When a quantity grows at a rate proportional to how much is already there, its differential equation is dy/dt = ky. Separating and integrating — exactly the method from the last lesson — gives the model in closed form.
Here k > 0 means growth and k < 0 means decay. The constant k sets the speed; y₀ is the value at t = 0.
Every solution of dy/dt = ky is an exponential. Slide k through positive and negative values and change the start y₀ — watch the curve climb away to infinity, or fall toward zero, always tangent to the slope field.
Real populations can't grow forever — food, space, and resources impose a ceiling called the carrying capacity L. The logistic equation slows growth as the population approaches L: dP/dt = kP(1 − P/L). When P is small the factor (1 − P/L) ≈ 1 and growth is nearly exponential; as P nears L that factor shrinks to zero and growth stalls.
The logistic curve isn't only theory. In 1913 the biologist Carlson counted the yeast cells in a culture every two hours, and the numbers trace an almost perfect S. You're the modeler: drag the carrying capacity, the rate, and the starting count to thread a logistic curve through the real measurements — watch the fit score climb — then reveal the best fit a computer finds.
Rate proportional to amount, dy/dt = ky, solves to unbounded growth or decay.
Growth throttled by a ceiling. The carrying capacity L is read straight off the equation.
A logistic population grows fastest when P = L/2 — the inflection point of the S-curve, where dP/dt peaks.
From any positive start, P → L as t → ∞. Below L it rises; above L it falls — both settle at the capacity.
Six problems across both models — writing them, solving the exponential one, and pulling the key facts off a logistic equation. Try each before revealing.
Six fast checks across the lesson. Pick an answer to see whether it's right and why — then work through all six.
The carrying capacity is L; the population grows fastest at L/2. Mixing these up is the most common logistic error on the exam.
Near the carrying capacity the rate is slowing to zero. Maximum dP/dt happens at the inflection point, exactly halfway up.
Decay is dy/dt = ky with k < 0, giving y = y₀e^(kt) with a negative exponent. Don't write a separate “minus” and a positive k both.
Carrying capacity comes from the (1 − P/L) factor. In dP/dt = kP(1 − P/L), the capacity is L — not k, and not the coefficient out front.
If the population starts above L it decreases toward L. The limit is still L; logistic solutions approach the capacity from whichever side they start.
The exam almost never needs the explicit logistic formula. It wants the capacity (L), where growth is fastest (L/2), and the limit (L) — read or reasoned, not derived.