Eine Modellierungsaufgabe zum Thema: "Munterer Partnertausch beim Marienkäfer"

Wachstums - bzw. Populationsmodelle sind in der Biomathematik weit verbreitet. Ausgehend von einer aktuellen Studie zur Fortpflanzung der zweigepunkteten Marienkäfer-Spezies Adalia Bipunctata werden zwei mathematische Modelle vorgestellt, die Schülerinnen und Schüler der Sekundarstufe II im Rahmen einer sogenannten Modellierungswoche erarbeitet haben. Der Ansatz der Schüler basiert in beiden Fällen im Wesentlichen auf einer zeitdiskreten Rekursion, wobei ein Ansatz sich mit dem direkten Aufschreiben der Rekursion beschäftigt und der andere aus einer Skizze zum möglichen Verlauf der Populationen hergeleitet wird. Im Folgenden wollen wir nun die Aufgabenstellung präzisieren und auf weitere wichtige Aspekte des Modellierungskreislaufes (Kaiser 1996, S. 68) wie z.B. das Beschaffen von Daten und das Herleiten eines Modells näher eingehen

A micro-macro hybrid model with application for material and pedestrian flow

In this paper, a hybrid modeling approach for granular flow-like applications is presented. The approach allows to switch for a priori fixed points in time between the different levels of description which are the microscopic and macroscopic scale, respectively. Based on the numerical discretization of the models, the switching procedure is able to interpret information on individual objects as density distributions and vice versa. In particular, the reverse direction, i.e. from a macroscopic to a microscopic perspective, requires the solution of a nonlinear least squares problem subject to further constraints. Simulation results are given and demonstrate the good performance of the algorithm in the case of material and pedestrian flow models

Microscopic and Macroscopic Traffic Flow Models including Random Accidents

We introduce microscopic and macroscopic stochastic traffic models including traffic accidents. The microscopic model is based on a Follow-the-Leader approach whereas the macroscopic model is described by a scalar conservation law with space dependent flux function. Accidents are introduced as interruptions of a deterministic evolution and are directly linked to the traffic situation. Based on a Lax-Friedrichs discretization convergence of the microscopic model to the macroscopic model is shown. Numerical simulations are presented to compare the above models and show their convergence behaviour.Comment: 32 pages, 6 figure

Density dependent diffusion models for the interaction of particle ensembles with boundaries

The transition from a microscopic model for the movement of many particles to a macroscopic continuum model for a density flow is studied. The microscopic model for the free flow is completely deterministic, described by an interaction potential that leads to a coherent motion where all particles move in the same direction with the same speed known as a flock. Interaction of the flock with boundaries, obstacles and other flocks leads to a temporary destruction of the coherent motion that macroscopically can be modeled through density dependent diffusion. The resulting macroscopic model is an advection-diffusion equation for the particle density whose diffusion coefficient is density dependent. Examples describing i) the interaction of material flow on a conveyor belt with an obstacle that redirects or restricts the material flow and ii) the interaction of flocks (of fish or birds) with boundaries and iii) the scattering of two flocks as they bounce off each other are discussed. In each case, the advection-diffusion equation is strictly hyperbolic before and after the interaction while the interaction phase is described by a parabolic equation. A numerical algorithm to solve the advection-diffusion equation through the transition is presented.Comment: 25 pages, 9 figure

The food seeking behavior of slime mold: a macroscopic approach

Starting from a particle model we derive a macroscopic aggregation-diffusion equation for the evolution of slime mold under the assumption of propagation of chaos in the large particle limit. We analyze properties of the macroscopic model in the stationary case and study the behavior of the slime mold between food sources. The efficient numerical simulation of the aggregation-diffusion equation allows for a detailed analysis of the interplay between the different regimes drift, interaction and diffusion.Comment: 23 pages, 11 figure