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Lectures6-7_4pp

# Lectures6-7_4pp - So you know something about how...

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So you know something about how individual organisms make a living… Collections of organisms of the same species living in the same place at the same time are called a population . Population density – number of individuals per unit area. Why care about population growth and limits to population size? Too much? Or too few? Population Growth and Demography MATH AHEAD A simple model of population growth •Open population - numbers change due to immigration and emigration from outside the population in addition to births and deaths •Closed population - numbers change only due to births and deaths N t + B B = # births N t+1 = N = population size t = time – D D = # deaths + I I = # immigrants – E E = # emigrants In one location, i.e., a local but open population

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Two types of population growth Exponential or geometric: the per-capita growth rate is constant or density independent Logistic (asymptotic or sigmoidal): the per-capita growth rate varies with population density or is density dependent Exponential Logistic Time Population Size Density-independent population growth in a closed population N (t+1) = N (t) + b . N (t) - d . N (t) B = bN (t) ; D = dN (t) " N/ " t = (b-d)N Let b = birth rate per individual (# births per individual per unit time) Let d = death rate per individual (# dying per individual per unit time) N (t+1) - N t = (b-d)N t dN/dt = (b-d)N (as time --> 0) N t+1 = N t + B – D + I – E This is a continuous time expression: works if organisms reproduce continuously (overlapping generations) dN/dt = rN b-d = (birth rate - death rate) = the instantaneous rate of increase lets call this r, the intrinsic growth rate of the population Integrate from 0 to time t to predict population size at time t: N t = N 0 e rt N t = # at some time in the future N 0 = # at time = 0 e = 2.717 (base of natural log, or ln) r = intrinsic rate of increase t = time Intuitively, a population increases in size when its birth rate exceeds its death rate: b > d or when r is positive; r = 0 when b = d r > 0 when b > d r < 0 when b < d Population size -200 0 200 400 600 800 1000 1200 0 1 2 3 4 5 6 7 8 9 10 11 time r = 0 r = -0.3 r = 0.3 r = 0.2 r = 0.1 Small changes in r make a big difference -100 0 100 200 300 400 500 600 700 800 Population Size 0 1 2 3 4 5 6 7 8 9 10 11 time N 0 = 5 r = 0.5 N 0 =50, r = 0.1 The influence of N 0 vs. r on population growth dN/dt = rN
Doubling time: time required for population size to double N t = N 0 e rt 2 N 0 = N 0 e rt

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