<p/>Predacious mites are considered to be important natural enemies of phytophagous mites. Their efficiency in the natural control of prey populations depends on the relationships of the number of prey killed per predator per time unit and the oviposition rate on the one hand and prey density on the other hand. These relationships determine the functional and the numerical response of the predator population to the prey density. The shape of these responses indicates to what extent the mortality of the prey population will be raised by an increase of the prey density. When the mortality rate is increased to such an extent that it exceeds the reproduction rate of the prey population at a certain value of the prey density, a necessary condition for the regulation of the prey would be satisfied. Through the operation of such a regulating mechanism the predator and the prey population may remain present at low levels, which gives protection against outbreaks of harmful prey species.<p/>Curves of the functional response of <em>Typhlodromus occidentalis</em> females to the density of <em>Tetrapychus urticae</em> eggs and males on disks of bean leaf have been obtained experimentally by Kuchlein (in preparation). The curves do not correspond with the fundamental types distinguished by Holling (1959a, 1961). This study is an attempt to explain these curves and to find out on which conditions <em>T. occidentalis</em> can regulate prey species such as <em>T.</em><em>urticae.</em> For that purpose simulation models were constructed to compute the relevant components of the behaviour of the predator and the prey, the state variables of the predator-prey system, and the resulting predation rates at different prey densities. In this system approach different system elements are distinguished. The components of behaviour considered are the proportion of encounters between the predator and the prey resulting in a capture (the success ratio), the length of handling periods, the coincidence in space, and the locomotion activity and velocity of the predator and the prey. The state variables considered are the engagement of the predator in searching and handling prey, the actual prey densities, the hunger of the predator, and the density of the webbing cover produced by the <em>T.</em><em>urticae</em> males. A quantitative measure of the degree of satiation of the predator was found in the gut content; a measure for the relative density of the webbing cover in the proportion of grains free from the leaf's surface when the leaf disk is sprayed with a fine powder.<p/>To compute the gut content in the simulation of the predation process it is necessary to know the rate of ingestion when the predator feeds on a prey, and the rate of evacuation of food from the gut (denoted by digestion). Feeding experiments with radioactive prey revealed that the ingestion rate is not constant, but depends on the gut content of the predator and on the availability of food in the prey. The rate is higher when the prey is a <em>T.</em><em>urticae</em> egg, than when it is a male. The maximum gut content was computed to be 1.08 prey egg equivalents, the maximum food contents of eggs and males found are 0.94 and 0.67 egg equivalents respectively. The digestion rate is assumed to be proportional with the gut content. The coefficient of this proportionality, obtained from the restoration of the behaviour during a long digestive pause, measured 0.435 hour <sup>-1</sup>.<p/>The relationship between the relative density of the webbing cover and the total distance walked by the males on the leaf disk has been defined quantitatively. The production of webbing is gradually reduced, or the males tend to walk along the same pathways.<p/>Leaf disks were prepared with a standardized adult female <em>T. occidentalis</em> (with an empty gut and two days old), or with a normal ovipositing female, and a different number of <em>T.</em><em>urticae</em> eggs or males. These disks were observed continuously and the events occurring were recorded chronologically. The locomotion velocity of the predator and the males, and the locomotion activity of the males was measured during this observation. From the records of events a computer program derived for a series of gut-content classes, the mean values of the resting period and the walking period of the predator, the coincidence in space and the success ratio for different encountering situations, the proportion of disturbance in encounters of active males and a resting predator, and the handling and feeding periods.<p/>The velocity of the standardized predators did not differ significantly before and after feeding on the first prey (1.15 and 1.12 m/hour respectively).<p/>Oviposition by the predator slightly increases the predation rate by raising the success ratio, but reduces the locomotion velocity. The success ratio of standard predators was four times higher than the success ratio in Kuchlein's experiments, which was accounted for in the simulation models.<p/>The multiple relationships between the components of behaviour and the state variables were evaluated by partial correlation analysis and polyfactor analysis. The latter method provided quantitative descriptions of the relevant, curvilinear relationships by an iterative procedure for multiple regression. These descriptions were used in the models for simulation of the predation process.