Introduction to Nonlinear Dynamics

Author: D. Kaplan et al.
Publisher:
Publication Date: 1997-09-01
Features: Excerpt: Finite-Difference Equations 1.1
An MYTHICAL FIELD
Imagine that a graduate student goes to a meadow on the first day of May, walks through the meadow waving a flynet, and counts the number of flies caught in the net. She repeats this ritual for several years, following up on the work of previous graduate students. The resulting measurements might look like the graph shown in Figure 1.1. The graduate student notes the variability in her measurements and wants to find out if they contain any important biological information. Several different approaches could be taken to study the data. The student could do statistical analyses of the data to calculate the mean value or to detect long-term trends. She could also try to develop a detailed and realistic model of the ecosystem, taking into account such factors as weather, predators, and the fly populations in previous years. Or she could construct a simplified theoretical model for fly population density. Sticking to what she knows, the student decides to model the population variability in terms of actual measurements. The number of flies in one summer depends on the number of eggs laid the previous year. The number of eggs laid depends on the number of flies that lived during that summer. Thus, the number of flies in one summer depends on the number of flies in the previous summer. In mathematical terms, this is a relationship, or function. This equation says simply that the number of flies in summer \( t \) is determined by (or is a function of) the number of flies in the previous summer \( t-1 \). Equations of this form, which relate values at discrete times (e.g., each May), are called finite-difference equations. \( N_t \) is called the state of the system at time \( t \). We are interested in how the state changes in time: the dynamics of the system. Since the real-world ecosystem is complicated and since the measurements are imperfect, we do not expect a model like Eq. 1.1 to be able to duplicate exactly the actual fly population measurements. For example, birds eat flies, so the population of flies is influenced by the bird population, which itself depends on a complicated array of factors. The assumption behind Eq. 1.1 is that the number of flies in year \( t \) depends solely on the number of flies in year \( t-1 \). While this is not strictly true, it may serve as a working approximation. The problem now is to figure out an appropriate form for this dependence that is consistent with the data and that encapsulates the important aspects of fly population biology.