An approach to the residence time distribution for stochastic multi-compartment models

Stochastic compartmental models are widely used in modeling processes such as drug kinetics in biological systems. This paper considers the distribution of the residence times for stochastic multi-compartment models, especially systems with non-exponential lifetime distributions. The paper first derives the moment generating function of the bivariate residence time distribution for the two-compartment model with general lifetimes and approximates the density of the residence time using the saddlepoint approximation. Then, it extends the distributional approach to the residence time for multi-compartment semi-Markov models combining the cofactor rule for a single destination and the analytic approach to the two-compartment model. This approach provides a complete specification of the residence time distribution based on the moment generating function and thus facilitates an easier calculation of high-order moments than the approach using the coefficient matrix. Applications to drug kinetics demonstrate the simplicity and usefulness of this approach.     

Speed of invasion in lattice population models: pair-edge approximation

 We propose a simple approach to approximating the speed of invasion in lattice population models. Approximate critical parameter values for successful invasion are then found by solving for zero wave speed. The approximation is based on describing the occupied region by the ordinary pair approximation, and using quasi-steady-state pair approximations to describe the leading edge of the wave front. We illustrate this idea using the basic contact process on the 1 and 2 dimensional lattice (with and without nearest-neighbor migration), finding very good agreement between the approximation and simulation results. The approximate critical values obtained by our approximation are significantly more accurate than those obtained by the ordinary pair approximation. Received 4 September 1996     

Optimal harvesting of a logistic population in an environment with stochastic jumps

Dynamic programming is employed to examine the effects of large, sudden changes in population size on the optimal harvest strategy of an exploited resource population. These changes are either adverse or favorable and are assumed to occur at times of events of a Poisson process. The amplitude of these jumps is assumed to be density independent. In between the jumps the population is assumed to grow logistically. The Bellman equation for the optimal discounted present value is solved numerically and the optimal feedback control computed for the random jump model. The results are compared to the corresponding results for the quasi-deterministic approximation. In addition, the sensitivity of the results to the discount rate, the total jump rate and the quadratic cost factor is investigated. The optimal results are most strongly sensitive to the rate of stochastic jumps and to the quadratic cost factor to a lesser extent when the deterministic bioeconomic parameters are taken from aggregate antarctic pelagic whaling data.Research supported in part by the National Science Foundation under grants MCS 81-01698 and MCS 83-00562.     

Genetic parameters of body weight in sheep estimated via random regression and multi-trait animal models

Variance components for five consecutive measurements of body weight in Polish sheep were estimated using random regression and multi-trait animal models. The data included between 7856 and 31694 body weight records at 5 age classes from birth to 150 days of age. The random additive genetic, maternal environmental and individual permanent environmental effects were fitted. All variance components were increasing over time (not at equal rates), which reflects increasing phenotypic mean and variance with age. Direct heritability tended to increase with age, whereas the effect of dam was reduced for older ages, and the proportion of permanent environmental component was relatively stable. Generally, similar tendencies were registered for estimates obtained via multi-trait animal model. The results confirm that there is a scope for genetic improvement in growth pattern in Polish sheep.