Lecture 26

Topics

• MacArthur and Rosensweig change the nature of the model

• Exploring the stability of the interactions.

Study Sources

• Chapter 23 in Ricklefs; especially pp. 466-468.

• Study Guide: Predation

The Main Points

• MacArthur and Rosensweig increased the "biological reality" of the Lotka-Volterra models by imposing environment limits on the both the predator and the prey (Figures 23-19 and 23-20).

• Note in these cases that the "isoclines" (the zero-growth lines for the prey and predator) are made to bend to impose K-values for both species.

• The effect of bending the isoclines is that the new geometrical relationships (i.e. no longer perpendicular to one another) impose trajectories that either converge if the predator isocline intersects the prey isocline to the right or diverge if the predator isocline intersects the prey isocline to the left of the "peak" of the prey isocline curve. This "peak" is associated with a density of prey that is about 1/2 of its K-value. In terms of growth rate or dN/dt of the prey, what is happening at K/2?

• On the right, the interaction is stable and spirals inward toward the intersection of the two isoclines because reduction of the prey population when it is near to K, results in the prey population producing more offspring. And, since the predator must have a high population of prey to make a living, it is bounded quite close to the intersection. This makes the prey population more resiliant, i.e. able to "bounce-back" immediately from having its population reduced.

• The opposite effect occurs on the left when the intersection of the isoclines is near the lower limit of the prey population. At these prey densities, any decrease in their numbers causes the prey population to have a lower rate of growth, i.e. dN/dt for the prey is small when the population is small). The prey is no longer resiliant to the predation and interaction is unstable. This shows up in the outward spiraling trajectories around the potential equilibrium point.

• What is happening though at an intersection of the isoclines near the K/2 peak of the prey isocline? Notice that the geometry of the intersection is most like that of the limit-cyle of the original perpendicular isoclines.