Developmental Models

During the simulation of the evolution of a plant population, we must decide in each case to what point of accuracy one wants to control the system be­ing produced, and how much effort is acceptable. The appropriate techniques discrete methods ^ are divided into continuous and discrete methods based on individuals (see [7, 76]). A comparison of both approaches, with special consideration of the propagation of alien plants in ecological systems is found in [86].

Simple Demographic Models

With a demographic model the future number of individuals of a population is predicted. In the simplest case, growth is an exponential function, whereby its

Подпись: Section 3.6 DEVELOPMENTAL MODELS change is proportional to the current number of plants. This case arises if noth­ing obstructs the propagation of a species. An exponential model is described by the following differential equation:








Подпись: (3.17)Here N represents the size of the population and t the time. The parameter r is the growth rate of the population and consists of different factors such as the maximum age, the reproduction rate per year, and the survival rate. Usually it is empirically determined, whereby precise measurements are difficult due to special conditions.

The logistic model limits environmental resources as additional boundary con­ditions. The growth rate is described here as a function of the current size of the population and its maximum size in the equilibrium state K:



Here the growth rate reduces itself continuously in the proximity of the equi­librium, until it levels off to the size K. In this model the density-dependent development of the population is not yet considered. Additionally, K is depen­dent upon the size of the plants. In Chap. 8 this effect is taken into consideration in the practical simulation of a population.

Подпись: logistic modelПодпись: logistic difference model

Подпись: Nt Developmental Models Подпись: Nt Подпись: (3.18)

In order to move from the differential equations of the above models to growth algorithms, these equations must be discretised. With dt =1, and the current size represented using Nt, after a time interval dt, we obtain in a logistic dif­ference model, the future quantity Nt+1:

The continuous models in general can be better described mathematically than the discrete models; however, for the graphical simulation of smaller land­scapes they are indispensable, since the individual locations of plants are needed here.

In the previous models it is assumed that the environmental conditions are con­stant over space and time and that there is no change in the behavior of the population. Variability is, however, a substantial component of natural systems, therefore the logistic difference model must be changed additionally using ran­dom parameters.

Подпись: stochastic modelOne obtains a stochastic model:


Nt+i = Nt + Nt(r + y(ti)) – r~K, (3.19)

where y(t) is a time-dependent variable with an average value of zero and a variance of one. Comparisons between deterministic and stochastic models

Подпись: regression modelПодпись: geometric modelПодпись: Markov modelПодпись: reaction-diffusion modelПодпись: du ~dt seem to show that stochastic models tendentiously predict a smaller population density than deterministic models, and that the difference represents a function of the variance of y(t) [86].