1. Population Ecology: Growth & Regulation
Please do not use the images in these PowerPoint slides without permission.
Wikipedia “Myxomatosis” page – 07/IX/2014
Photo of introduced (exotic) rabbits at “plague proportions” in Australia from Wikimedia Commons.

2. 1 2 3 Life Cycle Diagram age 2 4 0.3 0.8 survival fecundity1 2 3
age
Please do not use the images in these PowerPoint slides without permission.
A life cycle diagram could be constructed based on age-, stage-, or size-classes. This one is constructed for the age classes of a plant, from seed to 1-yr old adult and 2-yr old adult, as in Table 10.3 of the textbook.
survival
fecundity
1 to 2 yr old
adult
2 to 3 yr old
adult
seed
seedling

3. Life Table (a.k.a. Actuarial Table)
Demographic rates often vary with
age, size or stage
Please do not use the images in these PowerPoint slides without permission.
Note: these could be age-, size-, or stage-specific; the example in Table 10.3 is for an age-specific life table.
Bowman, Hacker & Cain (2017), Table 10.3

4. Life Table (a.k.a. Actuarial Table)
Cohort Life Table
Fates of individuals in a cohort are followed from birth to death
Static Life Table
Survival & reproduction of individuals of known age are
assessed for a given time period
Please do not use the images in these PowerPoint slides without permission.
Note: The resulting tables are the same; it’s the process to create them that determines whether or not you have a cohort or static life table!

5. Life Table (a.k.a. Actuarial Table)
Sx = Age-specific survival rate; prob. surviving from age x to x+1
lx = Survivorship; proportion surviving from birth (age 0) to age x
Fx = Age-specific fecundity; average number of offspring
produced by a female at age x
Please do not use the images in these PowerPoint slides without permission.
Note: these could be age-, size-, or stage-specific; the example in Table 10.3 is for an age-specific life table.
Bowman, Hacker & Cain (2017), Table 10.3

6. Life Table (a.k.a. Actuarial Table)
Population growth from t0 (beginning population size)
to t1 (one year later)
Please do not use the images in these PowerPoint slides without permission.
Each of the 6 1-yr olds has 2 offspring = 12; each of the 24 2-yr olds has 4 offspring = 96; = 108.
F1 = 2, so 6 x 2 = 12
F2 = 4, so 24 x 4 = 96
108 offspring
Bowman, Hacker & Cain (2017), Table 10.4

7. Life Table (a.k.a. Actuarial Table)
Population growth from t0 (beginning population size)
to t1 (one year later)
Please do not use the images in these PowerPoint slides without permission.
Nt+1
138
Population growth rate =  = =
= 1.38 Nt
100
Bowman, Hacker & Cain (2017), Table 10.4

8. Life Table (a.k.a. Actuarial Table)
If age-specific survival & fecundity remain constant, the population
settles into a stable age distribution and population growth rate
1 = 1.38
11 = 1.32
12 = 1.32
2 = 0.87
13 = 1.32
etc. = 1.32
Please do not use the images in these PowerPoint slides without permission.
Bowman, Hacker & Cain (2017), Fig B

9. Age-structured matrix model (L) of population growth parameters
Leslie Matrix
Age-structured matrix model (L) of population growth parameters
Please do not use the images in these PowerPoint slides without permission.
Wikipedia “Leslie matrix” page; accessed 17-IX-2014
“It was invented by and named after Patrick H. Leslie.”
Note that L (bold) is the matrix notation for the Leslie matrix. To multiply Nt by L, multiply the elements of the column Nt by the elements in each row of L.
Age structure at t+1
Age-specific
survival & fecundity
Age structure at t
Dominant Eigenvalue of L = 
Dominant Eigenvector of L = stable age distribution
Example of a Leslie matrix from Wikimedia Commons

10. Stage-structured matrix model (L) of population growth parameters
Lefkovitch Matrix
Stage-structured matrix model (L) of population growth parameters
Please do not use the images in these PowerPoint slides without permission.
Wikpedia “Leslie matrix” page; accessed 17-IX-2014
Note that L (bold) is the matrix notation for the Lefkovitch matrix. To multiply Nt by L, multiply the elements of the column Nt by the elements in each row of L.
Stage structure at t+1
Stage-specific
survival & fecundity
Stage structure at t
Dominant Eigenvalue of L = 
Dominant Eigenvector of L = stable stage distribution
Example of a Lefkovitch matrix adapted from Leslie matrix from Wikimedia Commons

11. Population Age Structure
Useful for predicting population growth
Please do not use the images in these PowerPoint slides without permission.
Wikpedia “Population pyramid” page; accessed 17-IX-2014
“China had an extreme youth bulge until the 1960s, when it sharply curbed partly as an effect of the one-child policy.”
The one-child policy was effective at reining in population growth, but is ethically deplorable.
Age structure for China in 2014 from Wikimedia Commons; China implemented a “one-child policy” in 1960s

