Research for Computational Methods

This paper considers a special class of characteristic first-order initial value problems. The initial value problem arises in the asymptotic solution of parabolic and elliptic equations. The problem is characterized by a singular, characteristic initial manifold. Namely, initial data is given on a characteristic curve. It is proven that such problems have unique solutions. The theorem also has an interesting interpretation of the calculus of variations.

In this paper, the following type of harvesting problem is considered. An animal population is divided into two stocks: an 'underlying' population and a 'surface' population. It is assumed that there is a natural exchange between the two population levels. The predator or harvestor affects only the 'surface' population and does not influence the 'underlying' population directly. Such a situation occurs, for example, in the off-shore Eastern Tropical Tuna Fishery. In this case, tuna associate with porpoise schools. The fishery harvests only those tuna associated with porpoise. Consequently, the underlying population of tuna is not sampled by the fishery. One may wonder what information measurements on the surface, harvested population provides about the unobservable underlying population. Furthermore, it is interesting and important to know if the standard, linear relationship between harvest and effort is valid in an aggregating population.

This paper considers the effects of small random perturbations on deterministic systems of differential equations. The systems of interest have a steady state that is a saddle point. A first exit problem is formulated. The quantity of basic interest is the probability of exit from a band around the deterministic separatrix through a specified boundary, conditioned on initial position. A technique for the approximate calculation of this probability is given. As an example, it is shown how the theory applies to the calculation of the probability of victory in a combat situation that has a stochastic component.