Logistic and Predator-Prey Models
Populations cannot grow forever, and two classic models show how limits and species interactions enter the math.
From exponential to logistic
Pure proportional growth, dP/dt equals r P, predicts unchecked exponential blowup, which real environments cannot support. The logistic model multiplies by the brake factor one minus P over K, where K is the carrying capacity. With r equal to 0.1 and K equal to 1000, a population at 500 grows at 0.1 times 500 times 0.5, or 25 individuals per year.
\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)Reading the S-curve
The logistic solution rises slowly, grows fastest at half the carrying capacity, then flattens toward K. The equilibria are P equal to 0 and P equal to K, and the first is unstable while the second is stable. Small populations act nearly exponential, and crowding only bites once P is a real fraction of K.
P = K/2Predator-prey cycles
The Lotka-Volterra system pairs a prey equation a x minus b x y with a predator equation negative c y plus d x y. The interaction term x y appears in both, helping the prey die and feeding the predators. For the classic parameters, prey alone grow, predators alone die out, and the two populations chase each other in cycles around a coexistence equilibrium.
\frac{dx}{dt} = ax - bxy,\quad \frac{dy}{dt} = -cy + dxyFlashcard
Logistic growth with a population cap
Check your understanding
For the logistic model dP/dt equals 0.2 times P times (1 minus P over 800), at what population is the population growing fastest?