133,294 research outputs found
From individual dynamics at the microscopic scale to continuum dynamics at the macroscopic scale: The ant colony paradigm
No full textParticular attention is being paid these days to the mathematical modelling of the social behaviour of individuals in a biological population, for different reasons; on one hand there is an intrinsic interest in population dynamics of herds, on the other hand agent based models are being used in complex optimization problems (ACO's, i.e. Ant Colony Optimization). Further decentralized/parallel computing is exploiting the capabilities of discretization of nonlinear reaction-diffusion systems by means of stochastic interacting particle systems.
Among other interesting features, these systems lead to self organization phenomena, which exhibit interesting spatial patterns.
As a working example, an interacting particle system modelling the social behaviour of ants is proposed here, based on a system of stochastic differential equations, driven by social aggregating/repelling "forces". Specific reference to observed species in nature will be made.
Current interest concerns how properties on the macroscopic level depend on interactions at the microscopic level. Among the scopes of the seminar, a relevant one is to show how to bridge different scales at which biological processes evolve; in particular suitable "laws of large numbers" are shown to imply convergence of the evolution equations for empirical spatial distributions of interacting individuals to nonlinear reaction-diffusion equations for a so called mean field, as the total number of individuals becomes sufficiently large.
In order to support a rigorous derivation of the asymptotic nonlinear integrodifferential equation, problems of existence of a weak/entropic solution will be analyzed.
Further the existence of a nontrivial invariant probability measure is analyzed for the stochastic system of interacting particles.
As a further application of the same paradigm, a multiscale model for tumour-driven angiogenesis will be presented.
REFERENCES
[1] Boi S., Capasso V., and Morale D., Modelling the aggregating behaviour of ants of the species Polyergus Rufescens. Spatial heterogeneity in ecological models. Nonlinear Analysis. Real World Appl. 1:163-176, 2000.
[2] Burger M., Capasso V., and Morale D., On an aggregating model with long and short range interactions. Nonlinear Analysis. Real World Appl. 2006.
[3] Morale D., Capasso V. and Oelschlaeger K., An interacting particle system modelling aggregation behaviour: from individuals to populations. J. Mathematical Biology. 50:49-66, 2005.
[4] Capasso V., and Morale D., Stochastic modelling of tumour-induced angiogenesi
Ant colonies: a nature inspired paradigm for the mathematical modelling of self-organizing systems
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Rescaling Stochastic Processes : Asymptotics
No full textIn this chapter the authors investigate the links among different scales, from a probabilistic point of view. Particular attention is being paid to the mathematical modelling of the social behavior of interacting individuals in a biological population, on one hand because there is an intrinsic interest in dynamics of population
herding, on the other hand since agent based models are being used in complex optimization problems. Among other interesting features, these systems lead to phenomena of self-organization, which exhibit interesting spatial patterns. Here we show how properties on the macroscopic level depend on interactions at the microscopic level; in particular suitable laws of large numbers are shown to imply convergence of the evolution equations for empirical spatial distributions of interacting individuals to nonlinear reaction–diffusion equations for a so called mean field, as the total number of individuals becomes sufficiently large. As a working example, an interacting particle system modelling social behavior has been proposed, based on a system
of stochastic differential equations, driven by both aggregating/repelling and external “forces”. In order to support a rigorous derivation of the asymptotic nonlinear integro-differential equation, compactness criteria for convergence in metric spaces of measures, and problems of existence of a weak/entropic solution have been analyzed.
Further the temporal asymptotic behavior of the stochastic system of a fixed number of interacting particles has been discussed. This leads to the problem of the existence of nontrivial invariant probability measure
On the generalized geometric densities of random closed sets. An application to growth processes
No full textIn recent literature the authors have introduced a Delta formalism, á la Dirac, for the description of random closed sets of lower dimension with respect to the environment space d. Mean densities can be introduced for expected measures associated with such sets, with respect to the usual Lebesgue measure. In this paper we offer a review of the main results; in particular approximating sequences for the quoted mean densities are provided, that are of interest in the concrete estimation of mean densities of fibre processes, surface processes, etc. For time dependent random closed sets, as the ones describing the evolution of birth-and-growth processes (of interest for many models in material science and in biomedicine), the Delta formalism provides a natural framework for deriving evolution equations for mean densities at any (integer) Hausdorff dimension, in terms of the relevant kinetic parameters. In this context connections with the concepts of hazard functions, and spherical contact functions are presented
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
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
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