CASE STUDY: The Herbivore–Algae Predator Prey Model

Source:

Dynamic Modeling, by Bruce Hannon

Goal:

To build a computational model of the relationship between a plant (algae) which grows at a certain rate and a herbivore, an animal which eats plants (in this case algae!) at a certain rate.

Overview of the model:

In this model, we are interested in the amount of algae and the number of herbivores in a closed system. Algae can do one of two things: it can grow at some rate, and it can be consumed by herbivores. Herbivores, on the other hand, do what most animals do -- they are born and they die.

Set your model up as a classic population dynamics model for each of the two populations. Use the values below for your model:

Initial number of algae: 210
Initial number of herbivores: 45

Algae growth rate: algae are born at a rate that depends on the number of algae. Use these numbers to build a graph of algae population to growth rate:

Algae population	Growth Rate
0	0.21
100	0.168
200	0.112
300	0.0902
400	0.0781
500	0.066
600	0.0572
700	0.0462
800	0.0363
900	0.0198
1000	0.000

Look first at the Algae section. The growth portion we have seen before. The growth rate is a function of the Algal density ("Algae"). This function is monotonic and declining. The Algal growth is just the product of the density and the growth rate. The Algae density is reduced through consumption by the herbivore. The consumption per head is a nonlinear function of the Algal density: the greater the density, the higher the consumption per head. The consumption rate is simply the product of the number of herbivore and the consumption per head.

The herbivore death rate is determined by their average life span which is a nonlinear function of the consumption per head: the higher the consumption per head, the longer the life span, within limits. Indirectly, the denser the algae, the lower the herbivore death rate. The herbivore growth rate is a product of the herbivore stock and the fractional herbivore growth rate FHG (fcn_herb_grow). To complicate matter (toward realism), I make the FHG a function of the algae density in the previous time period. This is done by producing an additional stock called "algae_min1". The program simply stores the Algae density in the current period into the algae_min1 box. That value is dumped in the next period by the variable called flow 1. We could continue this process indefinitely by creating more and more of these delaying stocks. In general, it makes sense to represent herbivore behavior in this way. Herbivore gestation time reflects the origin of this lagged behavior.

The first graph shows the wide swings in algal density and herbivore population. The second graph, a plot of the density vs the population, shows the limit cycle resulting from this particular choice of the variables.

Now you try changing things. Can you make the herbivore crash and not re–emerge? Can you adjust only the variable "fcn_herb_grow" , without changing the maximum and minimum rates, to maximize the average herbivore population? Can you induce chaos into this model by increasing the lag time?

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