1. Population Ecology: Population Dynamics
Global human population
United Nations projections (2004)
(red, orange, green)
U. S. Census Bureau modern (blue)
& historical (black) estimates
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Wikipedia “Malthusian catastrophe” page – 07/IX/2014
Image from Wikimedia Commons

2. 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
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“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

3. 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)
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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.

4. Deterministic logistic growth
Population Dynamics
Deterministic logistic growth
Invariant density-dependent
vital rates dN N
= rN
1 – r dt K
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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

5. Population Dynamics Deterministic vs. stochastic logistic growth
Invariant density-dependent
vital rates
“Fuzzy” density-dependent
vital rates r ri
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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 in the textbook).
Stable equilibrium
carrying capacity
Fluctuating abundance within a range of values for carrying capacity
Bowman, Hacker & Cain (2017), Fig. 11.5

6. 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-)
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For more detail see: May, Robert M Simple mathematical models with very complicated dynamics. Nature 261:
= rN
1 – dt K
If r is large,
stable limit cycle
Bowman, Hacker & Cain (2017), Fig

7. 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
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For more detail see: May, Robert M Simple mathematical models with very complicated dynamics. Nature 261:
Photo from

8. Population Dynamics Population cycles & chaos
(scaled to max. size attainable)
Population size
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For more detail see: May, Robert M Simple mathematical models with very complicated dynamics. Nature 261:
Per capita rate of increase

9. 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
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The expected long-term r is clearly not 0 (the arithmetic mean of rgood & rbad)!

10. 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
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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

11. 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
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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

12. 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
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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

13. 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
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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:
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.

14. 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.
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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.

15. Population Size & Extinction Risk Environmental stochasticity
How could the previous exercise be modified
to illustrate environmental stochasticity?
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16. 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?
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17. Population Size & Extinction Risk
Allee Effects occur when average per capita fitness
declines as a population becomes smaller
Birth (b) ?
Rate ?
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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

18. Spatially-Structured Populations
Patchy population
(High rates of inter-patch dispersal, i.e., patches are well-connected)
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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 Empirical evidence for metapopulation dynamics. In: Metapopulation Biology: Ecology, Genetics and Evolution (ed. Ilkka A. Hanski & Michael E. Gilpin) Academic Press, San Diego, CA.

19. 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 Empirical evidence for metapopulation dynamics. In: Metapopulation Biology: Ecology, Genetics and Evolution (ed. Ilkka A. Hanski & Michael E. Gilpin) Academic Press, San Diego, CA.

20. 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
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Metapopulation – spatially isolated populations linked by dispersal.
See: Levins, Richard Some demographic and genetic consequences of environmental heterogeneity for biological control. Bulletin of the Entomological Society of America 15:
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 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)

21. 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 Some demographic and genetic consequences of environmental heterogeneity for biological control. Bulletin of the Entomological Society of America 15:
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 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)

22. Source-Sink Population Dynamics
Habitats vary in habitat quality;
occupied sink habitats broaden the realized niche
sink
source
source
source
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See: Pulliam, H. Ronald Sources, sinks, and population regulation. American Naturalist 132:
sink
sink
Original source-sink idea from Pulliam (1988)