Exponential: dN/dt = rN; closed form N_t = N₀e^(rt). Logistic: dN/dt = rN(K−N)/K. N = population size; r = intrinsic growth rate; K = carrying capacity; t = time. Logistic includes the term (K−N)/K which approaches 0 as N → K, slowing growth.
B. The maximum population size that the environment can sustain indefinitely given resources, predators, disease.
r-selected: many offspring; little or no parental care; short lifespan; unstable/disturbed habitats (e.g., dandelions, mosquitoes). K-selected: few offspring; extensive parental care; long lifespan; stable, predictable habitats (e.g., elephants, oak trees, humans).
Density-dependent (intensify with crowding): disease, intraspecific competition, predation, parasitism. Density-independent (act regardless of density): drought, hurricane, fire, severe winter, volcanic eruption. Density-dependent factors are stabilizing; density-independent can cause sudden crashes.
B. Many young, very high juvenile mortality (e.g., oysters, fish, frogs); only a few survive to reproduce.
A. Primary: lifeless substrate, requires pioneers (lichens) to build soil → centuries. Secondary: existing soil after disturbance, faster (decades).
(a) N = N₀ e^(rt) → 12,800 = 200 e^(6r) → e^(6r) = 64 → 6r = ln 64 = 4.16 → r ≈ 0.69 hr⁻¹. (b) N(12) = 200 × e^(0.69 × 12) = 200 × e^8.28 ≈ 200 × 3960 ≈ 820,000. (c) Assumption: unlimited resources (exponential model holds, no logistic damping yet).
A: Σp² = 0.36 + 0.09 + 0.01 = 0.46; D_A = 1 − 0.46 = 0.54. B: Σp² = 4 × 0.0625 = 0.25; D_B = 1 − 0.25 = 0.75. Community B is more diverse — both higher richness (4 species) and even abundance.
N ≈ (M × C)/R = (80 × 100)/16 = 500 deer. Assumptions: (1) closed population (no births, deaths, immigration, emigration between samples); (2) marks don't fall off or affect survival; (3) each animal equally likely to be captured (no trap-shyness or trap-happiness); (4) marked animals mix randomly with unmarked between samples.
Exponential: J-shape, dN/dt always increasing. Logistic: S-shape (sigmoid). Lag phase (slow start) → exponential phase (steep, rapid growth) → deceleration as N approaches K → plateau at K. Maximum dN/dt occurs at N = K/2 (inflection point). Carrying capacity K shown as horizontal asymptote.
Hudson's Bay Co. fur-trade records show ~10-year cycles. Hare numbers rise → abundant food for lynx → lynx reproduce, lynx population rises (lagged by ~1–2 years). Heavy lynx predation depresses hare population; hare crash → lynx starve → lynx population crashes. Reduced predation lets hares recover, restarting the cycle. Mechanism: time-lagged density-dependent predation; also vegetation quality and disease contribute.
(1) Bare lava rock — no soil, no life. (2) Pioneer species: lichens and mosses break rock chemically/mechanically; trapped organic material begins forming soil. (3) Small herbaceous plants (grasses, ferns) colonize thin soil. (4) Shrubs and small trees follow as soil deepens. (5) Hardy fast-growing trees (alder, birch) form early forest. (6) Shade-tolerant climax species (e.g., maple, hemlock) dominate the mature, stable climax community. Process can take centuries.
t_double ≈ 70/r%. A: 70/1.4 = 50 years. B: 70/3.5 = 20 years. Rapidly doubling populations face accelerating resource demand (food, water, housing, energy), strain on infrastructure and services, and increased ecological footprint. Sustainable development requires balancing growth with capacity.
Initial growth: exponential — abundant food (plankton), few predators, no competitors → J-curve. Effects: outcompete native mussels (extirpations); filter-feed plankton, reducing food for fish larvae; clog water intakes, fouling infrastructure (~$ billions); alter clarity, favouring some species over others. Mitigation: ballast-water exchange/treatment regulations; "Clean, drain, dry" boater education; biocontrol research; chemical treatments at intakes.
Implication: Canadians use the equivalent of ~5 Earths per capita — unsustainable. Continued at this rate, biodiversity loss, climate change, depletion of fish stocks and forests will accelerate. Personal: reduce meat consumption (esp. beef); use active or public transit; insulate home and switch to heat pump; reduce flights. Policy: carbon pricing; protected areas (e.g., 30 × 30); subsidies for renewables; circular-economy regulations on packaging; investment in transit and densification.