Prey and Predator

We can use a fuzzy cognitive map to model the behaviour of an ecosystem formed by three elements: predators, preys and food for preys. An example of such a system could be a grassland area with zebras and lions. The predators feed on the preys and the preys feed on the prey's food, which can be grass, crops or other preys.

This system is a generalization and extension of the Grazing Fields sytem. If you have not already done so, it is advisable that you have a look first at the Grazing Fields fuzzy cognitive map here.

The graphical representation of the food-prey-predator system is as follows:

The "+" arrows indicate positive causality of the source factor onto the target factor. The "-" arrows indicate negative causality. The cause factor and the effect factor in a causality relationship are indicated by the direction of the arrow.

1. Loading the system

The Prey and Predator fuzzy cognitive map has been stored in the applet. To load the system, click the Load System button, then select the system and click Load in popup dialog box.

The three factors of the system are displayed in the main window of the application. Factors can be added, removed and edited by using the labelled buttons in the window.

2. Setting the parameters of the sytem

Let's suppose that the food-prey-predator system is used to represent an ecosystem formed by grassland, zebras and lions.

The biologists that study the ecosystem have determined that the sustainable and average levels of the three factors are a coverage of one half of the land by grass, 10 zebras per square kilometer of grassland and a ratio of lions and zebras of 20:1.

In our fuzzy cognitive map we can represent these values with levels of 50, out of a maximum of 100, for each of the three factors.

Our model is not very sophisticated, and we will not try to fine-tune it. Therefore we assume that the intensity of the causality relationships is the same for the four relationships. We set the intensities of effects at 50 in all cases.

3. Running the simulation

To run the simulation, we select the Run until convergece option and then click the Run simulation button in the main window. The results of the simulation are shown in a popup window. We see that the system is stable. There is no variation in the levels of the parameters.

4. Effects of variations of prey's food

We can expect that a variation in the level of prey's food will have an effect in the balance of the system.

A drop in the extension of grassland (prey's food) is represented by setting a lower level for that factor. The level of the factor can be modified by selecting the factor in the main window of the applet and then clicking the Edit button. In the Factor Editor dialog box, the level is modified with a scrollbar.

We set the level of the prey's food at 30, and then we run the simulation. In the results, we see that the level of prey's food drops quickly until it becomes zero. The population of preys cannot be sustained by the lower than average level of prey's food and consequently the population of preys also diminish and disappears, which in turn is followed by the extintion of predators.

An increase in the amount of prey's food, which can be represented by an initial level of 60, has the opposite effect. The prey population is not enough to consume the food. In absence of enough prey, the prey's food increaes steadily. The prey population also increases, pulled by the higher availability of food, and the predator population increases as well. The prey population does not catch up with the increase of prey's food, and therefore the prey's food increases to its maximum extent possible. The populations of prey and predator grow to match it.

5. Effects of variations of prey population

To see the effect of the variation of prey population on the system, we first drop the prey population to a level of 40, leaving the other two factors at 50, and then run the simulation.

The results show that there is not enough prey population to consume the prey's food, and therefore the prey's food level increases steadily.

There is an initial drop in the predator population because there are not enough preys to sustain its population. However, the steady increase in the prey's food level triggers a raise of the prey population, which in turn stops the drop in predator population.

Eventually, the prey and predator population increase to a higher than average value, pulled by the higher level of prey's food available.

An increase in the prey population can be represented with a level of, say, 60. If we run the simulation with this value (leaving the levels of prey's food and predator population at 50) we see that there is a quick drop in the amount of prey's food, which is not enough to sustain the prey population, and an initial increase in the predator population, pulled by the higher than average initial prey population.

However, the prey population diminishes down to extintion, following the exhaustion of prey's food, and after it comes the extintion of the predators.

6. Effects of variations of predator population

The results of a simulation using initial levels of prey's food, prey population and predator population at 50, 50 and 40, respectively, show that the low population of predators causes an increase in the prey population.

The increase in the prey population cannot be sustained by the average level of prey's food, which is depleted quickly. After the initial increase, the prey population starts decreasing following the lack of food.

The predator population also diminishes and eventually the prey's food, the preys and the predators disappear.

An increase in the initial level of predators, set at 60, shows opposite results. The initial high predator population causes a drop in the prey population. Following the drop in prey population the level of prey's food increases.

The prey population is recovered due to the higher availability of food, so much that it keeps increasing. The predator population also increases, following the high prey population.

Eventually, the prey's food, the prey population and predator population grow to very high levels.

7. Variations in the intensities of effect

In the model described above we have assumed that the intensity of effect is the same for all four causality relationships in the system. We have used an intensity of 50 for all of them. However, this may not be an accurate description of the ecosystem.

The biologists that study the ecosystem may have found that animal populations are strongly dependent on the availability of food for them in the ecosystem, and that they are not so dependent on the presence or absence of predators. An animal population would be undermined by lack of food, but would bear with predators unless the predator population raises too much.

We can modify the intensities of effect in our model to reflect the biologists' findings better. We assign an intensity of 60, out of 100, to the dependence of animals on food, and an intensity of 40 to their dependence on the presence or absence of predators.

The representation of the system is then as follows:

where the intensities of effect are indicated between brackets next to the arrows that represent relationships of causality. As before, the "+" and "-" symbols indicate positive and negative causality, respectively.

If we set the levels of the factor at 50 for all three factors and then we run the simulation we see that the system is stable, as we expected. There is not variation from the initial levels.

Now we can modify the levels of prey's food, prey population and predator population in turn, as we did in the previous sections, and see what the outcome is.

If we run the simulations we see that the results are the same as those we obtained when the intensities of effect were all set at 50. The only difference is in the number of iterations it takes to reach stability in the system (i.e. convergence).

This means that the variations in the intensities of effect do not play a significant role in the fate of the ecosystem.

We could try to modify the intensities of effect further, using values of 20 and 80 rather than 40 and 60, and see what the effect on the behaviour of the system is.

8. Validity of the results

We have seen that, after a variation of the level of any of the factors of the system, the fuzzy cognitive map for the food-prey-predator system shows trends that seem reasonable. However, the final results of the simulations are rather extreme: slight variations from the average values of any of the levels lead either to extintion of all species or to maximum proliferation.

In fact, the model used here is an oversimplification of the real thing. In a real ecosystem there are many more factors than those included in the fuzzy cognitive map. Also, in a real ecosystem variations off the average levels are temporary, often tied to seasonal changes. In our model variations are (wrongly) assumed to be persistent.

The important point to note is that even a fuzzy cognitive map containing only three factors may show somehow interesting trends in the system, subject to the limitations of the model.

If a more accurate description of the reality is needed the model can be refined as much as required.