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Population Growth and Regulation BIOL400 31 August 2015 Population Individuals of a single species sharing time and space Ecologists must define limits of populations they study Almost no population is closed to immigration and emigration Exponential Population Growth Equations: Nt = N0ert dN/dt = rN Model terms: r = per-individual rate of change (= b – d) = intrinsic capacity for increase, given the environmental conditions N = population size, at time t e = 2.718 Fig. 8.13 p. 131 Exponential Population Growth Assumptions of the Model: Constant per-capita rate of increase, regardless of how high N gets Continuous breeding Geometric Population Growth Model modified for discrete annual breeding Nt = N0t = er is the annual rate of increase in N Example: N0 = 1000, = 1.10 • • • • N1 = 1100 N3 = 1331 N5 = 1611 N25 = 10,834 N2 = 1210 N4 = 1464 N10 = 2594 N100 = 13,780,612 Q: Can you spot the oversimplification of nature here? Fig. 8.10 p. 129 Fig. 9.1 p. 144 R0 = per-generation multiplicative rate of increase Logistic Population Growth Equations: Nt = K/(1 + ea-rt) • K = karrying kapacity of the environment • a positions curve relative to origin dN/dt = Nr[(K-N)/K] Assumption: Growth rate will slow as N approaches K Fig. 9.4 p. 146 Table p. 148 As N increases, per-capita rate of increase declines, but the absolute rate of increase always peaks at ½ K Data from Populations in the Field Fig. 9.8 p. 150 Cormorants in Lake Huron Low numbers due to toxins Increase is not strongly sigmoid Fig. 9.9 p. 150 Ibex in Switzerland Reintroduced after elimination via hunting Roughly sigmoid (=logistic) but with big decline in 1960s Fig. 9.10 p. 151 Whooping cranes of single remaining wild population 15 in 1941, now over 200 r increased in 1950s Every mid-decade, there is a mini-crash Apparently related to predation cycles Fig. 9.15 p. 154 Cladocerans Predominant lake zooplankton No constant K; big swings seasonally Can We Improve Our Models? 1) Theta logistic model 2) Time-lag logistic model 3) Stochastic models 4) Population projection matrices Theta Logistic Model New term, , defines curve relating growth rate to N dN/dt = Nr[(K-N)/K] Fig. 9.12 p. 152 Fig. 9.13 p. 152 Time-Lag Models Logistic model in which population growth rate depends not on present N, but on N one (or more) time periods prior Assumes population’s demographic response to density may be delayed Fig. 9.14 p. 153 With time lag, stable ups and downs may occur Fig. 11.14 p. 170 Water fleas show stable approach to K at 18C Time lag effect occurs at 25C Daphnia store energy to use when food resources collapse Stochastic Models Predict a range of possible population projections, with calculation of the probability of each Fig. 9.17 p. Population Projection Matrices Use matrix algebra to project population growth, based on fecundity and agespecific survivorship • Fig. 9.18A p. 157 Application: Determining whether changes in one aspect or another of the life history of an organism have the greater impact on r (calculate “elasticity” of each life-history parameter) Fig. 9.19 p. 159 HANDOUT—Biek et al. 2002 Survivorship in a Population Three types of curves are recognized following Pearl (1928) Examination of the survivorship of various species shows that most have a mixed pattern Fig. 8.6 p. 124 Fig. 8.8 p. 126 Life Table Used to project population growth Can be used to determine R0, from which r or can be calculated 1) Vertical (=Static): useful if there is long-term stability in age-specific mortality and fecundity 2) Cohort: data taken from a population followed over time (ideally, a cohort followed until all have died) Observing year-year survivorship, or Collecting data on age at death Table 8.5 p. 128 Table 8.3 p. 122