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Population Ecology: Population Dynamics Image from Wikimedia Commons Global human population United Nations projections (2004) (red, orange, green) U.

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Presentation on theme: "Population Ecology: Population Dynamics Image from Wikimedia Commons Global human population United Nations projections (2004) (red, orange, green) U."— Presentation transcript:

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

2 The demographic processes that can change population size: Birth, Immigration, Death, Emigration B. I. D. E. (numbers of individuals in each category) Population Dynamics N t+1 = N t + B + I – D – E For an open population, observed at discrete time steps: For a closed population, observed through continuous time: dN dt = (b-d)N dN = rN (b-d) can be considered a proxy for average per capita fitness dt

3 Population Dynamics 5 main categories of population growth trajectories: Exponential growth Logistic growth Population fluctuations Regular population cycles Chaos

4 Population Dynamics Cain, Bowman & Hacker (2014), Fig Invariant density-dependent vital rates Stable equilibrium carrying capacity Deterministic logistic growth r dN dt = rN N K 1 –

5 Population Dynamics Cain, Bowman & Hacker (2014), Fig Deterministic vs. stochastic logistic growth Invariant density-dependent vital rates “Fuzzy” density-dependent vital rates Stable equilibrium carrying capacity Fluctuating abundance within a range of values for carrying capacity rriri

6 Population Dynamics Cain, Bowman & Hacker (2014), Fig dN dt = rN N (t-  ) K 1 – Instead of growth tracking current population size (as in logistic), growth tracks density at  units back in time Time lags can cause delayed density dependence, which can result in population cycles If r  is small, logistic If r  is intermediate, damped oscillations If r  is large, stable limit cycle

7 Sir Robert May, Baron of Oxford Population Dynamics Time lags can cause delayed density dependence, which can result in population cycles or chaos Photo from

8 Population Dynamics Per capita rate of increase Population size (scaled to max. size attainable) Population cycles & chaos

9 Is the long-term expected per capita growth rate (r) of a population simply an average across years? At t 0, N 0 =100 t 1 is a bad year, so N 1 = N 0 + (r bad * N 0 ) = 50 t 2 is a good year, so N 2 = N 1 + (r good *N 1 ) = 75 Consider this hypothetical example: r good = 0.5; r bad = -0.5 If the numbers of good & bad years are equal, is the following true? r expected = [r good + r bad ] / 2 Variation in r and population growth The expected long-term r is clearly not 0 (the arithmetic mean of r good & r bad )!

10 Variation in and population growth Cain, Bowman & Hacker (2014), Analyzing Data 11.1, pg. 258 N t+1 = N t = NtNt N t Arithmetic mean = 1.02 Geometric mean = 1.01 A fluctuating population

11 Variation in and population growth Cain, Bowman & Hacker (2014), Analyzing Data 11.1, pg. 258 N t+1 = N t = NtNt N t Arithmetic mean = 1.02 Geometric mean = 1.02 A steadily growing population

12 Variation in and population growth Cain, Bowman & Hacker (2014), Analyzing Data 11.1, pg. 258 N t+1 = N t = NtNt N t Arithmetic mean = 1.01 Geometric mean = 1.01 A steadily growing population Which mean (arithmetic or geometric) best captures the trajectory of the fluctuating population (the example given in the textbook)?

13 Deterministic  r < 0 Genetic stochasticity & inbreeding Small populations are especially prone to extinction from both deterministic and stochastic causes Population Size & Extinction Risk Demographic 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) Catastrophes

14 Each student is a sexually reproducing, hermaphroditic, out-crossing annual plant. Arrange the plants into small sub-populations (2-3 plants/pop.). In the first growing season (generation), each plant mates (if there is at least 1 other individual in the population) and produces 2 offspring. Offspring have a 50% chance of surviving to the next season. flip a coin for each offspring; “head” = lives, “tail” = dies. Note that average r = 0; each parent adds 2 births to the population and on average subtracts 2 deaths [self & 1 offspring – since 50% of offspring live and 50% die] prior to the next generation. Demographic stochasticity Population Size & Extinction Risk

15 Environmental stochasticity Population Size & Extinction Risk How could the previous exercise be modified to illustrate environmental stochasticity?

16 Natural catastrophes Population Size & Extinction Risk What are the likely consequences to populations of sizes: 10; 100; 1000; 1,000,000 if 90% of individuals die in a flood?

17 Density (N) K Zone of Allee Effects Birth (b) Death (d) Rate ? ? Population Size & Extinction Risk Allee Effects occur when average per capita fitness declines as a population becomes smaller

18 Spatially-Structured Populations Patchy population (High rates of inter-patch dispersal, i.e., patches are well-connected)

19 Spatially-Structured Populations Mainland-island model (Unidirectional dispersal from mainland to islands)

20 Spatially-Structured Populations Classic Levins-type metapopulation (collection of populations) model (Vacant patches are re-colonized from occupied patches at low to intermediate rates of dispersal ) Original metapopulation idea from Levins (1969) occupied unoccupied Assumptions of the basic model: 1. Infinite number of identical habitat patches 2. Patches have identical colonization probabilities (spatial arrangement is irrelevant) 3. Patches have identical local extinction (extirpation) probabilities 4. A colonized patch reaches K instantaneously (within-patch population dynamics are ignored)

21 Spatially-Structured Populations Classic Levins-type metapopulation (collection of populations) model (Vacant patches are re-colonized from occupied patches at low to intermediate rates of dispersal ) Original metapopulation idea from Levins (1969) occupied unoccupied dp dt = cp(1 - p) - ep c = patch colonization rate e = patch extinction rate p = proportion of patches occupied Key result: metapopulation persistence requires (e/c)<1

22 Habitats vary in habitat quality; occupied sink habitats broaden the realized niche Source-Sink Population Dynamics Original source-sink idea from Pulliam (1988) source sink


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