With a Carrying Capacity
Real prey cannot grow without limit: food and space run out. Add a carrying capacity and the endless cycles turn into swings that slowly die out — or the predators may not be able to survive at all.
Why the Cycles Die Out
With a carrying capacity, prey growth slows down as their numbers approach K. That acts like a brake on every swing: after each cycle the populations come back a little closer to the equilibrium, and they spiral inwards in the phase plane until they settle.
Predators need enough prey to replace their losses. If the prey would have to be more numerous than K for that, the predators die out and the prey settle at K. The smaller the gap between the prey equilibrium and K, the fewer predators the area can support.
Populations Over Time
yearsPhase Plane
spiralWorked Techniques
The prey equilibrium is set by the predators, as before; the carrying capacity lowers the number of predators.
The eigenvalues of the model at the equilibrium. A negative real part means the swings shrink; an imaginary part means they still oscillate while doing so.
Even with the prey at their maximum K, each predator must gain more than it loses.
Sources & Verification
- N. J. Gotelli, A Primer of Ecology, 4th ed., Sinauer, 2008 — chapter on predation and the chapter “Logistic Population Growth”.