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UI WLF 448 - Limited Growth

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9/27/2011 1 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 r Some final thoughts on exponential growth: time N •Exponential growth does not always mean ‘rapid’ growth •Be clear on the implications of the models you use / assumptions you make about where stochasticity occurs.9/27/2011 2 Population growth models Carrying capacity (k) Classic growth curve, unlimited resources Classic growth curve, limited resources (k) time time N N k Population Reproduction, births, natality (B) Mortality, death (D) Emigration (E) Immigration (I) Population growth n t+1 = n t + (B + I) - (D + E) Gains Losses r or lambda9/27/2011 3 How Birth rate and/or Mortality could change with density: Logistic Population Growth (Density dependent growth) • At higher densities these factors limit population growth b/c they influence one or some combination of BIDE. • This shows up as our measure of growth rate (lambda, r) changing as a function of population size (density). Growth rate is negatively affected by higher populations. • And results in a sigmoidal shaped function of population growth over time….most typically described as logistic growth. • Growth slows as pop size approaches the maximum number/density that a given area can sustain—the carrying capacity (K).9/27/2011 4 Logistic Population Growth (Density dependent growth) There are various models that describe different ways that r changes with increasing density. We will cover 3. Ricker (logistic) Model: Assumes a constant, linear decrease of r as population size increases. ln(n t+1) = ln (nt) + rmax + b(nt) + F Where: b is a parameter measuring the strength of intraspecific competition rmax is the populations maximum growth rate in the absence of density dependent competition (what we’ve dealt with up to now) 0.00000.10000.20000.30000.40000.50000.60000.70000.80000 50 100 150 200growth rate (rt) Abundance r max is the y-intercept of this line Rickers (logistic) Growth: ln(n t+1)/ ln (nt) = + rmax + b(nt) + F Constant linear decrease in r as nt increases b is the slope of this line K (carrying capacity) is the x-intercept of this line9/27/2011 5 Sometimes useful to think about recruitment, or yield. What is added to the population at each time step (product of rate and n) Logistic Population Growth (Density dependent growth) Gompertz Model: Similar to Rickers except for underlying assumption of how r changes with density. ln(n t+1) = ln (nt) + rmax + b( ln(nt)) + F Where: b is a parameter measuring the strength of intraspecific competition rmax is the populations maximum growth rate in the absence of density dependent competition (what we’ve dealt with up to now) Also produces sigmoidal curve9/27/2011 6 0.00000.10000.20000.30000.40000.50000.60000.70000 1 2 3 4 5 6Growth rate (rt) ln (Abundance) r max is the y-intercept of this line Gompertz model of DD growth: ln(n t+1)/ ln (nt) = + rmax + b(ln(nt))+ F Constant linear decrease in r as ln(nt ) increases b is the slope of this line K (carrying capacity) is the x-intercept of this line Gompertz model of DD growth: ln(n t+1)/ ln (nt) = + rmax + b(ln(nt))+ F Constant linear decrease in r as ln(nt ) increases 0.00000.10000.20000.30000.40000.50000.60000.70000 50 100 150 200growth rate (rt) Abundance Concave relationship between r and abundance Where is dd strongest?9/27/2011 7 Logistic Population Growth (Density dependent growth) Theta-logistic model: DD growth rate is a function of population size raised to the power of θ ln(n t+1)/ ln (nt) = rmax + b( nt θ) + F Where: b is a parameter measuring the strength of intraspecific competition rmax is the populations maximum growth rate in the absence of density dependent competition (what we’ve dealt with up to now) θ is a parameter describing the curvature of the r and n relationship n r t n r t n r t The shape of the relationship between growth rate and population size varies such that when θ is: θ = 1 0 < θ < 1 θ > 1 Theta-logistic model Rickers Lots of DD at small numbers Lots of DD at large


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UI WLF 448 - Limited Growth

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