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

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

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

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 –

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

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

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

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

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 )!

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

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

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)?

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

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

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

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?

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

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

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

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)

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

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