in such models the focus no longer is on the single plants and their total number, but on the surface used as well as the growth rate. The density of the individual species is determined here by real measurements in nature. interaction between the species takes place according to an ecological model. in the simplest case, the so-called regression model, the propagation of a population measured over the years and throughout regression, will be illustrated usually with an exponential function. Different studies exist for the propagation of alien plants in ecological systems, in which over many decades the changes were measured. The parameters can at least partially be used for simulation. in a geometrical model the propagation is described by a number of growth centers on an unrestricted two-dimensional surface, which enlarge with a defined rate. The original data for such a model can be obtained again through observation methods.
A so-called Markov model describes the change of a landscape of m cells over the matrix equation
Nt+i = PN„ (3.20)
where Nt = (nt1, …,ntm) is a column vector, that indicates the part of the m cells occupied by the population at the time t, and P is the matrix for the description of the change. The significance of the Markov model is twofold; on the one hand, the change is made dependent only on the current condition of the population, thus the system no longer has a “memory”; on the other hand the transition probabilities in P remain constant over time.