Modelling and Simulation of Ecology Systems

I have done some research concerning nonlinear partial integro-differential equations. These equations can be used to model systems with delay effects. Common examples might be ecology systems in an environment with a limited carrying capacity or viscoelasticity. This includes e.g. models with stress applied to a viscoelastic material. It is possible to simulate creep with such models. Creep is the tendency of a solid material to slowly move or deform permanently under the influence of stresses.

The modelling and simulation of ecology systems is a nice field to combine approaches more related to computer science like for example agent based models with equation based models. To give an example: If one uses a model that is based on partial differential equations the model will provide information about the density of a population in a given area. Even if the area is in a way discretized by an FEM-Mesh the result will be a continuous model for the area and the population. Thinking of bacteria, blood cells or krill it is a very good approach. Thinking of the Siberian tiger this will not make sense in all models. Depending on what you would like to simulate the density of such a rare species won't lead you anywhere. But if we consider the interaction of whales and krill it might make sense to combine the different worlds.

A spatial population model based on a nonlinear partial integro-differential equation

A spatial population model with a delay term is modelled by this Nonlinear Partial Integro-Differential Equation.

We call u the density of a population simulated over an environment with a limited carrying capacity and a memory or delay effect. The domain is a kind of square island. The reproduction rate is quite high but not unrealistic thinking e.g. of the European Rabbit with one doe having an offspring of about thirty young rabbits a year. The Figure illustrates the behavior of the solution. The initial condition are four peaks, a little bit later the population density starts to spread on the domain. Later on the effect of the high population at the beginning takes place and the density falls at the initial peak regions. After some ups and downs the density u becomes very smooth over the whole domain with a time changing amplitude.

last modified by Joerg Frochte on March 1st 2010