17.2: Phase Diagrams

The principles can be visualized in phase diagrams, with arrows showing how populations change. Earlier, Figure 10.11 showed strong competition with coexistence at equilibrium, as could be the case if the species were competing for two different resources. Nonetheless, each species restricts the other to much lower levels than it could maintain on its own. Weaker competition means that each species is restricted less so each can maintain higher levels, as in Figure 17.1.2. You can tell this is competition because the two parameters \(s_{1,2}\) and \(s_{2,1}\) are both negative. This is shown in the negative slopes of the two diagrams in the upper left. The carrying capacity of each species together is only slightly reduced from the individual carrying capacities, which would be about 1.2 for Species 1 living alone and about 0.8 for Species 2 living alone. (for example, an average of 1.2 individuals per square meter, or 1,200,000 individuals per square mile if measured in millions). However, for species living together, the carrying capacity of each is slightly reduced, perhaps 10 to 20 percent.

In this case, by the way, the two species together have a higher total population than would be the case if either was living alone. This is called “over yielding,” and is a recurrent theme in studies of plant communities.

Figure 17.1.3 shows a similar situation, but now with the inter-species interaction terms \(s_{1,2}\) and \(s_{2,1}\) both positive, shown by the positive slopes in the two upper-left diagrams of the figure. It looks quite similar to Figure 17.1.2, but the two together are each more abundant than they would be apart— the joint equilibrium is larger than the individual carrying capacities.

This joint equilibrium can be computed from the \(r_i\) and \(s_{i,j}\) parameters. It will occur where the growth of each species simultaneously reaches 0. You can find the numerical value for this equilibrium with pencil and paper by setting the first species growth rate to 0, solving for the populations of Species 1, substituting that into the equation for Species 2, and solving for when the growth of that species reaches 0. Alternatively, you can pose the problem to a symbolic mathematics program and ask it to solve the two equations simultaneously. In any case, you would start with both growth rates set to zero at equilibrium,

\[\frac{1}{N_1}\frac{dN_1}{dt}\,=\,r_1\,+\,s_{1,1}N_1\,+\,s_{1,2}N_2\,=\,1.2\,-\,1N_1\,+0.1N_2\,=\,0\]

\[\frac{1}{N_2}\frac{dN_2}{dt}\,=\,r_2\,+\,s_{2,2}N_2\,+\,s_{2,1}N_1\,=\,0.8\,-\,1N_2\,+0.1N_1\,=\,0\]

and end up with \(N_1\) = 1.2929 and \(N_2\) = 0.929.

As mutualisms become stronger—meaning that the interspecific interactions become more positive—the equilibrium point moves further out. It can be very large, as in in Figure 17.1.4, but, in restrained mutualism, the equilibrium is finite and computable from the parameters of the individual species.

On the other hand, when the inter-species enhancement terms are stronger still, a bifurcation occurs and the joint equilibrium ceases to exist at all. (Figure 17.1.5). The calculated equilibrium point has, in effect, moved to infinity, or in a sense beyond, meaning that the carrying capacity cannot be computed from the parameters of the species and their interactions. Some further information is needed about the system.

Beyond this, the mutualists can become more dependent on each other, so that the \(r_i\) terms become smaller, as in Figure 17.1.6, or negative, as in Figure 17.1.7. The mutualism can be unrestrained even if the intrinsic growth rates \(r_i\) are negative. What arises is a kind of Allee point, where the populations run away if they start above that point, but decline to extinction if they start below.