Population Growth
Two models — exponential and logistic
A population is a group of individuals of the same species living in the same area at the same time. Population size changes based on four parameters: births (B), deaths (D), immigration (I), and emigration (E). ΔN = B - D + I - E. Population ecology seeks to understand what determines these rates and how they interact to produce population dynamics over time.
💡 r vs K Selection — Life History Strategies
Species evolve different reproductive strategies depending on their ecological context:
r-selected species (opportunists): small body size, short lifespan, reproduce early and often, many small offspring with little parental care, high mortality, populations fluctuate widely. Adapted to unpredictable environments where conditions can improve suddenly (dandelions, mice, insects, bacteria). The 'r' refers to the intrinsic rate of natural increase — r-selected species maximize r.
K-selected species (equilibrium species): large body size, long lifespan, reproduce late and infrequently, few large offspring with extensive parental care, low mortality in stable environments. Adapted to stable environments where competition for resources near K is the primary selective pressure (elephants, humans, whales, eagles). K-selected species maximize competitive ability near carrying capacity.
Conservation implication: K-selected species recover slowly from population declines because their reproductive rates are inherently low. Elephants, whales, and large cats cannot quickly replace losses — which is why hunting pressure or habitat loss can rapidly push them toward extinction, while mouse or insect populations can recover within weeks.
Exp
Exponential growth — the J-curve
Exponential growth occurs when a population increases by a constant proportion each time period — when each individual produces the same number of offspring regardless of population density. The equation is dN/dt = rN, where N = population size and r = intrinsic rate of natural increase (birth rate minus death rate per individual). When r > 0, the population grows. When r = 0, it's stable. When r < 0, it declines.
Exponential growth produces a J-shaped curve — slow at first (when N is small), then increasingly steep as population size grows (because more individuals are reproducing). Exponential growth is seen in nature when populations colonize new environments with abundant resources and few predators — introduced species in new habitats, bacterial growth in fresh nutrient media, or human populations before the Industrial Revolution. Exponential growth cannot continue indefinitely in a finite world.
Memory trick: Exponential growth = J-curve. More individuals = more babies = faster growth = even more individuals. Like compound interest — starts slow, then explosive.
Log
Logistic growth — the S-curve
Logistic growth incorporates the reality that resources are limited. As population density increases, competition for food, space, and mates intensifies, disease spreads more easily, and predators become more efficient. The equation becomes dN/dt = rN × [(K-N)/K], where K = carrying capacity — the maximum population size the environment can sustainably support.
When N is much less than K, (K-N)/K ≈ 1 → growth is approximately exponential. As N approaches K, (K-N)/K → 0 → growth slows. At N = K, growth stops. The resulting S-shaped (sigmoidal) curve is the logistic growth model. Growth rate is maximum at N = K/2 — the inflection point of the S-curve. This is the basis for maximum sustainable yield in fisheries management: harvest populations at K/2 to maximize the rate of population replacement.
Memory trick: Logistic growth = S-curve. K = carrying capacity (the ceiling). Growth slows as population approaches K. Maximum growth rate at K/2 — fisheries harvest here.
Lim
Limiting factors — what sets the carrying capacity
Limiting factors are resources or conditions that prevent a population from growing indefinitely. Two categories:
Density-dependent factors become more intense as population density increases: food competition (more individuals competing for the same resources), disease (spreads more easily in crowded populations), predation (predators switch to the most abundant prey), waste accumulation, stress-induced hormonal changes that suppress reproduction. These factors provide negative feedback that stabilizes population size near K.
Density-independent factors affect the population regardless of its size: weather events (frosts, droughts, floods), wildfires, habitat destruction. A severe frost kills the same proportion of a small population and a large population — its effect doesn't depend on how crowded the population is. Density-independent factors cause population crashes regardless of whether the population is at or well below carrying capacity.
Memory trick: Density-dependent = crowding makes it worse (disease, competition, stress). Density-independent = doesn't care how crowded you are (flood, fire, frost). Real populations are regulated by both.
🔬 Clinical/Applied Scenario — Population Models in Conservation and Epidemiology
Population growth models are applied directly in conservation biology, fisheries management, and disease control:
A
Minimum viable population (MVP) in conservation. Below a critical population size, genetic diversity is lost (inbreeding depression), demographic stochasticity (random variation in births and deaths) can cause extinction, and Allee effects emerge (individuals have difficulty finding mates → per capita reproduction falls → population spiral downward). Conservation biologists use population viability analysis (PVA) to estimate the minimum population size needed for a species to persist over 100 years with >95% probability.
