Theory: Ecology / Population Dynamics
1. Population Growth Models
Two basic models: Exponential: dN/dt = rN → J-shaped curve (unlimited resources). Logistic: dN/dt = rN(K-N)/K → S-shaped curve (limited resources). J-curve: no upper limit. S-curve: levels off at K. Verhulst (1838) first developed logistic equation. Pearl and Reed (1920): applied to human population data. Both models are simplifications — real populations fluctuate around K rather than reaching a perfect plateau.
2. The Sigmoid Growth Curve
Phases of sigmoid (S-shaped) growth: Lag phase: slow initial growth (small population, may be establishing). Log/exponential phase: rapid growth (N<3. Factors Limiting Population Growth
Density-dependent factors (become more intense as population increases): Food/water competition, disease transmission (contact rate increases), predation (predators attracted by high prey density), intraspecific competition for nesting sites/territory. These are the factors that make growth approach K. Density-independent factors (affect population regardless of density): Natural disasters, extreme weather, climate change. Both shape population dynamics.
4. K-strategists vs r-strategists
r-strategists: high r, low K, many small offspring, short lifespan, high mortality. Examples: bacteria, insects, annual plants, mice. r-selected traits: fast development, early reproduction, many offspring, less parental care. K-strategists: low r, high K, few large offspring, long lifespan, low mortality. Examples: elephants, whales, humans, large trees. K-selected traits: delayed reproduction, few offspring, extensive parental care, competitive ability near K.
Frequently Asked Questions
1. What is logistic growth? ⌄
Logistic growth: population growth that slows as population size (N) approaches carrying capacity (K). Mathematical model: dN/dt = rN[(K-N)/K]. Where: r = intrinsic rate of natural increase, N = current population size, K = carrying capacity, (K-N)/K = unused resources (growth modifier). When N is small: (K-N)/K ≈ 1 → growth ≈ exponential (dN/dt ≈ rN). When N = K: (K-N)/K = 0 → dN/dt = 0 → no growth.
2. What is carrying capacity (K)? ⌄
Carrying capacity (K) = maximum population size that a given environment can sustainably support, given available resources (food, water, space, nesting sites). When N < K: resources available → population can grow. When N = K: resources exactly support existing population → no net growth. When N > K: resources insufficient → population declines back toward K.
3. What is the difference between exponential and logistic growth? ⌄
Exponential (J-shaped curve): dN/dt = rN. Growth rate constantly increases (more organisms → more offspring → more growth). Continues indefinitely (in theory). Only occurs when resources are unlimited. Logistic (S-shaped/sigmoid curve): dN/dt = rN(K-N)/K. Growth rate decelerates as N approaches K. Population stabilises at K. More realistic model for natural populations with limited resources.
4. What is r (intrinsic rate of natural increase)? ⌄
r = (birth rate − death rate) = per capita rate of increase. r > 0: population growing. r = 0: population stable. r < 0: population declining. r_max: maximum possible rate for that species under ideal conditions. In logistic growth, the effective growth rate is r × (K-N)/K, which decreases as N increases toward K.
5. Which organisms show logistic growth in nature? ⌄
Most natural populations follow logistic growth when resources are limited (which they always eventually are). Classic experimental examples: Paramecium caudatum grown in lab with fixed food supply (Pearl and Reed, 1920). Sheep population of Tasmania (showed classic S-curve after introduction). Human population has been debated — currently showing some features of logistic growth as resource limits approach, though complex.