Mathematical Modeling by Meerschaert Mark M

Author:Meerschaert, Mark M. [Meerschaert, Mark M.]
Language: eng
Format: epub
ISBN: 9780123869968
Publisher: Elsevier Science
Published: 2013-01-28T00:00:00+00:00

(6.17)

on the state space x1 ≥ 0, x2 ≥ 0, where x1 denotes the population of blue whales and x2 the population of fin whales. In order to simulate this model, we will transform to a set of difference equations

(6.18)

over the same state space. Then, for example, Δx1 represents the change in the population of Blue whales over the next Δt years. We will assume that α = 10−8 to start with, and later on we will do a sensitivity analysis on α. Our objective is to determine the behavior of solutions to the discrete time dynamical system in Eq. (6.18) and compare to what we know about solutions to the continuous time model in Eq. (6.17).

In step 4 we solve the problem by simulating the system in Eq. (6.17) using a computer implementation of the Euler method for several different values of h = Δt. We assume that x1(0) = 5, 000 and x2(0) = 70, 000, as in Example 6.2. Figure 6.27 illustrates the results of our simulation with N = 50 iterations and a time step of h = 1 years. In 50 years the fin whales grow back steadily but do not quite reach their eventual equilibrium level. In Figure 6.28 we increase the step size to h = 2 years. Now our simulation with N = 50 iterations shows the fin whale population approaching its equilibrium value. Using a larger time step h is an efficient way to project further into the future, but then something interesting happens.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.