Inhaltszusammenfassung:
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.
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.
Alignment , Nonlinear mathematical model , Systems of hyperbolic partial differential equations , Qualitative analysis
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