CSci 418/618, Spring, 2002

Continuous Simulation - Fundamental Differential Equation Models

A system is continuous if the activities of the system cause smooth changes in the system states. Typically, system states in continuous systems are defined by sets of algebraic, difference, or differential equations. For some simple cases, it is possible to analytically solve the equations in closed form. However, in most applications, closed form solutions are impossible, leaving numerical methods as the alternative. These methods invoke discrete approximations of the continuous process. There are many examples of simulations of this type, and several simulation languages that have been developed to solve them. We describe some fundamental models that serve as building blocks for the continuous systems simulation approach below.

Simple exponential growth. This model fits situations in which the rate at which a state variable x changes with respect to time is directly proportional to the value (or level) of that variable. A common example is an investment that grows at a specified interest rate. If the interest rate is an estimate of future performance of the investment, as it would be with a stock for example, then the differential equation below that models simple exponential growth is actually a simulation model.

dx/dt = k x

Intuitively, the rate at which x increases over a time increment is in direct proportion to the current value of x, so that for a fixed time increment size, the magnitude of the growth is ever increasing, assuming that the constant k is positive. There must also be a place to start, i. e., there must be an initial condition given by:

x = x₀ when t = 0

The solution to this differential equation is:

x = x₀ exp (kt)

To get a feeling for the solution, consider a table of the solution values for x₀ = 1; t = 0.0, 0.5, 1.0, 1.5, 2.0, …, 6.0:

t	0	0.5	1	1.5	2	2.5	3	3.5	4	4.5	5	5.5	6
k = 0.2	1.00	1.11	1.22	1.35	1.49	1.65	1.82	2.01	2.23	2.46	2.72	3.00	3.32
k =0.4	1.00	1.22	1.49	1.82	2.23	2.72	3.32	4.06	4.95	6.05	7.39	9.03	11.02
k =0.6	1.00	1.35	1.82	2.46	3.32	4.48	6.05	8.17	11.02	14.88	20.09	27.11	36.60

A corresponding graphical representation of the solution for the different values of k is given by:

Note that under exponential growth, ln,(x), the natural logarithm, increases linearly with t. If the constant k is negative, the values of x will decrease as time increases, a process known as exponential decay. The following table and graph illustrate exponential decay.

t	0	0.5	1	1.5	2	2.5	3	3.5	4	4.5	5	5.5	6
k = -0.2	1	0.90	0.82	0.74	0.67	0.61	0.55	0.50	0.45	0.41	0.37	0.33	0.30
k = -0.4	1	0.82	0.67	0.55	0.45	0.37	0.30	0.25	0.20	0.17	0.14	0.11	0.09
k = -0.6	1	0.74	0.55	0.41	0.30	0.22	0.17	0.12	0.09	0.07	0.05	0.04	0.03

Modified exponential growth. In some cases there is a maximum value or "ceiling" on the value that the variable x can assume. Modified exponential growth refers to the growth rate being proportional to the difference between the maximum value and current value of x. The maximum value can also be viewed as a “goal” to which the system consistently strives. The following differential equation models the situation:

dx/dt = k (X -x) with

x - x₀ when t = 0

With x₀ = 0, the solution is given by:

x = X ( 1 - exp (-kt) )

In marketing, modified exponential growth is used a basic model of how many units of a product might be sold over time, under circumstances in which there is a well-understood maximum number of total buyers. For example, suppose that there are 100 million households in a population, each of which is a potential buyer for cable modems for high-bandwidth access to the internet. The following table shows values of 100 (1 - exp (-kt)) and the associated, showing the "market penetration" for cable modems over time under modified exponential growth. Note how the marketing adage "Sales rate drops as market penetration increases" fits the behavior of this model.

t	0	1	2	3	4	5	6	7	8	9	10	11	12
k = 0.2	0	18.13	32.97	45.12	55.07	63.21	69.88	75.34	79.81	83.47	86.47	88.92	90.93
k =0.4	0	32.97	55.07	69.88	79.81	86.47	90.93	93.92	95.92	97.27	98.17	98.77	99.18
k =0.6	0	45.12	69.88	83.47	90.93	95.02	97.27	98.50	99.18	99.55	99.75	99.86	99.93

Logistics curves. In some situations there is a known maximum value for the variable of interest, but it is also the case that early on the process the growth depends fundamentally on the level of the variable of interest. Only later in the growth process does the influence of the maximum value become dominant. Such a model would be similar to exponential growth early on, and similar to modified exponential growth later. This might be appropriate,, for example, in modeling market penetration of a product for which sales start slowly while the product is becoming known, and later become more dependent on the maximum market penetration. Other applications include population growth of a species, and modeling the spread of a disease. Such situations can be modeled by logistics functions, as described by the following differential equation:

dx/dt = k x ( X - x)

Note that when x is much smaller than X, (X - x) is close in value to X itself, so that the equation is approximated by dx/dy = k X x, which is the exponential growth model with constant k X. For larger values of x that are close to X in value, the equation is approximated by dx/dt = k X ( X - x ) , which is modified exponential growth with a constant of kX. The true closed form solution to the differential equation is complicated, because it is not linear in x, so we do not provide it here. In simulation practice, continuous simulations are rarely solvable in closed form, and require a numerical method. The following graph illustrates a typical type of solution for a logistics function.