Population Ecology: Population Dynamics 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/2014 Image from Wikimedia Commons

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

Population fluctuations Population Dynamics 5 main categories of population growth trajectories: Exponential growth Logistic growth Population fluctuations (all pops. fluctuate – some erratically, some very regularly) Population cycles (regular fluctuations) Chaos (appears noise-like, but is constrained) Please do not use the images in these PowerPoint slides without permission. Regular population cycles can be thought of as special cases of population fluctuations. In addition, these are not mutually exclusive – e.g., population fluctuations can occur with logistic growth. Ask students to draw each of these. The textbook considers all but chaos.

Deterministic logistic growth Population Dynamics Deterministic logistic growth Invariant density-dependent vital rates dN N = rN 1 – r dt K 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. Note that the term (1-N/K) in the equation is the density-dependent dampener of population growth rate – commonly interpreted to result from intraspecific competition, but could occur for other reasons (predation, parasitism, etc.). One can interpret r as average fitness relative to fitness at carrying capacity equilibrium. Stable equilibrium carrying capacity Bowman, Hacker & Cain (2017), Fig. 11.5

Population Dynamics Deterministic vs. stochastic logistic growth Invariant density-dependent vital rates “Fuzzy” density-dependent vital rates r ri 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. Note that the second figure would result in a general logistic-growth pattern, with fluctuations (as in Fig. 11.4 in the textbook). Stable equilibrium carrying capacity Fluctuating abundance within a range of values for carrying capacity Bowman, Hacker & Cain (2017), Fig. 11.5

Population Dynamics Time lags can cause delayed density dependence, which can result in population cycles If r is small, logistic Instead of growth tracking current population size (as in logistic), growth tracks density at  units back in time If r is intermediate, damped oscillations dN N(t-) Please 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. = rN 1 – dt K If r is large, stable limit cycle Bowman, Hacker & Cain (2017), Fig. 11.10

Sir Robert May, Baron of Oxford Population Dynamics Time lags can cause delayed density dependence, which can result in population cycles or chaos Sir Robert May, Baron of Oxford Please 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. Photo from http://www.topbritishinnovations.org/PastInnovations/BiologicalChaos.aspx

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

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

Variation in  and population growth Nt+1 = Nt Nt+1  = Nt A fluctuating population Arithmetic mean  = 1.02 Geometric mean  = 1.01 1.21 0.87 Please 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))/n Geometric mean = (Product(xi))^(1/n) -- i.e., the nth root of the product 1.17 1.02 1.13 Bowman, Hacker & Cain (2017), Analyzing Data 11.1, pg. 258

Variation in  and population growth Nt+1 = Nt Nt+1  = Nt A steadily growing population 1000 1.02 Arithmetic mean  = 1.02 1020 1.02 Geometric mean  = 1.02 1040 1.02 1061 1.02 Please do not use the images in these PowerPoint slides without permission. Arithmetic mean = Sum(xi)/n Geometric mean = Product(xi)^(1/n) -- i.e., the nth root of the product 1082 1.02 1104 1.02 1126 1.02 1148 Bowman, Hacker & Cain (2017), Analyzing Data 11.1, pg. 258

Variation in  and population growth Nt+1 = Nt Nt+1  = Nt A steadily growing population 1000 1.01 Arithmetic mean  = 1.01 1010 1.01 Geometric mean  = 1.01 1020 1.01 1030 1.01 Please do not use the images in these PowerPoint slides without permission. Arithmetic mean = Sum(xi)/n Geometric mean = Product(xi)^(1/n) -- i.e., the nth root of the product Which mean (arithmetic or geometric) best captures the trajectory of the fluctuating population (the example given in the textbook)? 1040 1.01 1051 1.01 1061 1.01 1072 Bowman, Hacker & Cain (2017), Analyzing Data 11.1, pg. 258

Population Size & Extinction Risk Small populations are especially prone to extinction from both deterministic and stochastic causes Deterministic  r < 0 Genetic stochasticity & inbreeding 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 Please 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 Holsinger’s Conservation Biology Web site, especially re “Biology of Small Populations” Deterministic threats cause r<0. Stochastic threats cause variability in r: demographic stochasticity (or uncertainty) concerns chance events that affect indiv. mort. and reprod., whereas environmental stochasticity (or uncertainty) concerns temporal fluctuations in prob. of mort. or reprod. experienced simultaneously by all indivs.

Population Size & Extinction Risk Demographic stochasticity 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 summer-time growing season (i.e., first generation), each plant mates (if there is at least 1 other individual in the population), produces 2 seeds (offspring) in the fall, and dies before winter. Offspring have a 50% chance of surviving to the next growing season. Flip a coin for each offspring; “head” = lives, “tail” = dies. Note that on average: (birth – death) = 0 Each parent adds 2 births to the population and subtracts 2 deaths [self & 1 offspring – since 50% of offspring live and 50% die] prior to the next generation. Please do not use the images in these PowerPoint slides without permission. As a good class-wide comparison, run one round deterministically, then run one round with coin-flipping to add stochasticity. In the first growing season (generation), each student 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. In a large pop. (e.g., whole class), heads and tails average out to give r=0 (no change in pop. size). When class is sub-divided into small sub-populations (e.g., 2 individuals each with no migration), some will have less than 2 live individuals after the coins are flipped to determine survivorship to the next growing season (the next generation). It would be especially instructive to compare the population trajectory for the class as a whole for the same number of generations as there are groups of 2 in the class (since some of the groups of 2 will perish in 1 generation [and the overall meta-population is likely to decline], whereas for the whole class as a single panmictic population many generations can be expected before extinction would become likely). Remember also that even this does not take into account individual variability in r itself; as advocated by Dan Doak and Bill Morris one should use a distribution of values for r.

