# American Institute of Mathematical Sciences

ISSN:
1551-0018

eISSN:
1547-1063

All Issues

## Mathematical Biosciences & Engineering

2015 , Volume 12 , Issue 2

Special issue on Modeling with Measures

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2015, 12(2): i-ii doi: 10.3934/mbe.2015.12.2i +[Abstract](1591) +[PDF](93.4KB)
Abstract:
Different communities met in the research workshop Modeling with Measures" that took place at the Lorentz Center (Leiden, The Netherlands) during 26th--30th of August 2013. They were groups of researchers active in the following fields:

2015, 12(2): 233-258 doi: 10.3934/mbe.2015.12.233 +[Abstract](2185) +[PDF](675.0KB)
Abstract:
We study a quasilinear hierarchically size-structured population model presented in [4]. In this model the growth, mortality and reproduction rates are assumed to depend on a function of the population density. In [4] we showed that solutions to this model can become singular (measure-valued) in finite time even if all the individual parameters are smooth. Therefore, in this paper we develop a first order finite difference scheme to compute these measure-valued solutions. Convergence analysis for this method is provided. We also develop a high resolution second order scheme to compute the measure-valued solution of the model and perform a comparative study between the two schemes.
2015, 12(2): 259-278 doi: 10.3934/mbe.2015.12.259 +[Abstract](2534) +[PDF](643.2KB)
Abstract:
In the present note we discuss in details the Riemann problem for a one-dimensional hyperbolic conservation law subject to a point constraint. We investigate how the regularity of the constraint operator impacts the well--posedness of the problem, namely in the case, relevant for numerical applications, of a discretized exit capacity. We devote particular attention to the case in which the constraint is given by a non--local operator depending on the solution itself. We provide several explicit examples.
We also give the detailed proof of some results announced in the paper [Andreianov, Donadello, Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop], which is devoted to existence and stability for a more general class of Cauchy problems subject to Lipschitz continuous non--local point constraints.
2015, 12(2): 279-290 doi: 10.3934/mbe.2015.12.279 +[Abstract](1847) +[PDF](378.6KB)
Abstract:
In this note we explain by an example what we understand by a balance situation and by a balance equation in terms of measures.
The latter ones are an attempt to start modelling of (not only) diffusion-reaction or mass-conservation scenarios in terms of measures rather than by derivatives and other rates.
By means of three examples this concept is extended to two-features (= two-traits-) balance situations, which, e.g., combine features like aging and physical motion in populations or physical motion and formation of polymers by means of a single model equation.
2015, 12(2): 291-310 doi: 10.3934/mbe.2015.12.291 +[Abstract](1998) +[PDF](491.3KB)
Abstract:
The ideas and techniques developed in [12,3] are extended to a basic stage structured model. Each strategy consists of two stages: a Juvenile (L for larvae), and Adult (A). A general model of this basic stage structure is formulated as a dynamical system on the state space of finite signed measures. Nonnegativity, well-posedness and uniform eventual boundedness are established under biologically natural conditions on the rates. Similar to [12] we also have the unifying of discrete and continuous systems and the containment of the classic nonlinearities.
2015, 12(2): 311-335 doi: 10.3934/mbe.2015.12.311 +[Abstract](1998) +[PDF](9396.4KB)
Abstract:
We prove existence and uniqueness of solutions, continuous dependence from the initial datum and stability with respect to the boundary condition in a class of initial--boundary value problems for systems of balance laws. The particular choice of the boundary condition allows to comprehend models with very different structures. In particular, we consider a juvenile-adult model, the problem of the optimal mating ratio and a model for the optimal management of biological resources. The stability result obtained allows to tackle various optimal management/control problems, providing sufficient conditions for the existence of optimal choices/controls.
2015, 12(2): 337-356 doi: 10.3934/mbe.2015.12.337 +[Abstract](1911) +[PDF](1189.9KB)
Abstract:
Focusing on a specific crowd dynamics situation, including real life experiments and measurements, our paper targets a twofold aim: (1) we present a Bayesian probabilistic method to estimate the value and the uncertainty (in the form of a probability density function) of parameters in crowd dynamic models from the experimental data; and (2) we introduce a fitness measure for the models to classify a couple of model structures (forces) according to their fitness to the experimental data, preparing the stage for a more general model-selection and validation strategy inspired by probabilistic data analysis. Finally, we review the essential aspects of our experimental setup and measurement technique.
2015, 12(2): 357-373 doi: 10.3934/mbe.2015.12.357 +[Abstract](1760) +[PDF](459.2KB)
Abstract:
We consider a linear diffusion equation on $\Omega:=\mathbb{R}^2\setminus\overline{\Omega_\mathcal{o}}$, where $\Omega_\mathcal{o}$ is a bounded domain. The time-dependent flux on the boundary $\Gamma:=∂\Omega_\mathcal{o}$ is prescribed. The aim of the paper is to approximate the dynamics by the solution of the diffusion equation on the whole of $\mathbb{R}^2$ with a measure-valued point source in the origin and provide estimates for the quality of approximation. For all time $t$, we derive an $L^2([0,t];L^2(\Gamma))$-bound on the difference in flux on the boundary. Moreover, we derive for all $t>0$ an $L^2(\Omega)$-bound and an $L^2([0,t];H^1(\Omega))$-bound for the difference of the solutions to the two models.
2015, 12(2): 375-392 doi: 10.3934/mbe.2015.12.375 +[Abstract](1892) +[PDF](832.4KB)
Abstract:
In this article, we present a simplified model to describe the dynamics of two groups of pedestrians moving in opposite directions in a corridor. The model consists of a $2\times 2$ system of conservation laws of mixed hyperbolic-elliptic type. We study the basic properties of the system to understand why and how bounded oscillations in numerical simulations arise. We show that Lax-Friedrichs scheme ensures the invariance of the domain and we investigate the existence of measure-valued solutions as limit of a subsequence of approximate solutions.
2015, 12(2): 393-413 doi: 10.3934/mbe.2015.12.393 +[Abstract](1930) +[PDF](844.6KB)
Abstract:
Based on an established mathematical model for the behavior of large crowds, a new model is derived that is able to take into account the statistical variation of individual maximum walking speeds. The same model is shown to be valid also in traffic flow situations, where for instance the statistical variation of preferred maximum speeds can be considered. The model involves explicit bounds on the state variables, such that a special Riemann solver is derived that is proved to respect the state constraints. Some care is devoted to a valid construction of random initial data, necessary for the use of the new model. The article also includes a numerical method that is shown to respect the bounds on the state variables and illustrative numerical examples, explaining the properties of the new model in comparison with established models.

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