Exponential (Malthusian) growth: dN/dt = rN. N = population size. r = intrinsic rate of natural increase (births - deaths per individual per time). Integrated form: Nt = N0 e^rt (continuous) or Nt = N0 lambda^t (discrete, lambda = finite rate of increase). J-shaped curve. No resource limitation. Occurs when: resources unlimited, no predators/competitors, new colonisation. Examples: bacteria in ideal conditions, introduced species in new habitat, human population historically. Logistic growth (Verhulst, 1838): dN/dt = rN[(K-N)/K]. K = carrying capacity = max sustainable population. (K-N)/K = unutilised reproductive capacity. When N << K: growth nearly exponential. When N = K/2: growth rate maximum (inflection point of S-curve). When N = K: dN/dt = 0. S-shaped (sigmoid) curve. Real populations rarely perfectly logistic but useful model.

Carrying capacity (K): determined by available resources (food, water, space, nesting sites, light for plants). Density-dependent factors: intensity increases as population density increases. Competition, predation, disease, parasitism, food shortage. Regulate populations around K. Negative feedback. Density-independent factors: affect population regardless of density. Weather events, natural disasters, temperature extremes. Can cause sudden population crashes. Intraspecific competition: between individuals of same species. Strongest type of competition (same niche). Interspecific competition: between different species. May lead to competitive exclusion or niche differentiation. Predator-prey cycles: Lotka-Volterra equations describe oscillating predator-prey dynamics. Lynx-snowshoe hare cycles (Canada, ~10 year cycles). Evidence for both density-dependent and density-independent regulation.

Life table: age-specific survival and fecundity data. Cohort (horizontal) life table: follows group of same-age individuals from birth to death. Static (vertical) life table: age structure at one time point. Parameters: lx = survivorship (fraction surviving to age x). mx = fecundity at age x (female offspring per female). Net reproductive rate R0 = sum(lx*mx). R0 > 1: population growing. R0 = 1: stable. R0 < 1: declining. Generation time T = sum(x*lx*mx)/R0. Intrinsic rate of increase r = ln(R0)/T (approximately). Survivorship curves: Type I: low juvenile mortality, most die old (humans, elephants, K-strategists). Type II: constant mortality at all ages (birds, many lizards). Type III: high juvenile mortality, survivors live long (oysters, fish, r-strategists). Population age structure: growing population: many young (triangular pyramid). Stable: even distribution. Declining: few young.

Competition (-/-): both species harmed. Competitive exclusion principle (Gause): two species competing for identical resources cannot coexist. One will outcompete and eliminate the other. Character displacement: competing species diverge in resource use when sympatric. Beak size divergence in Galapagos finches. Predation (+/-): predator benefits, prey harmed. Predator adaptations: speed, camouflage, venom, cooperative hunting. Prey adaptations: speed, armor, spines, warning coloration, mimicry, schooling behaviour. Herbivory: plant (+/-): secondary metabolites (alkaloids, tannins, terpenes) as defence. Mutualism (+/+): both benefit. Mycorrhizae: fungus provides minerals, plant provides sugar. Nitrogen-fixation: Rhizobium provides fixed nitrogen to legumes, gets carbon. Cleaner fish: removes parasites from larger fish (both benefit). Commensalism (+/0): one benefits, other unaffected. Epiphytes (orchids on trees: use tree for support, tree unaffected). Amensalism (-/0): one harmed, other unaffected. Antibiotic production by bacteria kills nearby bacteria.

Top-down regulation (trophic cascade): predators regulate prey populations, which in turn regulate lower trophic levels. Removal of wolves from Yellowstone: elk populations exploded, overgrazing devastated riparian vegetation, which altered river courses, reduced beaver populations. Wolf reintroduction reversed effects (1995). Bottom-up regulation: resource availability controls producer populations, which cascade upward through food web. Nutrient addition (eutrophication) increases algal growth, algae supports more herbivores, etc. Regulation usually involves both mechanisms. r vs K selection (MacArthur and Wilson): r-selected species: short lifespan, many small offspring, little parental care, boom-bust populations, colonisers. K-selected species: long lifespan, few large offspring, extensive parental care, stable populations near K. Continuum not a strict dichotomy.

Species richness: number of species in an area. Species diversity: richness + evenness (relative abundance). Shannon index H = -sum(pi * ln pi). Simpson's index D = 1 - sum(pi^2). Diversity-stability relationship: more diverse communities are generally more stable (resilient and resistant). Insurance hypothesis: more species = more functional redundancy = community can withstand species losses without losing function. Portfolio effect: diverse portfolio of species with different responses to environmental variability leads to stable overall community function (like diversified investment portfolio). Keystone species: disproportionately large impact on ecosystem relative to their abundance. Sea otter, wolf, fig tree (provides food and habitat for many species). Removal = ecosystem collapse. Foundation species: create and maintain habitat structures. Kelp forests (support hundreds of species), beaver dams (create wetlands). Ecosystem engineers: modify physical environment creating habitats.

Ecological pyramids: graphical representation of trophic levels. Pyramid of numbers: number of individuals at each trophic level. Usually decreases upward. Inverted in parasitic food chain (one large tree supports many insects, many insects support more parasites). Pyramid of biomass: total dry weight of organisms at each trophic level. Usually decreases upward. Can be inverted in aquatic ecosystems (phytoplankton biomass less than zooplankton at any one time due to rapid turnover). Pyramid of energy: always upright. 10% energy transfer between trophic levels. Never inverted. Most fundamental representation of ecosystem function. Energy is always lost as heat at each trophic level (respiration). Global energy flow: solar energy captured by photosynthesis = ~1-2% efficiency. GPP of terrestrial ecosystems: ~120 Gt C/year. GPP of oceans: ~50-55 Gt C/year. Human appropriation of NPP (HANPP): ~25-40% of all terrestrial NPP.

Global warming and climate change: CO2 from fossil fuels + deforestation. Methane from cattle, rice paddies, landfills. Nitrous oxide from fertilisers. Greenhouse effect traps heat. IPCC: 1.1°C warming above pre-industrial already occurred. 1.5°C expected by 2030s. Consequences: sea level rise (melting glaciers + thermal expansion), extreme weather events, species range shifts, coral bleaching, food security threats. Ozone depletion: CFC (chlorofluorocarbon) from aerosols, refrigerants. Cl atoms catalytically destroy ozone in stratosphere. Antarctic ozone hole seasonal. UV-B increases: skin cancer, cataracts, damage to phytoplankton. Montreal Protocol (1987): international ban on CFCs. Ozone layer slowly recovering. Acid rain: SO2 and NOx from burning fossil fuels + water = H2SO4, HNO3. pH below 5.6. Destroys forests, acidifies lakes, corrodes buildings. Eutrophication: nutrient pollution (nitrogen, phosphorus from agriculture, sewage) causes algal blooms, oxygen depletion, fish kills.