What is a simulation model?

Thus far, you have seen how to describe the structure of models by diagrams. To perform simulations to find out how variables change over time, we must describe cause and effect relationships by mathematical equations. You do not need to understand fully the mathematics of simulation models in the rest of the course. However, try to understand the logic of the equations. If you do, you will feel more confident about simulation models, they will no longer appear “mysterious”.

Here you will see a full description of the population model for an animal, for which you saw a CLD in step 1.3. The above SFD completes the CLD by distinguishing between instantaneous and accumulating relationships. To be able to simulate the population behavior, the model also needs exact descriptions of the mathematical equations. In this table you see mathematical descriptions of all the relationships in the population model.

Before discussing each relationship, a few words about how the model simulates. As a start, recall question 2 in graphical integration about the size of the change in the stock variable.

Stock after 5 hours = Average net flow * 5 hours + Initial Stock

These are operations that the computer can do repeatedly and very fast. However, there is a problem with graphical integration. In interlinked systems, you do not know what the flows will be into the future. When “everything influences everything” the flows depend on the stock variables that you have not yet calculated. Nor can the computer know what the flows will be before the stocks are known.

However, there is a trick that solves this problem. The computer can calculate the stock development for very small time intervals, with length dt. Over such small time intervals, the flows that are caused by the stocks will not change much. Therefore, as an approximation, the last estimate of the flow values at time t-dt can be used to update the stock value. In fact, if you make dt short enough, the simulation program will simulate with whatever accuracy you need. This method is called Euler integration. In the above equations listing you see the exact equation that the simulation program uses to calculate the next stock value at time t.

Stock(t) = (Births(t-dt) – Deaths(t-dt)) * dt + Stock(t-dt)

To be clear, t denotes the time when the next value of the stock will be calculated while t – dt denotes the last known values of the stock and the flows. In the simulation program dt will typically be very short compared to the time horizon for the simulation. It follows from the Euler equation that in the initial year of the simulation, the stock must be given an initial value. In the equation listing you see that INIT Population is equal to 2 animals. Before simulating one must also specify the time horizon for the simulation, the stop time, and the length of dt.

Once you understand how the stocks update from time to time, the rest is simple. This is because all other variables including the flows depend instantaneously on the updated values of the stocks.

You have already seen examples of mathematical descriptions of instantaneous relationships in step 1.4 on stock and flow diagrams. For instance, recall the linear equation for the cost of apples in a bag:

Cost = Price * Apples + Cost of bag

Some of the model equations are logical and are simply defined by the modeller. Births are given by the Fractional birth rate times the size of the Population. If each animal in the population gives 2 births per year, the birth rate will be 2 births per animal per year times the population size. Check that the same logic applies to the equation for the death rate.

The carrying capacity is given by a constant. This value will stay the same throughout the simulation. The size of the carrying capacity should be based on knowledge about the availability of food, weather conditions etc., or on historical observations of some upper limit for the population. It is important to be critical of assuming that carrying capacity is constant. You may ask, will the carrying capacity change if the animal population destroys the food source? Still, assuming a constant carrying capacity can be useful in a process of learning by adding complexity step by step. Crowding is given by the ratio between the population and the carrying capacity.

The relationships for the Fractional birth and the Fractional death rates are more complicated. These relationships are not given by definition. Both of them reflect knowledge about how crowding influences births and deaths. This knowledge could originate from observations of birth and death rates at different population and crowding levels. Information could also be provided by laboratory experiments. Such information typically follows from different sciences that take systems apart in order to study the parts in detail. Such knowledge is very useful when you create a model to study the behavior of a complete system.

The relationships for the Fractional birth and the Fractional death rates could have been formulated by some mathematical functions of Crowding. The tiny graphs in the equations listing suggest that these equations could be very complicated. Therefore, simulation software typically allow you to formulate the relationships by graph functions. The equations listing show the data points in these graph functions. For instance, for a crowding of 0.000 the fractional birth rate is 1.213, a crowding of 0.500 gives a slightly lower fractional birth rate of 1.170 and so on and so forth. In very crowded situations, the fractional birth rates are zero and the fractional death rates become very high.

These non-linear relationships are sufficient to simulate the model, which we will do in the next step.

To summarize:

Formulations for stocks are very simple, either net flows are added to or subtracted from the current stock values. Still, it is the stock behaviors that people have the greatest difficulty dealing with. Recall the tendency people have to think in terms of correlations and instantaneous relationships. This is why it is important to learn and practice graphical integration. In WEEK 2 you will learn more about heuristics people use to simplify complex problems, heuristics that often lead to biases and poor policies.

Formulations for instantaneous cause and effect relationships can be very simple and very complicated. They may rely on logic, definitions, prior information, or empirical data. They can be linear or nonlinear. In spite of how difficult it may be to formulate representative mathematical functions of instantaneous relationships, they are closer to the way people tend to think about cause and effect than stock accumulation.

Finally, here is a quick addition for those who know mathematics, and in particular calculus. A System Dynamics simulation model is used to study continuous change – just as calculus. There are two main differences. System Dynamics models are presented in a less demanding language than calculus. And, the need for mathematical knowledge is much less since System Dynamics models are simulated. Hence, there is no need to solve complex differential equations. Similar to calculus, System Dynamics is a language in which the laws of nature can be represented, understood, and used to improve our understanding of natural and human systems. A great advantage of simulation models is that they produce reliable results for highly nonlinear functions, which calculus fails to deal with. (My experience from learning how to solve differential equations was that I got more occupied with solving equations than to understand the time behavior and importance for decision-making.)