Lecture 20

Topics

• Intraspecific vs. Interspecific competition

• The Competition Coefficient

• Graphical methods of modeling

Study Sources

• Chapter 21 in Ricklefs; especially pp. 403-414.

• Study Guide: Competition

The Main Points

• The logistic model for a single population shows us the effect of intraspecific competition on the biotic potential. What we will do for the next couple of weeks is consider the effects of adding a second species, i.e. an interspecific competitor.

• Ricklefs shows how Volterra modified the logistic equation to add this effect (Equations 21-2 to 21-4). Ricklefs uses a slightly different notation for the competition coefficient than I presented in lecture. Whereas, I used a to show the "effect of Species 2 on Species 1", Ricklefs uses a_ij (the "effect of Species j on species i"). Similarly, I used ß to show the "effect of Species 1 on Species 2 and Ricklefs uses a_ji to show the effect of Species i on Species j. The latter notation is that commonly used in ecological modeling because it simplifies the list of symbols for the model. In this case, we see a single symbol used for a class of variables (in this case, competition coefficients) where the magnitudes and/or direction of the effect are different.

• I like the a and ß notation here because it tends to emphasize the point that the reciprocal effects of the two species upon one another need not be equal. In fact, you wouldn't expect two species to utilize any resource supply in exactly the same way. Therefore, when we examine the effects on the biotic potential (r_max) of Species 1 when growing in the presence of Species 2, we need to add both the effect of adding one individual of Species 1 and the Species 1 equivalent of Species 2 to compare with K₁. Read pp. 407-409. Ricklefs does a superb job of discussing this problem.

• Since we are considering a case of two species utilizing a common resource which is in short supply, we need a second equation to show the growth rate of the Species 2. Note how the subscripts change in the second equation.

• We can solve these simultaneous equations for equilibrium values of the two variables (N₁ and N₂ by graphing these equations for the condition when both species have a zero rate of growth (i.e. when the biotic potentials of both are 0). Read especially pp. 410-413 and study Fig. 21-4. Here Ricklefs shows effect of competition on the the growth rate of Species i by adding a third axis to show the effect on r_i. I didn't do this in lecture, but it's an effective way to visualize the dynamics.

• Near the end of the lecture we discussed the kinds of outcomes that one might expect when two species come into competition. Four outcomes were possible at equilibrium:
  1. Species 1 could "win" all encounters.
  2. Species 2 could "win" all encounters.
  3. Each Species could "win" a fraction of the encounters.
  4. The two species could coexist.

• I finished the lecture by developing the first case, where Species 1 "wins" all encounters. This is identical to Fig. 21-5 where Ricklefs presents the case for Species i. Note especially how Ricklefs shows the dynamics of the competition with the vectors resolved in each sector of the graphic.

  See also Fig. 21-6 where Ricklefs presents this same model accompanied with growth curves for each species. Each of the growth curves shows how each species departs from the growth to be expected (the dotted lines) if each was growing alone.

• Now, you should develop graphical models for the other three cases.