# population ecology: population dynamics

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Population Ecology: Population Dynamics. Global human population United Nations projections (2004) ( red , orange , green ) U. S. Census Bureau modern ( blue ) & historical ( black ) estimates. Image from Wikimedia Commons. Population Dynamics. - PowerPoint PPT PresentationTRANSCRIPT

Slide 1

Population Ecology: Population DynamicsImage from Wikimedia Commons

Global human population

United Nations projections (2004) (red, orange, green)

U. S. Census Bureau modern (blue) & historical (black) estimates

Please do not use the images in these PowerPoint slides without permission.

Wikipedia Malthusian catastrophe page 07/IX/20141The demographic processes that can change population size:Birth, Immigration, Death, EmigrationB. I. D. E. (numbers of individuals in each category)Population DynamicsNt+1= Nt + B + I D E For an open population, observed at discrete time steps: For a closed population, observed through continuous time: dNdt = (b-d)NdN = rN(b-d) can be considered a proxy for average per capita fitnessdtPlease do not use the images in these PowerPoint slides without permission.

Open is with respect to immigration / emigration.

For discrete time periods in open population (each variable is the number of individuals in that particular category):Nt+1 = Nt + B + I D E

For rates in continuous time in closed population:b = per capita birth rated = per capita death rate

2Population Dynamics5 main categories of population growth trajectories:

Exponential growthLogistic growthPopulation fluctuationsRegular population cyclesChaosPlease do not use the images in these PowerPoint slides without permission.

Regular population cycles can be thought of as special cases of population fluctuations.

Draw each of these on the chalkboard.

Your textbook considers all but chaos.3Population Dynamics

Cain, Bowman & Hacker (2014), Fig. 11.5Invariant density-dependent vital ratesStable equilibrium carrying capacityDeterministic logistic growthrdNdt = rNNK 1 Please do not use the images in these PowerPoint slides without permission.

Note that either birth or death could be density dependent; they do not both need to be for a population to be regulated.4Population Dynamics

Cain, Bowman & Hacker (2014), Fig. 11.5Deterministic vs. stochastic logistic growthInvariant density-dependent vital ratesFuzzy density-dependent vital ratesStable equilibrium carrying capacityFluctuating abundance within a range of values for carrying capacityrri Please do not use the images in these PowerPoint slides without permission.

Note that either birth or death could be density dependent; they do not both need to be for a population to be regulated.

I am using ri to indicate multiple values of r, either across various individuals or among seasons of good and bad conditions, etc.5Population Dynamics

Cain, Bowman & Hacker (2014), Fig. 11.10dNdt = rNN(t-)K 1 Instead of growth tracking current population size (as in logistic), growth tracks density at units back in timeTime lags can cause delayed density dependence,which can result in population cyclesIf r is small, logisticIf r is intermediate,damped oscillationsIf r is large,stable limit cyclePlease do not use the images in these PowerPoint slides without permission.

For more detail see: May, Robert M. 1976. Simple mathematical models with very complicated dynamics. Nature 261:459-467.

6Sir Robert May, Baron of Oxford

Population DynamicsTime lags can cause delayed density dependence, which can result in population cycles or chaosPhoto from http://www.topbritishinnovations.org/PastInnovations/BiologicalChaos.aspxPlease do not use the images in these PowerPoint slides without permission.

For more detail see: May, Robert M. 1976. Simple mathematical models with very complicated dynamics. Nature 261:459-467.

7Population Dynamics

Per capita rate of increasePopulation size (scaled to max. size attainable)Population cycles & chaosPlease do not use the images in these PowerPoint slides without permission.

For more detail see: May, Robert M. 1976. Simple mathematical models with very complicated dynamics. Nature 261:459-467.

8Is the long-term expected per capita growth rate (r) of a population simply an average across years? At t0, N0=100t1 is a bad year, so N1 = N0 + (rbad* N0) = 50t2 is a good year, so N2 = N1 + (rgood*N1) = 75Consider this hypothetical example:rgood = 0.5; rbad = -0.5If the numbers of good & bad years are equal, is the following true?rexpected = [rgood + rbad] / 2Variation in r and population growthThe expected long-term r is clearly not 0 (the arithmetic mean of rgood & rbad)!Please do not use the images in these PowerPoint slides without permission.

9Variation in and population growth

Cain, Bowman & Hacker (2014), Analyzing Data 11.1, pg. 258Nt+1 = Nt =NtNt+11.210.871.171.021.13Arithmetic mean = 1.02Geometric mean = 1.01A fluctuating populationPlease do not use the images in these PowerPoint slides without permission.

I filled in the missing values in the table from the textbook.

Arithmetic mean = Sum(xi)/nGeometric mean = Product(xi)^(1/n) -- i.e., the nth root of the product

10Variation in and population growth

Cain, Bowman & Hacker (2014), Analyzing Data 11.1, pg. 258Nt+1 = Nt =NtNt+11.021.021.021.021.02Arithmetic mean = 1.02Geometric mean = 1.02A steadily growingpopulation1.021.0210401061108211041126102010001148Please do not use the images in these PowerPoint slides without permission.

Arithmetic mean = Sum(xi)/nGeometric mean = Product(xi)^(1/n) -- i.e., the nth root of the product

11Variation in and population growth

Cain, Bowman & Hacker (2014), Analyzing Data 11.1, pg. 258Nt+1 = Nt =NtNt+11.011.011.011.011.01Arithmetic mean = 1.01Geometric mean = 1.01A steadily growingpopulation1.011.0110201030104010511061101010001072Which mean (arithmetic or geometric) best captures the trajectory of the fluctuating population (the example given in the textbook)?Please do not use the images in these PowerPoint slides without permission.

Arithmetic mean = Sum(xi)/nGeometric mean = Product(xi)^(1/n) -- i.e., the nth root of the product

12Deterministic r < 0Genetic stochasticity & inbreedingSmall populations are especially prone to extinction from both deterministic and stochastic causesPopulation Size & Extinction RiskDemographic stochasticity individual variability around r (e.g., variance at any given time)

Environmental stochasticity temporal fluctuations of r (e.g., change in mean with time)

CatastrophesPlease do not use the images in these PowerPoint slides without permission.

Genetic stochasticity mostly involves genetic drift, founder effects, etc.

Note that demographic stochasticity differs from environmental stochasticity in that demographic stochasticity concerns the variance around r, whereas environmental stochasticity concerns the temporal variation in mean r.

For more info. on deterministic and stochastic causes of population change, see: M. S. Boyce. Population viability analysis. Annual Review of Ecology and Systematics 23:481-506.

Also see: Kent Holsingers Conservation Biology Web site, especially re Biology of Small Populations

Deterministic threats cause r

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