Mathematical modeling in the sustainable use of natural resources.
The sustainable use of natural resources is of utmost importance for every community. In particular, it is important for every given generation to plan in such a way that proper provision is made for future generations. The scientific understanding of resources use and appreciation for its life-supporting capacity is therefore essential. Mathematical modeling has proved useful to inform the planning and management of strategies for sustainable use of natural resources. Some specific topics in resource management has been studied intensively through many decades. In particular, mining, fisheries, forestry and water resources are among these. Instead of presenting a study of the latter topics, this dissertation presents a variety of cases of mathematical modeling in resource management. The aim is to improve the general understanding of the relevant problems. We expand on existing literature, papers of other authors, and add to such studies by focusing on specific items in the work, illuminating it with further explanations and graphs, or by modifying the models through the introduction of stochastic perturbations. In particular this dissertation makes contributions by giving more explanation, on the so-called environmental Fisher information or EFI for brevity (Section 2.4 and Chapter 6), and by introducing stochasticity into a pest control model (Chapter 4) and into a savanna vegetation model (Chapter 5). In Chapter 3 we present a model from the literature pertaining to the problem of shifting cultivation, i.e, the use of forest land when used for subsistence level agricultural purposes, until the land is so degraded that the occupants abandon it and move on to a new stand. The model used to study the shifting period is similar to the forest rotation problem. A model, already in the literature, for biological control of a pest is studied in Chapter 4. Onto the deterministic model we impose a stochastic perturii bation, so that we obtain a stochastic differential equation model. We prove stochastic stability of the disease-free state, when the basic reproduction number of the pest is below unity. We have performed simulations of solutions of the stochastic system. In Chapter 5 we review an existing ordinary differential equation model for the competition between trees and grass in savanna environment. The competition between them is for soil water, fed by annual rainfall. On the other hand, trees and grass are perturbed by fire, and some other environmental forcings such as herbivores. For this ODE model, we introduce stochastic perturbations. The stochastic perturbations are in the form of three mutually independent Brownian motions. Simulations to illustrate the effect of the stochasticity are shown. We present a three-tiered predator-prey model and consider its stability in terms of Fisher information. This appears as Chapter 6. The Fisher information is defined on the basis of the so-called sustainable measures hypotheses. The model is already in the literature and in the dissertation we present several computations to show the influence of carrying capacity of prey and of mortality rate on EFI. Another problem that we consider, in Chapter 7, is that of lake eutrophication caused by excessive phosphorus inflow. The computation illustrates the management of the runoff nutrients into or out of the lake. Necessary and the sufficient conditions for an optimal utility management are obtained using standard optimal control theory. The results of this dissertation demonstrate the modeling techniques in the sustainable use of natural resources. Sustainability is the quest for equal opportunities over all generations. The manner in which this sustainability is quantified in models is being debated and improved all the time. The discourse on sustainability is especially important in view of a growing world population, and with forcings such as climate change. The most important original contribution in this dissertation is the stochastic analysis on the pest control model and the savanna model.