Exponential Growth and Decay

MODEL GUIDE INVESTIGATIONS REFERENCE FOR TECHIES

Introduction

This is a simple population model using an exponential growth model. The rate of increase (or decrease) of a population is assumed to be proportional to the current population.

Rate of Change of Population= Rate_Constant * Population

Rate of Change of Z = A * Z

This model applies to all processes where the rate of change of a parameter is proportional to the parameter value: money in a fixed rate bank account, economic growth, radioactive decay,...

Here are the parameters that affect the performance of the model:

Population (POP)
Population Level Default value = 1
Net Birth Rate (NET_B_RATE)
(Birth Rate - Death Rate) of Organism (1/time unit) Default value = .2

Exponential Growth and Decay Model



Step
Interval
 Run
Interval
 Elapsed
Interval
 Random%
Variation

NET_B_RATE
POP

Guide To The Model

Getting Started
Go to the Model, press the "Set defaults" button to initialize the model with typical starting values and then press the Make A Run button to run the model. Wait for the results to be calculated. The run time may be set to any desired time interval.
Investigations
Think First!
Think of the real-life situation that might follow the model on this page. Make a list of factors that affect the real-life situation.
Population
  • What causes changes in this parameter. What makes it increase? What makes it decrease? What effect does an increase (or decrease) have on the rest of the system?
    Make a sketch that shows how the various factors influence the outcomes. ? What are the differences and similarities?
    Default Run
    Make a long-term run starting with the default values. Make a table and record all output parameters (blue boxes) after each activation of the "run" button. Explain the changes that you observe in the numbers and discuss how the numbers relate to model behavior. Does the model behave realistically? What are unrealistic features of the model? Assume the default settings of the model relate to a real case, describe the history of the case based on your model values.
    Sharpening Your Intuition
    Pick one input parameter (green boxes) and change its value (increase, decrease). Run the model for each new parameter setting and record all output parameter values (blue boxes). Does the model respond to changes as you would expect?
    • Testing The Model
    You cannot just assume that the numbers computed by a quantitative model have any predictive value. Follow the tests suggested by the attached guide to verify that the model passes the basic tests of:
    • Reproducibility: Running the model starting from the same conditions should give identical results. Running with smaller step sizes should yield the same calculated result.
    • Sensitivity: Randomize your starting values. Find out by what percent your calculated result changes. Huge changes in the calculated results for small changes in starting values may mean the model may be unstable. Reduce the Run Interval and the Step Interval and see if you observe more continuous change in your model performance.
    Basic Rate Equations The following equations are used to compute the changes that occur in values that are shown on the screen.

    • Rate of Population = NET_B_RATE * POP

    Reference

    This model is based on "Exponential Growth and Decay" from Modeling and Simulation, Harmut Bossel, A. K. Peters, 1994.

    Technical Details

    This model uses a fourth-order Runge-Kutta methods with fixed step size to solve the model's dynamic rate equations. The random number generator is seeded by the current time, so that the sequences of random numbers do not repeat. A Gaussian or Normal distribution is simulated for the "Randomize" button by adding 48 uniformly-distributed random numbers. These are generated and new model parameter values are calulated assuming a starting value taken from the screen (screenValue) and a distribution standard deviation (SD) equal to "Random%" (SD = screenValue * Random% / 100.0). The model on-screen values may be changed during the course of a run at any time to more realistically simulate changing conditions.

    Some effort has been made to provide modest bulletproofing for this model. Choice of unusual parameter values may cause the model to delay for large amounts of time and possibly cause your WWW browser to freeze. If this happens, you may need to restart your computer!

    Copyright © 1998 Gallaudet University

    All Rights Reserved
    03-01-2000

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