In this module we continue looking at the three models you looked at in the previous module.
p' = 0.50 (1 - 0.005 p - 0.01 q) p
q' = 0.50 (1 - 0.01 p - 0.005 q) q
p' = 0.50 (1 - 0.01 p - 0.005 q) p
q' = 0.50 (1 - 0.001 p - 0.002 q) q
p' = 0.50 (1 - 0.005 p + 0.002 q) p
q' = 0.50 (1 + 0.002 p - 0.005 q) q
The key to understanding these models lies in the coefficients circled below. These coefficients represent how sensitive the relative growth rates for p and q are to the p and q populations. The coefficients circled in red represent the intraspecies interactions and the coefficients circled in blue represent the interpsecies interactions.
In the first model the interspecies interactions are stronger than the intraspecies interactions, so this model represents a very aggressive kind of interaction.
In the third model the interspecies coefficients are positive. This means that each species has a positive effect on the relative population growth rate for the other species. This model represents two species that cooperate in some way or that have a mutually symbiotic relationship.
The second model is a bit more subtle. If we look at the difference between the relative population growth rate for the species q and the relative population growth rate for the species p we see that
is always positive. This means that the relative population growth rate for the species q is always greater than the relative population growth rate for the species p and that q is a stronger or more vital species. This is an example of weak-strong interaction.
Click here to open a new window with four Java applets. When the window is open arrange the two windows so that they overlap and you can move back-and-forth between them by clicking on the exposed portion of the inactive window to make it active.
This window contains four Java applets that are similar to the Java applet in the last module. In fact, the first of the Java applets in this window is identical to the Java applet at the end of the last module. The four Java applets simulate lakes and currents for the four models above -- the competitive model, the aggressive model, the weak-strong model, and the coopoerative model. At the end of the last module you were supposed to draw rough diagrams by hand and determine the behavior of these models on the basis of your diagrams. Now you can check your work by "dropping corks" at various different initial points in these applets.
If necessary modify your description of these models from the previous module.
