Population Ecology — Print · Grade 12 Science

Population Ecology

Learning goals

• I can model population change with exponential and logistic growth and locate the carrying capacity.
• I can distinguish density-dependent from density-independent limiting factors and interpret survivorship curves.

Readings for this unit

• A food-web arrow and a survey count both need a definition

A population is a group of one species in one place, and its size is bookkeeping: it rises with births and immigration and falls with deaths and emigration. Ecologists summarize the balance with a per-capita growth rate, r. When r is positive the population grows, when it is negative it shrinks, and the beauty of the model is that the same small handful of numbers describes a colony of bacteria, a herd of deer, or the human species.

When resources are effectively unlimited, growth is exponential: each individual reproduces at the same rate, so the more individuals there are, the faster the total climbs, tracing a J-shaped curve that gets ever steeper. Written as a rate, dN/dt = rN. Bacteria in fresh nutrient broth, an invasive species arriving in a new land, and the recent explosion of the human population all show this runaway pattern. But no J-curve can last: multiply anything long enough and it collides with the finite limits of the real world.

The continuous logistic-growth model dN/dt=rN(1−N/K) reduces per-capita growth as N increases toward a fixed conditional carrying-capacity parameter K. It approaches K smoothly under its simple assumptions; overshoot, oscillations and collapse require changing conditions, delays or other model extensions. Real carrying capacity can change with resources and habitat. A measured population trajectory is not required to match one ideal S-curve, and K is not an eternal fixed biological maximum.

What actually applies the brakes are limiting factors, and sorting them into two kinds is one of the most useful ideas in population ecology. *Density-dependent* factors bite harder as a population gets crowded — competition for food, predation, and the rapid spread of disease all intensify with density, so they naturally pull a population back toward K. *Density-independent* factors strike regardless of how many individuals are present: a hurricane, a hard freeze, a drought, or a wildfire can flatten a sparse population and a dense one alike. Real populations are shaped by both, which is why their numbers fluctuate rather than holding perfectly steady.

Species also differ in their whole life strategy, and survivorship curves make the differences visible by plotting how many of a cohort remain alive across the lifespan. *Type I* species (humans, elephants, other large mammals) invest heavily in a few offspring, so most survive to old age and mortality is late. *Type III* species (many fish, insects, oysters, plants) flood the environment with huge numbers of offspring and provide little care, so the vast majority die young while a lucky few survive. *Type II* species (many birds and small mammals) fall in between, facing a roughly constant risk of death at every age.

For the ideal fixed-parameter logistic model, maximum instantaneous total growth occurs at N=K/2. That mathematical result does not by itself establish a safe real harvest limit: variable recruitment, observation error, age structure, ecosystem effects and delayed management responses matter. Use the model as a conditional baseline, compare evidence and uncertainty, and avoid treating maximum modeled production as a guaranteed sustainable policy.

Model check: rN(1−N/K) is an instantaneous rate in the continuous model, not an exact finite-year population increase. dN/dt=.4×500×(1−.2)=160individuals/year at that moment; future N changes the rate.

Exponential growth

dN/dt = rN

N is the population size and r is the per-capita growth rate; produces a J-shaped curve when resources are unlimited.

Logistic growth

dN/dt = rN((K - N)/K)

K is the carrying capacity; the (K - N)/K term shrinks toward 0 as N approaches K, slowing growth into an S-shaped curve.

Population growth rate

growth rate = (births - deaths) / N

the per-capita rate of change (also add immigration and subtract emigration when present); multiply by 100 to express it as a percent.

Rule of 70 (doubling time)

doubling time (years) ≈ 70 / (percent growth rate)

a quick estimate of how long a steadily growing population takes to double; e.g., a 2%/yr rate doubles in about 35 years.

Worked examples

Avoid these mistakes

• rN(1−N/K) is an instantaneous rate in the continuous model, not an exact finite-year population increase.

Words to know

population

A group of individuals of the same species living in the same area, whose size changes with births, deaths, immigration, and emigration.

exponential growth

Growth at a constant per-capita rate when resources are unlimited, producing a J-shaped curve (dN/dt = rN).

logistic growth

Growth that slows as a population nears the limits of its environment, producing an S-shaped curve that levels off at carrying capacity.

carrying capacity (K)

The maximum population size an environment can sustain over the long term, given its resources and limiting factors.

limiting factor

A resource or condition that restricts population growth; density-dependent factors intensify with crowding, while density-independent factors act regardless of density.

survivorship curve

A graph of how many individuals of a cohort survive over their lifespan, classed as Type I, II, or III.

Live fast or live long: r- and K-selection

Ecologists describe two broad life strategies. r-selected species (many insects, weeds, bacteria) reproduce early and prolifically with little parental care; they thrive in unstable or empty habitats and show boom-and-bust, near-exponential growth — the Type III survivorship pattern. K-selected species (elephants, whales, humans, oak trees) produce few offspring, invest heavily in each, mature slowly, and live near carrying capacity with Type I survivorship. The labels are two ends of a continuum, but they neatly connect growth curves, survivorship, and reproductive strategy into one story.

The lynx and the hare: a century of coupled cycles

Some of the best long-term population data on Earth came from an unlikely source: the fur-trapping records of the Hudson's Bay Company, which logged snowshoe hare and Canada lynx pelts for nearly two centuries. The counts reveal a striking pattern — hare numbers rise and fall on a roughly 10-year cycle, and lynx numbers follow a step behind, a real-world example of coupled predator-prey cycles. More hares feed more lynx; more lynx then crash the hare population; fewer hares then starve the lynx, letting hares rebound. (The cycle is also driven partly by the hares' own food supply, showing that nature is rarely a clean two-species model.)

Does the human population have a carrying capacity?

For most of history human numbers crept along, then exploded into a textbook J-curve after the Industrial and Agricultural Revolutions, reaching 8 billion. Yet growth rates peaked in the 1960 s and have been falling since, as countries pass through the demographic transition from high birth-and-death rates to low ones. Demographers now project the curve bending toward an S-shape, leveling off later this century. Whether Earth's carrying capacity for humans is 9 billion or far more is fiercely debated — because, uniquely, humans can raise K with technology like fertilizer and irrigation, and lower it by degrading the very resources they depend on.

Overshoot and collapse on St. Matthew Island

Populations do not always glide gently to carrying capacity. In 1944, 29 reindeer were introduced to remote St. Matthew Island with abundant lichen and no predators. Freed of limits, the herd grew explosively to about 6,000 by 1963 — then, having stripped the slow-growing lichen bare, it crashed catastrophically to fewer than 50 animals in a single brutal winter. It is a stark real-world lesson that a population can overshoot its carrying capacity, degrade the resource base, and collapse far below where it started, rather than settling smoothly at K.

Why maximum sustainable yield sits near half of K

The logistic model has a practical payoff. Because the growth term rN((K - N)/K) is a downward parabola in N, a population adds the most new individuals per year when N is about K/2 — that is where the curve is steepest. Harvesting a fishery or forest down to roughly half its carrying capacity therefore lets it regrow fastest, defining the maximum sustainable yield. In practice, managers treat this as a ceiling, not a target: uncertainty in the data means aiming exactly at MSY has repeatedly tipped real fisheries, like the Atlantic cod, into collapse.

In the real world

Fisheries managers use logistic-growth models to set catch limits near a stock's maximum sustainable yield (around N = K/2, where the population grows fastest), harvesting enough to feed people while leaving the population able to rebuild.

Exit check

N=500,r=.4/year,K=2500. Find instantaneous growth rate.