Run sensitivity analysis for this model with respect to the Half-saturation parameter SS. What do you observe when SS varies in the [0.5, 1.5] interval? How can you explain that?
|Run 1 in red is SS = 0.5; run 5 in purple is SS = 1.5|
As SS increases, the system converges toward the stable, non-trivial equilibrium more rapidly; that is, increasing SS dampens the extremes of rabbit and wolf populations as they approach their equilibrium values. The equilibrium populations are unaffected. This can be explained on several levels.
Mathematically, since SS is in the denominator of the predation function, as SS increases, predation slows, which effectively stabilizes the system.
Thinking about the system ecologically, if wolves consume rabbits very rapidly—that is, the slope of V vs. x is steep, which is a consequence of lower values of SS—then an increase in rabbit populations will produce a rapid upward swing in wolf population, because predation will increase their numbers before wolf mortality stabilizes their population. As a result of the increased number of wolves, the rabbit population will plummet, which will lead to a wolf die-off, and the cycle will repeat. On the other hand, when SS is large, predation is slow, so the wolf population reacts more gradually to fluctuations in the rabbit population, which effectively stabilizes both populations.