<p/>CSMP programs for three kinds of models are described to simulate a predation process on a computer, each dealing with chance variables in a different way. A deterministic model gives erroneous results as soon as the predation rate is a curvilinear function of stochastic variables, whereas a stochastic model consumes too much computer time. An intermediate approach applies deterministic simulation to classes of the stochastic variables in a hypothetical population of predators. The classes in this method of compound simulation are chosen in such a way that within the classes the relationship with the predation rate is approximately linear. The classes contribute to the expectation value of the output variables on the base of the relative frequency distribution of the predators over the classes. The three types of models were applied to simulate the predation process with prey eggs only. The most comprehensive model for the predation of prey eggs and males was built according to the principles of compound simulation. It computes expectation values of the numbers of prey eggs and males destroyed, the biomass consumed, the gut content and occupation of the predator, the actual prey densities, the instantaneous prey mortality and prey utility, and the relative density of the webbing cover as a function of time, the prey density maintained, and the prey replenishment interval. Compound simulation can be applied to study all sorts of stochastic processes.<p/>The results of simulation lead to the following conclusions with respect to <em>T. occidentalis</em> and <em>T. urticae</em> :<p/>All functional response curves obtained by simulation deviate from the fundamental types of Holling. These deviations are discussed and explained with the help of the values of the components of behaviour at the different prey densities.<p/>Hunger exerts an important influence on predation. The success ratio decreases with increasing gut content, which relationship is the main determinant of the functional response to prey density. There is no distinct threshold-value of the gut content for the evocation of attacks. Hunger induces a long feeding time and a high ingestion rate, and hence a better utilization of the prey captured. At all hunger levels a prey egg provides more food than a prey male.<p/>In mites, which are deprived of food for a different number of days, the difference in hunger level will be negligible. Differences in behaviour demonstrated by such mites will not be effected by hunger, but rather by starvation.<p/>The success ratio seems to be reduced by a high frequency of encounters with prey. Probably predation by mites can be inhibited at a high prey density by stimulus satiation.<p/>Webbing produced by the prey reduces the encountering rate of the predator and the prey, the locomotion velocity and activity of the predator, and the handling time for prey males. It increases the velocity of the prey males, but in general reduces the predation rate, because it has a distinct barrier effect. Other factors related to prey density are the disturbance by active males and the aggregation of the males. Disturbance reduces the mean period of resting and extends the mean period of walking, increasing the predator activity and the encountering rate. Disturbance of feeding predators does not affect the success ratio. Aggregations of males reduce the rate of encounters between the predator and active males, and the activity of the males. Satiation, prey aggregation and the production of webbing are the main factors determining the shape of the functional response of <em>T.</em><em>occidentalis</em> to the density of <em>T. urticae</em> males on leaf disks.<p/>The prey mortality rate and the prey utility depend on the gut content of the predator. The ratio of the mortality rate or the utility of different prey species is not constant, but changes with the gut content. This induces switching from one kind of prey to another at increasing prey density.<p/>The influence of prey replenishment on the functional response curves is almost negligible for an exposition time of six hours. In general the predation rate will be underestimated for low prey densities when the prey captured is not replenished in experiments.<p/>The number of eggs laid per 24 hour by the predator is about four times its mean gut content. This linear relationship between the oviposition rate and the gut content implies a numerical response of the predator population to prey density, which is most intensive at prey densities below five eggs or males per cm <sup>2</sup>.<p/>The combined effect of the functional and the numerical response to prey density is indicated by the prey risk induced by the next generation of predators as a function of current prey densities. Such total response curves show a steep rise to a maximum of 2.4 prey eggs or 2.0 prey males per cm <sup>2</sup>, and a gradual decline beyond these prey densities. It has been computed that an equilibrium density of at least 0.006 predators per cm <sup>2</sup>at a prey density below 2.4 eggs per cm <sup>2</sup>is a necessary condition for 'short term' regulation of the prey population by predation of eggs. A high predator mortality can inhibit such a regulation. The expectation value of the reproductive period of the newborn predators must not be lower than 0.95 days for predation on eggs, and 1.92 days for predation on males.<p/>Some conclusions derived in this book concern general aspects of predation:<p/>It seems to be indispensable in the study of predation to take account of the stochastic character of predation processes with a single predator. Deterministic simulation models can give incorrect results.<p/>In natural systems the functional responses of predators to prey density will be multiform, and very probably most of them will differ from the fundamental types described by Holling (1959a, 1961).<p/>Predators need some time to adapt their mean gut content to a new prey density. The adaptation time is longer at a low prey density, than at a high one. This may lead to an overestimation of the predation rate at low prey densities, when starved predators are used in experiments with a short exposition time.<p/>Hunger dependent switching can contribute to the regulation and stabilization of the prey species preferred at a low hunger level.<p/>The contribution of chemical control measures to the mortality of predacious mites can release populations of their prey species.<p/>It can be advantageous to introduce, simultaneously with the predator, a harmless and rather unattractive alternative prey species.<p/>Finally some suggestions are given to build a submodel of the multiple functional response to the densities of several kinds of prey into models for simulation of the dynamics of predator and prey populations.