12. Which is most likely to characterize an
Survivorship Curves
Which is most likely to characterize an
r-selected species?
K-selected species?
Please do not use the images in these PowerPoint slides without permission.
Note: The y-axis is number of survivors or survivorship, NOT survival rate (think through what that curve would look like for Type III)!
Bowman, Hacker & Cain (2017), Fig. 10.5

13. Geometric population growth rate
Exponential Growth
Geometric growth when reprod. occurs at regular time intervals
Population grows by a constant proportion in each time step
Nt+1 = Nt
Nt = tN0
 =
Geometric population growth rate or
Per capita finite
rate of increase
Please do not use the images in these PowerPoint slides without permission.
The blue dots plot geometric growth.
The first equation gives the change in population size over one time interval.
See: Aronhime, B. et al Teaching exponential and logistic growth in a variety of classroom and laboratory settings. Teaching Issues and Experiments in Ecology 9:1-26.
Bowman, Hacker & Cain (2017), Fig

14. Exponential Growth
Exponential growth when reproduction occurs “continuously”
Reproduction is not synchronous in discrete time periods dN
= rN dt
N(t) = N(0)ert
r =
Exponential population growth rate or
Per capita intrinsic rate of increase
Please do not use the images in these PowerPoint slides without permission.
The red line plot exponential growth.
dN/dt represents the rate of change in pop. size at each instant in time.
Exponential growth occurs when the rate of change is proportional to the current population size.
See: Aronhime, B. et al Teaching exponential and logistic growth in a variety of classroom and laboratory settings. Teaching Issues and Experiments in Ecology 9:1-26.
Bowman, Hacker & Cain (2017), Fig

15. Exponential Growth  = er Nt = tN0 Geometric Exponential
N(t) = N(0)ert = ertN(0)
 = er
r = ln()
Exponential
decline /
decay
Constant
population
size
Exponential
growth
Please do not use the images in these PowerPoint slides without permission.
By rearranging we see how  is related to r.
See: Aronhime, B. et al Teaching exponential and logistic growth in a variety of classroom and laboratory settings. Teaching Issues and Experiments in Ecology 9:1-26.
Bowman, Hacker & Cain (2017), Fig

16. The Fundamental Law of Population Ecology
Peter Turchin
“A population will grow… exponentially as long as the environment
experienced by all individuals in the population remains constant.”
In other words, as long as the amount of
resources (energy & material) necessary for survival & reproduction
continues expanding indefinitely as the population expands towards infinity.
Please do not use the images in these PowerPoint slides without permission.
Peter Turchin Does population ecology have general laws? Oikos 94:17-26.
Original idea from Turchin (2001) Oikos

17. Laws of Thermodynamics
1st Law of Thermodynamics  Law of Conservation of Energy
Related to Law of Conservation of Mass
E=mc2
Earth
(finite mass & energy)
Please do not use the images in these PowerPoint slides without permission.
Wikipedia “Carnot heat engine” page; downloaded 12/IX/2014
Energy (E) and mass (m) are oddly interconvertible (i.e., E=mc2).
“Carnot engine diagram (modern) - where an amount of heat QH flows from a high temperature TH furnace through the fluid of the "working body" (working substance) and the remaining heat QC flows into the cold sink TC, thus forcing the working substance to do mechanical work W on the surroundings, via cycles of contractions and expansions.”
Sun
Bio-geo-chemical processes
Image of Carnot engine from Wikimedia Commons

18. Limited Scope for Population Increase
“No population can increase in size forever.”
Number of atoms
in the universe
(finite mass)  
<
<
Exponential growth potential of E. coli 
beginning with 1 cell, 6 days for population > 1080 cells
Please do not use the images in these PowerPoint slides without permission.
Most estimates of the number of particles in the universe are less than a googol (10 raised to the 100th power); however large, a googol is a finite number; i.e., there are always finite resources to build living organisms.
Quote from Bowman, Hacker & Cain (2017), pg. 227

19. Limits to Exponential Growth
Density independent
Density-independent factors can limit population size
Density dependent
Please do not use the images in these PowerPoint slides without permission.
Bowman, Hacker & Cain (2017), Fig

20. Limits to Exponential Growth
Density independent
Density-independent factors can limit population size
Density dependent
Density-dependent factors can regulate population size
Please do not use the images in these PowerPoint slides without permission.
Bowman, Hacker & Cain (2017), Fig

21. Logistic Growth r = Intrinsic Rate of Increase K = Carrying Capacity
Please do not use the images in these PowerPoint slides without permission.
Does r change along the time axis in logistic growth?
No; r is constant – it determines the exponential growth potential of the pop. 
The density-dependent term [1-(N/K)] then determines how much of the exponential growth potential the population experiences.
Bowman, Hacker & Cain (2017), Fig

22. r- vs. K-selection r = Intrinsic Rate of Increase K = Carrying
Capacity
Please do not use the images in these PowerPoint slides without permission.
Note that r- & K-selection are named owing to the general tendency for individuals in r-selected species to often find themselves in populations at the left-hand side of the figure, whereas individuals in K-selected species generally constitute populations at the right-hand side of the figure (blue curve).
Bowman, Hacker & Cain (2017), Fig