Johan Grasman

Asymptotic methods for the Fokker-Planck equation and the exit problem in applications

I The Fokker-Planck Equation.- 1. Dynamical Systems Perturbed by Noise: the Langevin Equation.- 2. The Fokker-Planck Equation: First Exit from a Domain.- 3. The Fokker-Planck Equation: One Dimension.- II Asymptotic Solution of the Exit Problem.- 4. Singular Perturbation Analysis of the Differential Equations for the Exit Probability and Exit Time in One Dimension.- 5. The Fokker-Planck Equation in Several Dimensions: the Asymptotic Exit Problem.- III Applications.- 6. Dispersive Groundwater Flow and Pollution.- 7. Extinction in Systems of Interacting Biological Populations.- 8. Stochastic Oscillation.- 9. Confidence Domain, Return Time and Control.- 10. A Markov Chain Approximation of the Stochastic Dynamical System.- Literature.- Answers to Exercises.- Author Index.

A two-component model of host–parasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control

A two-component differential equation model is formulated for a host-parasitoid interaction. Transient dynamics and population crashes of this system are analysed using differential inequalities. Two different cases can be distinguished: either the intrinsic growth rate of the host population is smaller than the maximum growth rate of the parasitoid or vice versa. In the latter case, the initial ratio of parasitoids to hosts should exceed a given threshold, in order to (temporarily) halt the growth of the host population. When not only oviposition but also host-feeding occurs the dynamics do not change qualitatively. In the case that the maximum growth rate of the parasitoid population is smaller than the intrinsic growth rate of the host, a threshold still exists for the number of parasitoids in an inundative release in order to limit the growth of the host population. The size of an inundative release of parasitoids, which is necessary to keep the host population below a certain level, can be determined from the two-component model. When parameter values for hosts and parasitoids are known, an effective control of pests can be found. First it is determined whether the parasitoids are able to suppress their hosts fully. Moreover, using our simple rule of thumb it can be assessed whether suppression is also possible when the relative growth rate of the host population exceeds that of the parasitoid population. With a numerical investigation of our simple system the design of parasitoid release strategies for specific situations can be computed.

Stochastic epidemics: major outbreaks and the duration of the endemic period.

A study is made of a two-dimensional stochastic system that models the spread of an infectious disease in a population. An asymptotic expression is derived for the probability that a major outbreak of the disease will occur in case the number of infectives is small. For the case that a major outbreak has occurred, an asymptotic approximation is derived for the expected time that the disease is in the population. The analytical expressions are obtained by asymptotically solving Dirichlet problems based on the Fokker-Planck equation for the stochastic system. Results of numerical calculations for the analytical expressions are compared with simulation results.

Reconstruction of the seasonally varying contact rate for measles

With the extended Kalman filter the time dependent contact rate for measles in the SEIR-model is reconstructed from data of the incidence of this infectious disease in the city of New York. It is concluded that although these data show through the years an irregular change in the number of infected children the contact rate is definitely periodic and follows the season. The analysis gives improved values of the parameters in the SEIR-model for this special problem.

Stochastic epidemics: the expected duration of the endemic period in higher dimensional models

A method is presented to approximate the long-term stochastic dynamics of an epidemic modelled by state variables denoting the various classes of the population such as in SIR and SEIR model. The modelling includes epidemics in populations at different locations with migration between these populations. A logistic stochastic process for the total infectious population is formulated; it fits the long-term stochastic behaviour of the total infectious population in the full model. A good approximation is obtained if only the dynamics near the equilibria is fit.

On local extinction in a metapopulation.

Abstract A method is presented to express for a metapopulation the expected local extinction time in an area in the parameters of the full model for the metapopulation. For that purpose the long-term stochastic dynamics of the local population is approximated by a stochastic logistic process fitting the behavior of the local population as it acts in the full system. Moreover, a formula for the occupation rate of the area is derived.

Dispersive Groundwater Flow and Pollution

In this chapter we apply the asymptotic approximation method of the previous chapter to the problem of arrival of contaminants in the groundwater at a pumping well. For the study of groundwater pollution it is not sufficient to model the transport of particles by advection only. In addition the mechanism of macroscopic dispersion has to be taken in consideration. It accounts for the random motion of individual particles in the flow. In this study we assume that the hydrodynamic dispersion is proportional to the velocity with coefficients a L in the longitudinal direction and a T in the transversal direction, see Bear and Verruijt (1987). We analyse the properties of the random walk model for particles from the Fokker—Planck equation. The concentration function for a pollutant is interpreted as the space-time probability density function for a contaminated water particle as it makes a random walk. We will take a L and a T constant. The method also applies to spatial heterogeneous dispersion constants, see Van Kooten (1996).

Sustainable yields from seasonally fluctuating biological populations

An algorithm is developed to determine the optimum sustainable yield subject to the condition that the resulting solution has a stable limit cycle. Periodic harvesting is considered and the algorithm is applied to a logistic population and a prey-predator system.

The dynamics of addiction : Craving versus self-control

This study deals with addictive acts that exhibit a stable pattern not intervening with the normal routine of daily life. Nevertheless, in the long term such behaviour may result in health damage. Alcohol consumption is an example of such addictive habit. The aim is to describe the process of addiction as a dynamical system in the way this is done in the natural and technological sciences. The dynamics of the addictive behaviour is described by a mathematical model consisting of two coupled difference equations. They determine the change in time of two state variables, craving and self-control. The model equations contain terms that represent external forces such as societal rules, peer influences and cues. The latter are formulated as events that are Poisson distributed in time. With the model it is shown how a person can get addicted when changing lifestyle. Although craving is the dominant variable in the process of addiction, the moment of getting dependent is clearly marked by a switch in a variable that fits the definition of addiction vulnerability in the literature. Furthermore, the way chance affects a therapeutic addiction intervention is analysed by carrying out a Monte Carlo simulation. Essential in the dynamical model is a nonlinear component which determines the configuration of the two stable states of the system: being dependent or not dependent. Under identical external conditions both may be stable (hysteresis). With the dynamical systems approach possible switches between the two states are explored (repeated relapses).

Reconstruction of the drive underlying food intake and its control by leptin and dieting

The intake of food and the expenditure of calories is modelled by a system of differential equations. The state variables are the amount of calories stored in adipose tissue and the level of plasma leptin. The model has as input a drive that controls the intake of food. This drive consists of a collective of physiological and psychological incentives to eat or to stop eating. An individual based approach is presented by which the parameters of the system can be set using data of a subject. The method of analysis is fully worked out using weight data of two persons. The model is prone to extensions by transferring incentives being part of the input to the collection of state variables.