Introduction to the Modeling and Analysis of Complex Systems

56 CHAPTER 4. DISCRETE-TIME MODELS I: MODELINGPreyPopulation : xPredatorsPopulation : yNaturally growsto carrying capacityif isolated+ -Naturally decaysif isolatedFigure 4.6: Inherent dynamics of each of the prey and predator populations illustratedin a causal loop diagram.• Predators die if there are no prey.These assumptions can be diagrammatically illustrated in Fig. 4.6.This type of diagram is called a causal loop diagram in System Dynamics [26]. Eachcircle, or node, in this diagram represents a state variable in the system. The self-looparrow attached to each node represents the effect of the variables on itself (e.g., the moreprey there are, the faster their growth will be, etc.). The plus/minus signs next to thearrows show whether the effect is positive or negative.We can now consider the interactions between the two variables, i.e., how one variableinfluences the other and vice versa. Naturally, there should be the following effects:• The prey’s death rate increases as the predator population increases.• The predators’ growth rate increases as the prey population increases.These interactions can be illustrated as arrows between nodes in the causal loop diagram(Fig. 4.7).Now is the time to translate the structure of the system illustrated in the diagram intomathematical equations. Each arrow in the diagram tells us whether the effect is positiveor negative, but they don’t give any exact mathematical form, so we will need to create amathematical representation for each (possibly using the aforementioned tips).