Two main models of population growth: Exponential (J-shaped): dN/dt = rN. Unlimited resources. No population regulation. Seen in organisms introduced to new environments (pest outbreaks, invasive species). Logistic (S-shaped): dN/dt = rN(K-N)/K. Limited resources. Population regulated by carrying capacity K. More realistic model for natural populations. The logistic model was developed by Pierre Verhulst (1838) and later re-derived by Pearl and Reed (1920) — hence Verhulst-Pearl logistic growth. Key parameter K (carrying capacity): max sustainable population size. Parameters r and K are important life history traits. r-selected species: high r, low K (rabbits, insects, bacteria). K-selected species: low r, high K (elephants, whales, humans).

The logistic growth curve (S-curve or sigmoid curve) has three phases: Lag phase: slow initial growth (small N, few individuals reproducing). Log/exponential phase: rapid growth (population far below K, resources abundant). Stationary phase: growth slows and levels off at K. The inflection point (maximum growth rate) occurs at N = K/2. At N = K/2: growth rate is maximum = rK/4. Above K: (K-N)/K becomes negative → population declines back to K. Key feature of logistic growth: density-dependent growth rate. As density increases, per capita growth rate (r-effective) decreases. Mechanisms: intraspecific competition for food, space, territory, mates. Predation, disease increase with density.

r-selected species: selected for high intrinsic rate of increase (r). Early maturity, many small offspring, little parental care, short lifespan. Adapted to unstable/unpredictable environments. Examples: bacteria, insects, annual plants, dandelion, mice, rabbits. K-selected species: selected for high competitive ability near K. Late maturity, few large offspring, extensive parental care, long lifespan. Adapted to stable, predictable environments (near carrying capacity). Examples: elephants, whales, humans, oak trees, eagles. Contrast: r-strategists produce many offspring hoping some survive. K-strategists produce few offspring and invest heavily in each. This spectrum (r to K) predicts life history tradeoffs. Modern ecology uses a more nuanced framework (pace of life, slow-fast continuum) but r/K selection remains conceptually useful.

Population regulation: Density-dependent factors: their effect increases with population density. Intraspecific competition (food, territory, mates). Predation (predator response to prey density). Disease and parasitism (spread faster in dense populations). Emigration increases, immigration decreases. These create negative feedback → limit population growth → stabilise near K. Density-independent factors: their effect is the same regardless of population density. Abiotic factors: temperature, floods, droughts, fires, storms, pollution. A severe frost kills the same fraction of a large or small population. These cause population fluctuations independent of density. Natural populations are regulated by both: density-dependent factors tend to stabilise near K, density-independent factors cause random fluctuations. Human populations largely escaped density-dependent regulation through agriculture, medicine → exponential growth approaching 8 billion.

Population: group of individuals of the same species living in the same area at the same time. Natality (birth rate): number of new individuals added per unit time per unit population. Mortality (death rate): number of individuals dying per unit time per unit population. r = b - d (b = birth rate, d = death rate). Emigration: individuals leaving. Immigration: individuals arriving. Net population change = (b - d) + (immigration - emigration). Age structure (age pyramid): proportion of individuals in different age groups (pre-reproductive, reproductive, post-reproductive). Expanding population: wide base pyramid. Stable population: rectangular pyramid. Declining population: narrow base. Sex ratio: proportion of males to females. Life table: age-specific survival and reproduction data. Survivorship curves: Type I (humans — high survival until old age), Type II (birds — constant mortality), Type III (fish, oysters — high early mortality, survivors live long).

Interspecific interactions affect population growth. Lotka-Volterra predator-prey model: predator and prey populations oscillate in cycles. Classic example: Canadian lynx and snowshoe hare cycles (10-year cycles). Competition (Lotka-Volterra competition model): two species competing for same resource. Competitive exclusion principle (Gause): two species competing for identical niche cannot coexist indefinitely — one excludes the other. Character displacement: competing species evolve differences to reduce competition. Predation: increases prey mortality → potentially reduces prey population → but also drives evolution of prey defenses. Mutualism: both species benefit → can increase both populations. Amensalism: one species suppressed, other unaffected. Commensalism: one benefits, other unaffected. These interactions determine community composition and diversity.

Human population has grown exponentially due to: agricultural revolution (~10,000 years ago) → food supply increased dramatically. Industrial revolution (1750s) → improved living standards, sanitation, medicine → reduced mortality. Medical advances: antibiotics, vaccines, improved surgery → further mortality reduction. Green Revolution (1960s) → increased food production prevented Malthusian famine predictions. Global population: 1 billion in 1800, 2 billion in 1927, 4 billion in 1974, 7 billion in 2011, 8 billion in 2022. Growth rate peaked ~1963 (2.2% per year), now ~0.9% per year. Total Fertility Rate (TFR): replacement level = 2.1. Many developed countries below 2.1 (declining population without immigration). India surpassed China as most populous country in 2023 (~1.43 billion). Demographic transition: population growth pattern from high birth+high death → high birth+low death (rapid growth) → low birth+low death (stable). Most developed countries in stage 4 (stable or declining).

Metapopulation: a population of populations connected by occasional migration between habitat patches. Levins metapopulation model: patches can be occupied or empty. Extinction and colonisation events determine fraction of occupied patches. Rescue effect: migration from occupied patches prevents local extinction. Significance for conservation: fragmented habitats create metapopulations. Corridors between habitat patches crucial for maintaining gene flow and allowing recolonisation after local extinction. Without corridors: isolated patches → local extinction not replaced → eventual regional extinction. Minimum Viable Population (MVP): smallest population size with a good probability (usually 99%) of surviving for a given time (usually 100 years) despite demographic, environmental, and genetic stochasticity. Used to set conservation targets. Population Viability Analysis (PVA): computer modelling of extinction probability based on population size, growth rate, habitat quality. Used in recovery plans for endangered species.