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Population Dynamics I: Exponential Growth • Assumptions – All organisms have at least one parent Population Dynamics I: Exponential Growth • Assumptions – All organisms have at least one parent – Population size increases through births and immigration Population Dynamics I: Exponential Growth • Assumptions – All organisms have at least one parent – Population size increases through births and immigration – Population size decreases through deaths and emigration Population Dynamics I: Exponential Growth • Assumptions – All organisms have at least one parent – Population size increases through births and immigration – Population size decreases through deaths and emigration – For simplicity’s sake, set immigration = emigration = 0 Modeling Exponential Growth Nt+1 = Nt + Births Deaths Modeling Exponential Growth Nt+1 = Nt + Births - Deaths In words: The population size N at at the next time sampled (t+1) equals the population size now plus the births in the population minus the deaths that occur in the population during one time interval (1 unit of t) Modeling Exponential Growth Nt+1 = Nt + Births Deaths How can we use this equation to estimate population size N at any future time t from population size now (N0), without resorting to recursion? Modeling Exponential Growth Nt+1 = Nt + Births - Deaths First, subtract Nt from both sides of this equation: Nt+1 - Nt = Nt - Nt + Births - Deaths or DN = Births - Deaths Read: “the change in population size N = births - deaths” Modeling Exponential Growth DN = Births - Deaths Assume that the population growth is continuous, and let the time interval t to t+1 get very small... dN/dt = Births - Deaths Read: “the change in population numbers (dN) with respect to changes in time (dt) equals births - deaths” Modeling Exponential Growth dN/dt = Births - Deaths Assume that each individual produces the same number of offspring in each time interval (we’ll violate this assumption later), so total births B equals per capita (each “head” or per person) births b times the number of individuals in the population N: B = bN Similarly, total deaths D equals per capita deaths d times the number of individuals in the population N: D = dN. dN/dt = (b - d)N Modeling Exponential Growth dN/dt = (b - d)N By convention, let r = (births-deaths). This parameter, r, is commonly referred to as the Malthusian parameter, the intrinsic rate of increase of a population, or the instantaneous rate of increase. Then, our equation becomes: dN/dt = rN Modeling Exponential Growth dN/dt = rN What was the question, again? Modeling Exponential Growth dN/dt = rN How can we use this equation to estimate population size N at any future time t from population size now (N0), without resorting to recursion? Modeling Exponential Growth dN/dt = rN How can we use this equation to estimate population size N at any future time t from population size now (N0), without resorting to recursion? and then a miracle happened... Modeling Exponential Growth Nt = N0ert How’d that happen? (calculus aficionados click here) What’s this mean? (keep going) Modeling Exponential Growth Nt = N0ert What’s this mean? You can use this equation to predict population size at any given time t in the future by multiplying the population size now (at time t=0) times the exponential operator e raised to the rt power (your calculator can do this for you). how about some examples...? Modeling Exponential Growth Nt = N0ert an example: Can I do this myself…? Modeling Exponential Growth You can simulate exponential growth using Microsoft Excel (or any other spreadsheet software), or... Modeling Exponential Growth To get software for modeling population dynamics: Download Populus free from the internet! (it’s not Mac-compatible ) It’s also on the campus network, in the folder Populus on the network drive //slab/biology why should I care...? Modeling Exponential Growth • Many human populations are still growing exponentially Modeling Exponential Growth • Many human populations are still growing exponentially • You can use this equation to estimate population doubling time Modeling Exponential Growth • Many human populations are still growing exponentially • You can use this equation to estimate population doubling time • You can also use this equation to sensibly choose a bank account Modeling Exponential Growth • Many human populations are still growing exponentially • You can use this equation to estimate population doubling time • You can also use this equation to sensibly choose a bank account • It will be on the first exam Modeling Exponential Growth Computing population doubling time Recall: Nt = N0ert For the population to double, Nt = 2N0 Modeling Exponential Growth Computing population doubling time Recall: Nt = N0ert For the population to double, Nt = 2N0 So, 2N0 = N0ert , where t will be doubling time. Modeling Exponential Growth Computing population doubling time Recall: Nt = N0ert For the population to double, Nt = 2N0 So, 2N0 = N0ert , where t will be doubling time. By division, 2 = ert , and taking logarithms of both sides gives ln(2) = rt, and by division again ln(2)/r = time to doubling. Modeling Exponential Growth Computing population doubling time Recall: Nt = N0ert For the population to double, Nt = 2N0 So, 2N0 = N0ert , where t will be doubling time. By division, 2 = ert , and taking logarithms of both sides gives ln(2) = rt, and by division again ln(2)/r = time to doubling. Note that doubling time is independent of population size! A brief assignment • Determine the population sizes and the approximate intrinsic rates of increase, r, for your home town, your home state, and your home country (note: many databases give r as numbers per 1000, not per capita. What do you have to do to correct for that?) • Compute the estimated doubling time for your home town, your home state, and your home country. Next: violating assumptions and approaching reality