The essentials
Population density changes because of four processes. Natality (B) and immigration (I) increase density; mortality (D) and emigration (E) decrease it. If N is the density at time t, then at time t + 1:
Nₜ₊₁ = Nₜ + [(B + I) − (D + E)]
Under normal conditions births and deaths matter most; immigration may matter when a new habitat is colonised.
Exponential growth: When resources are unlimited, each species can realise its full innate potential to grow. The rate of change is dN/dt = rN, where r is the intrinsic rate of natural increase (r = b − d, birth rate minus death rate). The integral form is Nₜ = N₀eʳᵗ. Plotting N against time gives a J-shaped curve. Any species growing exponentially with unlimited resources can reach enormous numbers in a short time.
Logistic growth: In nature resources are limited, so there is competition. A habitat can support only a maximum number of individuals, the carrying capacity (K). A population with limited resources shows an initial lag phase, then acceleration and deceleration, and finally an asymptote when N reaches K. The curve is sigmoid (S-shaped), described by the Verhulst–Pearl logistic growth equation: dN/dt = rN(K − N)/K. This is considered more realistic.
Life history variation: Populations evolve to maximise their reproductive fitness (Darwinian fitness) in their habitat. Some organisms breed only once in their lifetime (Pacific salmon, bamboo), others breed many times (most birds and mammals). Some produce many small offspring (oysters, pelagic fishes), others a few large offspring (birds, mammals).
Key points
- J-shaped curve: unlimited resources; S-shaped curve: limited resources and carrying capacity.
- When N = K, the term (K − N)/K is zero and growth stops.
- r for human populations differs between countries and over time.