Modellierung von Gruppen sich bewegender, gleich ausgerichteter Tiere und Zellen

Abstract

The striking patterns which can be found in moving polarized groups such as schools of fish or flocks of birds result from a twofold adaptation process: Individuals adapt their orientation of movement to that of their neighbors, a process which is called alignment. Within a moving group individuals also adapt their speed to the speed of the group.

Several models for this behavior are derived. They take the form of systems of nonlinear partial differential equations. First, the speed of an individual is assumed constant and in the simplest model individuals move in one dimensional space. They change direction depending on the direction of their neighbors. Still assuming constant speed, the model is generalized to movement in several space dimensions. Then the speed adaptation process is modeled for movement in one dimension. Finally, the two models for alignment and speed adaptation are combined in one dimension.

The qualitative behavior of solutions is examined analytically and numerically. Analytical results comprise existence of solutions, stability conditions, invariant domains and description of limit sets. Mathematical tools are dynamical systems theory, linear and nonlinear partial differential equations, a priori estimates, Lyapunov functions, vanishing viscosity solutions. Numerical simulations show that the behavior of solutions can be interpreted as schooling behavior of individuals.

The striking patterns which can be found in moving polarized groups such as schools of fish or flocks of birds result from a twofold adaptation process: Individuals adapt their orientation of movement to that of their neighbors, a process which is called alignment. Within a moving group individuals also adapt their speed to the speed of the group.

Several models for this behavior are derived. They take the form of systems of nonlinear partial differential equations. First, the speed of an individual is assumed constant and in the simplest model individuals move in one dimensional space. They change direction depending on the direction of their neighbors. Still assuming constant speed, the model is generalized to movement in several space dimensions. Then the speed adaptation process is modeled for movement in one dimension. Finally, the two models for alignment and speed adaptation are combined in one dimension.

The qualitative behavior of solutions is examined analytically and numerically. Analytical results comprise existence of solutions, stability conditions, invariant domains and description of limit sets. Mathematical tools are dynamical systems theory, linear and nonlinear partial differential equations, a priori estimates, Lyapunov functions, vanishing viscosity solutions. Numerical simulations show that the behavior of solutions can be interpreted as schooling behavior of individuals.