chalkdust

Computing −r.

Computing −f.

with diﬀerent signs. While the rabbit population would grow in the absence of predators due to the unlimited food supply, the fox population would die out when there are no rabbits and thus no food, hence the term −β1 f. The second term, β2 rf, describes the increase in the fox population due to rabbits being caught for food. Equations (3) and (4) can now easily be set up on an analogue computer by creating two circuits, as shown in the diagrams above. The circuit for (3) has two inputs: an initial condition r0 representing the initial size of the rabbit population, and the value rf which is not yet available. The second circuit looks similar with an initial fox population of f0 (please keep in mind that integrators and summers both perform a change of sign that can be used to simplify the circuits a bit, thus saving us from having to use two summers). All that is necessary now is a multiplier to generate rf from the outputs −r and −f of these two circuits. This product is then fed back into the circuits, thereby creating the feedback loop of this simple ecosystem. The setup of this circuit on a classical desktop analogue computer weighs in at 105 kg and requires quite a stable desk! Seing the parameters α1 , α2 , β1 and β2 to suitable values (it is not too easy to find a stable ecosystem) yields an output like the one shown here. One of the most fascinating properties of an analogue computer is its extremely high degree of interactivity: one can change values just by turning the dial of a potentiometer while a simulation is running and the eﬀects are instantaneously visible. It is not only easy to get a “feeling” for the properties of some diﬀerential equations, it is also incredibly addictive, as the following quote from John H McLeod and Suzee McLeod shows: “An analogue computer is a thing of beauty and a joy forever.”

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58

Results of the predatorprey simulation. Prey are on the top and predators on the boom.

Chalkdust, Issue 03

Popular mathematics magazine from UCL

Chalkdust, Issue 03

Popular mathematics magazine from UCL