Population Growth & Regulation
- Contrast exponential and logistic growth and compute the growth rate from each model
- Relate carrying capacity, r/K selection, and survivorship curves to a species’ life strategy
- Distinguish density-dependent from density-independent limiting factors
Growth with no brakes: exponential
When resources are unlimited, a population grows faster and faster as it gets bigger — more individuals means more births per unit time. This is exponential growth, and its curve is the classic J-shape. The per-capita growth rate r (births minus deaths per individual) stays constant, but because it multiplies an ever-larger N, the total increase accelerates. Exponential growth is real but temporary: it happens when a species colonizes new habitat, recovers after a crash, or blooms in a nutrient-rich pulse — until something runs short.
Growth with limits: logistic and carrying capacity
No environment supplies unlimited food, space, or water. Carrying capacity (K) is the maximum population size an environment can sustain over time. Logistic growth builds this ceiling into the model: growth is fast when the population is small, slows as it approaches K, and levels off at K, tracing an S-shaped curve. The extra term (K − N)/K is the fraction of resources still unused — when N is tiny it is near 1 (growth ≈ exponential), and when N nears K it approaches 0 (growth stalls).
A deer population has r = 0.5 per year and carrying capacity K = 1,000. Compare the growth rate dN/dt when N = 100 (early) with when N = 800 (near capacity).
- 1.At N = 100: (K − N)/K = (1,000 − 100)/1,000 = 900/1,000 = 0.9. Then dN/dt = rN × 0.9 = 0.5 × 100 × 0.9 = 50 × 0.9 = 45 deer per year.
- 2.At N = 800: (K − N)/K = (1,000 − 800)/1,000 = 200/1,000 = 0.2. Then dN/dt = rN × 0.2 = 0.5 × 800 × 0.2 = 400 × 0.2 = 80 deer per year.
- 3.Even though N is much larger at 800, the growth per year (80) is not eight times the growth at 100 (45) — the (K − N)/K brake has cut the per-capita output. As N climbs past K/2 the growth rate itself begins to fall back toward zero.
A bacterial population grows exponentially with r = 0.1 per hour. When the population is N = 500, what is the instantaneous growth rate dN/dt?
Do not confuse r with dN/dt. r is the per-capita rate — one number describing each individual. dN/dt is the whole population’s change per unit time, which is r × N (times the logistic brake, if any). A small constant r can still yield a huge dN/dt once N is large.
Two life strategies: r-selected and K-selected
Species tend toward one of two strategies. r-selected species (like insects, weeds, bacteria) maximize r: many small offspring, little or no parental care, early reproduction, short lives. They boom in unstable or empty habitats and tolerate high offspring mortality. K-selected species (like elephants, whales, humans) do well at densities near K: few large offspring, heavy parental investment, late reproduction, long lives. They thrive in stable, crowded environments where competition — not colonization — is the challenge.
Survivorship curves
A survivorship curve plots how many of a cohort survive at each age. Type I (humans, large mammals) is flat then plunges — low death rate early, most die old; these are K-selected. Type III (oysters, many plants, insects) plunges early then flattens — enormous early mortality, but the few survivors live long; these are r-selected. Type II (many birds, rodents, lizards) is a straight diagonal — a roughly constant death rate at every age.
A frog lays thousands of eggs at once, provides no parental care, and most tadpoles die young — but a few survive to full adulthood. This species is best described as:
What limits a population
Density-dependent factors hit harder as the population gets more crowded: competition for food and space, predation, and the faster spread of disease and parasites. These push a population back toward K and produce the leveling of the logistic curve. Density-independent factors affect the same fraction of a population regardless of its size — a hurricane, wildfire, flood, or hard freeze kills roughly the same proportion whether the population is dense or sparse. Most real populations are shaped by both.
A sudden winter freeze kills about the same percentage of a beetle population whether that population is dense or sparse. This is an example of a:
On the exam, tie the pieces together: exponential = J-curve, no limits (dN/dt = rN); logistic = S-curve leveling at K (dN/dt = rN((K − N)/K)). The (K − N)/K term is the density-dependent brake in equation form — link the graph, the equation, and the limiting factors in one answer.
Answer the 3 checkpoints as you read.
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