If a system is modeled by a differential equation, and if the equations are numerically solved on a digital computer or calculator, the system is said to be simulated on the computer. If the model is valid and the numerical methods accurate, experiments can be performed on the computer simulation that might be impossible to conduct otherwise.

Consider several examples that use the models already discussed. If a population is governed by a linear first-order equation

one would not be able to "solve" this equation on a computer. If, however, we use Euler's method as was done in (14) by approximating the derivative as

where time is considered at intervals of

This gives for (1) .EQ (34)

If we include the time interval

then (3) becomes

which is now in a form that one can easily calculate successive values
of

Next, consider the nonlinear equation that models a population with a simple limit given by (23).

Using Euler's method again gives

This equation is complex enough to illustrate several points;
therefore, we will examine several numerical solutions.
Equation 8 was programmed on a Tektronix 31 programmable calculator
with a plotter automatically plotting the solutions by drawing straight
lines between successive

First, consider a low-density growth rate
of

The curves in Figure A are the output of the simulation for the above parameters and also for other growth rates of 5% and 20%. Note the solution always approaching the same limit but requiring different amounts of time.

In Figure B, the model is run assuming several different initial
populations.
Again, the solutions always approach the limit of

Figure C shows the effects of various amounts of limiting by
considering various values for the factor

These examples illustrate the kinds of questions that can be pursued
by running experiments on the computer simulation.
There is one more point that should be considered.
It has nothing to do with the differential equation model (23) but with
the numerical procedure, Euler's method.
Consider the effects of using various time intervals

One last point should be made concerning this numerical simulation. Euler's method is the only approach to numerically solve (23) that has been discussed. That is not because it is the best – there are far more efficient and sophisticated methods – but that is not our subject here, so we will continue with the straightforward algorithm of Euler.

The super-exponential logistic equation of (31) was simulated on the
calculator and run with a low population growth rate
of