# Spatial Mechanistic Models

contrary to the phenomenological models, mechanistic models are based on ecological parameters of individual system components, which are defined in­dependently. The result of the simulation is determined by these parameters, by the interaction characteristics of the plants involved, and by the model as­sumptions.

The first category of such models is reaction-diffusion models, which describe the behavior through partial differential equations. The most simple dispersion model describes a homogeneous population, which spreads without restraint, and, thus exponentially on a two-dimensional surface:

d2u d2u dx2 + dy2

Here u(x, y, t) is the density of the plants at location (x, y) at time t, and E is the rate of the movements of the individuals of a population, also called the diffusion. In plants this occurs only with the seed; usually E is assumed to be a normal distributed random variable. There are many points of criticism with respect to reaction diffusion models, since some interaction forms cannot be described. However, the models have been successfully used in some places. Often the growth of a species is the combination of the growth of many local populations. The total system in this case is called a meta-population. The so- called population dynamic metapopulation models attempt to describe such systems. These are individual based simulation models, presented through the difference equation: n n

Ni(t + 1) = Ni(t) ri (t) — Eij + Eji, (3.22)

j=1 j=1

where n describes the number of populations and Ni (t) the number of individ­uals in population i at time t. The parameter ri(t) describes the growth rate of the population i at time t. Eij is the number of individuals changing from pop­ulation i to population j. Saturation and the development of new populations can be brought about using simple extensions.

Metapopulations divide the total population into subpopulations, and thus par­tition the surface of the total population. If the borders of the subpopulations do not align, however, with the heterogeneity, due to environmental factors of the soil and other local conditions, errors can occur. Discrete cellular automata can avoid this effect. They are particularly used in places where local effects have strong influence on a population. The models divide the surface into a discrete field of cells that can in each case accept a finite number of conditions (0,1,…, n). In order to obtain the value of cell i at time t + 1, a transition rule Ci(t + 1) is defined, which is dependent on the cur­rent condition of the cell and other cells (usually the geometrical neighbors):

Ci(t +1) = F (Ci(s),Cj (s)). (3.23)

Here s <t is an earlier point in time and Cj symbolizes the adjoining cells (see also ). These models are used in many places in ecology, where, aside from their simple implementation, the advantage is that many different data sources can be included in the simulation without large expenditure. Both the demographic models in their discrete form and the metapopulation models produce the positions and attributes of individual plants, and thus are also discrete models. The original data is empirically assessed through enu­meration methods in nature . By means of simulation methods, we attempt to copy the propagation procedures and interactions between species and to determine individual positions for plants. In Chap. 8 two implementations of such discrete models are discussed more deeply.