Population Size & Extinction Risk Environmental stochasticity How could the previous exercise be modified to illustrate environmental stochasticity? Please do not use the images in these PowerPoint slides without permission.

Population Size & Extinction Risk Natural catastrophes What are the likely consequences to populations of sizes: 10; 100; 1000; 1,000,000 if 90% of individuals die in a flood? Please do not use the images in these PowerPoint slides without permission.

Population Size & Extinction Risk Allee Effects occur when average per capita fitness declines as a population becomes smaller Birth (b) ? Rate ? Please do not use the images in these PowerPoint slides without permission. Note that if the b & d curves cross in the Allee-effect zone there is a minimum finite positive population size below which the population is never viable. Potential causes of Allee effects: (1) difficulty finding mates at low pop. densities; (2) difficulty avoiding predators in group-living species; etc. Death (d) K Density (N) Zone of Allee Effects

Spatially-Structured Populations Patchy population (High rates of inter-patch dispersal, i.e., patches are well-connected) Please do not use the images in these PowerPoint slides without permission. Classic Levins-type metapopulations represent just one type of spatially-structured population. Others include: mainland-island (in which dispersal is entirely or nearly entirely from mainland to islands); patchy population (with high enough rates of dispersal among patches that the population acts as a single entity, which is why Levins stipulated “low to intermediate rates of dispersal” for a metapopulation); etc. See: Harrison S. & A. D. Taylor. 1997. Empirical evidence for metapopulation dynamics. In: Metapopulation Biology: Ecology, Genetics and Evolution (ed. Ilkka A. Hanski & Michael E. Gilpin) Academic Press, San Diego, CA.

Spatially-Structured Populations Mainland-island model (Unidirectional dispersal from mainland to islands) Please do not use the images in these PowerPoint slides without permission. Classic Levins-type metapopulations represent just one type of spatially-structured population. Others include: mainland-island (in which dispersal is entirely or nearly entirely from mainland to islands); patchy population (with high enough rates of dispersal among patches that the population acts as a single entity, which is why Levins stipulated “low to intermediate rates of dispersal” for a metapopulation); etc. See: Harrison S. & A. D. Taylor. 1997. Empirical evidence for metapopulation dynamics. In: Metapopulation Biology: Ecology, Genetics and Evolution (ed. Ilkka A. Hanski & Michael E. Gilpin) Academic Press, San Diego, CA.

Spatially-Structured Populations Classic Levins-type metapopulation (collection of populations) model (Vacant or vacated patches are re-colonized from occupied patches at low to intermediate rates of dispersal) Assumptions of the basic model: occupied 1. Infinite number of identical habitat patches embedded in inhospitable matrix occupied unoccupied 2. Patches have identical colonization probabilities (spatial arrangement is irrelevant) unoccupied Please do not use the images in these PowerPoint slides without permission. Metapopulation – spatially isolated populations linked by dispersal. See: Levins, Richard. 1969. Some demographic and genetic consequences of environmental heterogeneity for biological control. Bulletin of the Entomological Society of America 15:237-240. Classic Levins-type metapopulations represent just one type of spatially-structured population. Others include: mainland-island (in which dispersal is entirely or nearly entirely from mainland to islands); patchy population (with high enough rates of dispersal among patches that the population acts as a single entity, which is why Levins stipulated “low to intermediate rates of dispersal” for a metapopulation); etc. See: Harrison S. & A. D. Taylor. 1997. Empirical evidence for metapopulation dynamics. In: Metapopulation Biology: Ecology, Genetics and Evolution (ed. Ilkka A. Hanski & Michael E. Gilpin) Academic Press, San Diego, CA. occupied 3. Patches have identical local extinction (extirpation) probabilities occupied 4. A colonized patch reaches K instantaneously (within-patch population dynamics are ignored) Original metapopulation idea from Levins (1969)

Spatially-Structured Populations Classic Levins-type metapopulation (collection of populations) model (Vacant or vacated patches are re-colonized from occupied patches at low to intermediate rates of dispersal) dp = cp(1 - p) - ep occupied dt occupied p = proportion of patches occupied unoccupied c = patch colonization rate unoccupied e = patch extinction rate Please do not use the images in these PowerPoint slides without permission. See: Levins, Richard. 1969. Some demographic and genetic consequences of environmental heterogeneity for biological control. Bulletin of the Entomological Society of America 15:237-240. Classic Levins-type metapopulations represent just one type of spatially-structured population. Others include: mainland-island (in which dispersal is entirely or nearly entirely from mainland to islands); patchy population (with high enough rates of dispersal among patches that the population acts as a single entity, which is why Levins stipulated “low to intermediate rates of dispersal” for a metapopulation); etc. See: Harrison S. & A. D. Taylor. 1997. Empirical evidence for metapopulation dynamics. In: Metapopulation Biology: Ecology, Genetics and Evolution (ed. Ilkka A. Hanski & Michael E. Gilpin) Academic Press, San Diego, CA. occupied Key results: Metapopulation persistence requires (e/c)<1 At equilibrium it is generally the case that not all patches are occupied occupied Original metapopulation idea from Levins (1969)

Source-Sink Population Dynamics Habitats vary in habitat quality; occupied sink habitats broaden the realized niche sink source source source Please do not use the images in these PowerPoint slides without permission. See: Pulliam, H. Ronald. 1988. Sources, sinks, and population regulation. American Naturalist 132:652-661. sink sink Original source-sink idea from Pulliam (1988)