The graph shows the next 50 years of a population that grows logistically and may be harvested at a constant rate. dN/dt = rN(1 − N/K) − H. With harvest at zero the curve is the familiar S-shape and levels off at the carrying capacity K. The greatest natural increase is rK/4, at half the carrying capacity.
Unlimited growth, dN/dt = rN, makes the population keep multiplying. Food, space and nesting sites are finite, so that model is only a short-term description. Logistic growth adds a carrying capacity and makes the per-capita growth rate fall as N approaches K.
When N is much smaller than K, growth is close to exponential. The increase dN/dt is greatest at N = K/2. At N = K the natural increase is zero. Above K the population declines toward K. The dashed line on the graph is that carrying capacity.
A constant harvest H is subtracted every year. The largest harvest that can be replaced indefinitely is the maximum of rN(1 − N/K), which is rK/4, taken when the population is held near K/2. Harvesting above that maximum sustainable yield pulls the population down.
Natural increase is only about rN when N is small. A fixed harvest that a large population could spare can be larger than rN for a small population, so N keeps falling. The advanced outlook label reports a collapse when the 50-year result is essentially zero.
GCSE and IGCSE ecology uses population curves, limiting factors and the idea of a carrying capacity. A level and IB, including the logistic equation and sustainable yield, are the quantitative version. The harvest term is the simplest constant-quota model.
The equations are continuous averages. They omit age structure, seasons, migration, predators and random bad years. A real population fluctuates around curves like these rather than following them exactly.