systems modeling aggregation behavior among populations
No full textA cellular automaton model is presented in order to describe mutual interactions among the individuals of a population
due to social decisions. The scheme is used for getting qualitative results, comparable to field experiments
carried out on a population of ants which present an aggregative behavior.
We also present a second description of a biological spatially structured population of individuals by
a system of stochastic differential equations of It\^o type. A ''law of large numbers'' to a continuum dynamics described by an integro-differential
equation is given
Stochastic modelling of tumour-induced angiogenesis
No full textA major source of complexity in the mathematical modelling of an angiogenic process derives from the strong coupling of the kinetic parameters of the relevant stochastic branching-and-growth of the capillary network with a family of interacting underlying fields. The aim of this paper is to propose a novel mathematical approach for reducing complexity by (locally) averaging the stochastic cell, or vessel densities in the evolution equations of the underlying fields, at the mesoscale, while keeping stochasticity at lower scales, possibly at the level of individual cells or vessels. This method leads to models which are known as hybrid models. In this paper, as a working example, we apply our method to a simplified stochastic geometric model, inspired by the relevant literature, for a spatially distributed angiogenic process. The branching mechanism of blood vessels is modelled as a stochastic marked counting process describing the branching of new tips, while the network of vessels is modelled as the union of the trajectories developed by tips, according to a system of stochastic differential equations `a la Langevin
Asymptotic behavior of a system of stochastic particles subject to nonlocal interactions
No full textIn this article, we present a rigorous mathematical derivation of a
macroscopic model of aggregation, scaling up from a microscopic description of
a family of individuals subject to aggregation/repulsion, described by a system
of Itô type stochastic differential equations. We analyze the asymptotics of the
system for both a large number of particles on a bounded time interval, and its
long time behavior, for a fixed number of particles. As far as this second part
is concerned, we show that a suitable localizing potential is required, in order
that the system may admit a nontrivial invariant distribution
Long time behavior of a system of stochastic differential equations modelling aggregation
No full textIn many biological settings it is possible to observe phenomena of pattern formation and clustering by cooperative individuals of a population. In biology and medicine there is a wide spectrum of examples which exhibit collective behavior, leading to self organization, with pattern formation. Aggregation patterns are usually explained in terms of forces, external and/or internal, acting upon individuals. Over the past couple of decades, a large amount of literature has been devoted to the mathematical modelling of self-organizing populations, based on the concepts of short-range/long-range social interaction at the individual level. The main interest has been in catching the main features of the interaction at the lower scale of single individuals that are responsible, at a larger scale, for a more complex behavior that leads to the formation of aggregating patterns
Randomness in self-organized phenomena. A case study: retinal angiogenesis
No full textThis note presents a review of recent work by the authors on angiogenesis, as a case study for analyzing the role of randomness in the formation of biological patterns. The mathematical description of the formation of new vessels is presented, based on a system of stochastic differential equations, coupled with a branching process, both of them driven by a set of relevant chemotactic underlying fields. A discussion follows about the possible reduction of complexity of the above approach, by mean field approximations of the underlying fields. The crucial role of randomness at the microscale is observed in order to obtain nontrivial realistic vessel networks
On an aggregation model with long and short range interactions
No full textIn recent papers the authors had proposed a stochastic model for swarm aggregation, based on individuals subject to long range attraction and short range repulsion, in addition to a classical Brownian random dispersal. Under suitable laws of large numbers they showed that, for a large number of individuals, the evolution of the empirical distribution of the population can be expressed in terms of an approximating nonlinear degenerate and nonlocal parabolic equation, which describes the limit.
In this paper the well-posedness of such evolution equation is investigated, which invokes a notion of entropy solutions extended to the nonlocal case. We motivate entropy solutions from the discrete particle system and use them to prove uniqueness. Moreover, we provide existence results and discuss some basic properties of solutions. Finally, we apply a Lagrangian numerical scheme to perform numerical simulations in spatial dimension one
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