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UI WLF 448 - Population Growth

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9/22/2011 1 Population Reproduction, births, natality (B) Mortality, death (D) Emigration (E) Immigration (I) Population growth Population parameters that influence population size and change (for now we will consider these all wrapped up into a measure of growth rate) Time 1 2 3 0 Exponential Population Growth in Unlimited Environments9/22/2011 2 Population growth Classic exponential growth, unlimited resources time n nt = n0 * λt Where t = time, t0 = 0 nt = population abundance (or index) at time t λ = finite rate of increase (nt+1/nt) λ > 1, increasing population λ < 1, decreasing population λ = 1, stable population Population growth Classic exponential growth, unlimited resources semi-log transformed time Ln (n) Ln (nt ) = ln(n0 ) + rt Where t = time, t0 = 0 nt = population abundance (or index) at time t r = instantaneous rate of increase ln(nt+1/nt) , or ln (λ) r > 0, increasing population r < 0, decreasing population r = 0, stable population9/22/2011 3 Population estimates fluctuate due to Observation Error • Missed animals / Overcounted animals • Equipment/survey limitations Here growth (λ, r) is deterministic and deviations from this is due to counting/estimation errors. ln[Nt] = ln [N0] + r*t + E Where E is observation error and is a normal random deviate with mean = 0, and standard deviation = tau E~normal (0, tau) 01234567198019811982198319841985198619871988198919901991199219931994199519961997199819992000Ln Abundance Year DeterministicExponential GrowthObserved numbers-0.6-0.4-0.200.20.40.61980 1985 1990 1995 2000Deviation from true count due to Observation Error (ln abundance) Deviations from exponential model predicted values due solely to observation error. Growth rate can be estimated from the slope a regression line fitted to observed data.9/22/2011 4 Population estimates fluctuate due to Process Noise Growth rate varies from year to year due to natural factors(food weather etc), ‘good’ and ‘bad’ years. Here deviations from exponential model are due to variation in growth rate (r ) from yr to yr ln[Nt+1] = ln [Nt] + r + F Where F is Process noise and is a normal random deviate with mean = 0, and standard deviation = sigma F~normal (0, sigma) 14 aug 2007 Deviations from exponential model predicted values due solely to variations in r. Growth rate can be estimated by averaging r’s from smaller intervals (each time step). 2.533.544.555.5198019811982198319841985198619871988198919901991199219931994199519961997199819992000Ln (Abundance) Year DeterministicExponential GrowthStochastic Growth 1Stochastic Growth 2Stochastic Growth 3-0.2-0.15-0.1-0.0500.050.10.151980 1985 1990 1995 2000Deviation from mean r9/22/2011 5 Do we see often see unlimited, exponential growth? • Is it common, realistic? – Why not? • When do we see it? – Examples? Population growth models Carrying capacity (k) Classic growth curve, unlimited resources Classic growth curve, limited resources (k) time time N N


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UI WLF 448 - Population Growth

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