Introduction to the Modeling and Analysis of Complex Systems

4.6. BUILDING YOUR OWN MODEL EQUATIONS WITH MULTIPLE VARIABLES 57PreyPopulation : xNaturally growsto carrying capacityif isolatedPositive influencePredators’ growth rateis higher withmore prey+-Negative influencePrey’s survival rate+ -is lower withmore predatorsPredatorsPopulation : yNaturally decaysif isolatedFigure 4.7: Interactions between the prey and predator populations illustrated in acausal loop diagram.The inherent dynamics of the two variables are quite straightforward to model. Sincewe already know how to model growth and decay, we can just borrow those existingmodels as building components, like this:x t = x t−1 + r x x t−1 (1 − x t−1 /K) (4.28)y t = y t−1 − d y y t−1 (4.29)Here, I used the logistic growth model for the prey (x) while using the exponential decaymodel for the predators (y). r x is the growth rate of the prey, and d y is the death rate ofthe predators (0 < d y < 1).To implement additional assumptions about the predator-prey interactions, we need tofigure out which part of the equations should be modified. In this example it is obvious,because we already know that the interactions should change the death rate of the preyand the growth rate of the predators. These terms are not yet present in the equationsabove, so we can simply add a new unknown term to each equation:x t = x t−1 + rx t−1 (1 − x t−1 /K) − d x (y t−1 )x t−1 (4.30)y t = y t−1 − dy t−1 + r y (x t−1 )y t−1 (4.31)Now the problems are much better defined. We just need to come up with a mathematicalform for d x and r y .