Biology Reference
In-Depth Information
increment used on the time variable t is often denoted by DT and is 0.5
in this example. Changing the value of DT allows for creating a specific
mesh of points at which the value of the function P(t) will be calculated.
For DT
¼
1, the time values for which P(t) will be calculated will be
¼
¼
¼
t
0, 0.2, 0.4, 0.6, etc.
There are a number of software products that can be used to model and
analyze dynamical systems and obtain numerical solutions of
differential equations, including MATLAB
0,1,2,3, etc. For DT
0.2, the time values will be t
, BERKELEY MADONNA,
W
Stella
, and others. For the rest of the chapter, we refer to the
specific syntax of BERKELEY MADONNA, although any of the above
software packages can be employed instead. A functional version (with
some limitations on saving and printing) of BERKELEY MADONNA can
be downloaded at no charge from the Web site, listed in Internet
Resources at the end of this chapter. The remaining chapters of the text
do not pertain to particular software, although references to relevant
programs are provided at the end of each chapter, where appropriate.
, Vensim
W
W
The initial and final values of the time interval over which we would like
to know the values of the solution should be specified. In BERKELEY
MADONNA, they are called STARTTIME and STOPTIME. In the
example above, we had the values of P(t) calculated over the interval [0,
6], corresponding to STARTTIME
¼
0 and STOPTIME
¼
6.
We nowgive a basic introduction that will allowyou to entermathematical
models in BERKELEY MADONNA and obtain their numerical solutions.
We shall use the models developed in this chapter as examples.
XII. SOME BERKELEY MADONNA SPECIFICS
When you start BERKELEY MADONNA, the screen that will contain your
model appears. There is even some code that has already been written:
METHOD RK4
STARTTIME
¼
0
STOPTIME
¼
10
DT
¼
0.02
The first line specifies the numerical method that will be used by the
program for computing the numerical solution. You can safely ignore
this for now and accept the default algorithm.
5
The remaining lines
5. For those readers familiar with the theory of numerical methods for solving
ordinary differential equations, we would add that BERKELEY MADONNA
allows you to choose from a set of built-in algorithms, including Euler's method
and two types of Runge-Kutta methods. More details on this and other specifics
related to the software can be found in BERKELEY MADONNA's brief
documentation accessible under the Help menu.