B
Fisheries management — maximum sustainable yield. The logistic model predicts maximum population growth rate at N = K/2. Harvesting a population down to K/2 and then maintaining it there maximizes sustainable yield. Atlantic cod was fished well below K/2 in the 1980s → population collapsed in 1992 → moratorium declared → population has still not recovered 30 years later (demonstrating that K-selected species cannot rebound quickly).
C
Invasive species and exponential growth. When a species is introduced to a new environment without its natural predators, parasites, or competitors, it experiences exponential growth — effectively, its r is very high and its K in the new environment is much larger than in its native range. Cane toads in Australia, kudzu vine in the American Southeast, Nile perch in Lake Victoria, and zebra mussels in the Great Lakes all demonstrate exponential population growth following introduction.
D
R₀ in epidemiology — the basic reproductive number. The same exponential growth logic applies to infectious disease. R₀ (R-naught) is the average number of secondary infections caused by one infected individual in a fully susceptible population. If R₀ > 1 → epidemic grows exponentially. If R₀ < 1 → epidemic dies out. COVID-19 had R₀ ≈ 2.5–3 for the original strain (Omicron ~8–15). Herd immunity threshold = 1 - (1/R₀). For measles (R₀ = 12–18), herd immunity requires >94% immunity in the population.
📌 Exam Application
Population ecology is heavily tested — master these equations and concepts:
1. Exponential growth: dN/dt = rN. J-shaped curve. Occurs when resources are unlimited. r > 0 = growth, r < 0 = decline.
2. Logistic growth: dN/dt = rN[(K-N)/K]. S-shaped curve. K = carrying capacity. Maximum growth rate at N = K/2.
3. Density-dependent vs density-independent limiting factors. Density-dependent: food, disease, predation, stress. Density-independent: weather, fire, flood.
4. r vs K selection: r = many small offspring, fast reproduction, unstable environments. K = few large offspring, slow reproduction, stable environments. K-selected species are conservation-vulnerable.
5. Allee effect: per capita population growth rate DECREASES at very LOW population densities (opposite of normal density dependence) — difficulty finding mates, reduced cooperative defense. Important for extinction risk of small populations.
⚠️ The Most Common Population Ecology Mistakes
Carrying capacity (K) is not fixed. K changes as the environment changes. A drought reduces food availability → K drops → population crashes. Habitat improvement → K rises. Students often treat K as a constant feature of a species rather than a property of the interaction between the species and its environment at a particular time and place.
Maximum growth rate in logistic growth is at K/2, not at K. At K, growth rate is zero (population is at equilibrium). Maximum population growth RATE occurs at K/2 — halfway to carrying capacity — because that's where the balance of available resources and existing population size produces the fastest absolute increase. Fisheries manage for K/2 for this reason.
r-selection and K-selection are endpoints on a continuum. Most species fall somewhere between the extremes. Also, a single species can exhibit r-selected traits in some conditions and K-selected traits in others (e.g., a species may produce fewer offspring at high density). The r/K framework is a useful heuristic but an oversimplification of real life history variation.
✓ Quick Self-Test
1. What is the difference between exponential and logistic population growth?
2. What is carrying capacity (K) and what determines it?
3. What is the difference between density-dependent and density-independent limiting factors?
4. Compare r-selected and K-selected species in terms of reproductive strategy and conservation vulnerability.
5. Why does maximum population growth rate in the logistic model occur at N = K/2?
Answers:
1. Exponential growth (J-curve): population grows at a constant per capita rate regardless of density — dN/dt = rN. Occurs when resources are unlimited. Logistic growth (S-curve): growth rate decreases as population density increases toward carrying capacity — dN/dt = rN[(K-N)/K]. Occurs in resource-limited environments.
2. Carrying capacity (K) is the maximum population size that an environment can sustainably support given available resources (food, water, space, shelter). It is determined by the availability of limiting resources and the intensity of density-dependent factors. K is not fixed — it changes as environmental conditions change.
3. Density-dependent factors intensify as population density increases (food competition, disease transmission, predation, stress) — they provide negative feedback that stabilizes population near K. Density-independent factors affect populations regardless of density (weather events, fire, floods, earthquakes) — they can cause population crashes at any population size.
4. r-selected: small, short-lived, early reproduction, many small offspring, little parental care, high mortality, thrive in unpredictable environments. K-selected: large, long-lived, late reproduction, few large offspring, extensive parental care, low mortality in stable environments. K-selected species are conservation-vulnerable because their low reproductive rates mean slow recovery from population declines.
5. In the logistic equation dN/dt = rN[(K-N)/K], growth rate is a product of two terms: rN (increases with N) and (K-N)/K (decreases with N). This product is maximized at N = K/2 — where the two opposing effects are balanced. Below K/2, adding more individuals increases growth rate. Above K/2, increasing density slows growth more than adding individuals increases it.