Modeling and Minimization of Extinction in Volterra–Lotka Type Equations with Free Boundaries
✍ Scribed by Srdjan Stojanovic
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 555 KB
- Volume
- 134
- Category
- Article
- ISSN
- 0022-0396
No coin nor oath required. For personal study only.
✦ Synopsis
An equation of the distributed Volterra Lotka type, with free boundary of the obstacle type, with possible applications in ecology, when extinction of the biological species is of particular concern, is introduced and solved. An optimal control problem for such an equation, and in particular the problem of minimization of the area of extinction of the species, is introduced and to some extent solved.
1997 Academic Press
1. Introduction
Modeling of distributed population dynamics was studied in the literature (see [14] and the references given there). The optimal control of distributed population dynamics, by the method of monotone iterations, 1 was introduced by the author and others in [8,9,17]. 2 The crucial assumption in those studies is that the harvesting rate of the species is linear with respect to the size of the population. The consequence of such an assumption in a model is that the species is either totally extinct, or its density is strictly positive in all of the considered region. The partial extinction never occurs.
The present study is about modeling and optimal control of Volterra Lotka type equations with stronger than linear harvesting rates when the size of the population is small. Mathematically, the problem becomes a free boundary problem of the elliptic or parabolic 3 (nonlinear) obstacle type.