Population Ecology: Growth & Regulation
Photo of introduced (exotic) rabbits at “plague proportions” in Australia from Wikimedia Commons.
Life Cycle Diagram
4
2
0.8
0.3
0
1
0
2
3
age
survival
fecundity
seed
seedling
1 to 2 yr old
adult
2 to 3 yr old
adult
Life Table (a.k.a. Actuarial Table)
Demographic rates often vary with
age, size or stage
Cain, Bowman & Hacker (2014), Table 10.3
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
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
Cain, Bowman & Hacker (2014), Table 10.3
Life Table (a.k.a. Actuarial Table)
Population growth from t0 (beginning population size)
to t1 (one year later)
F1 = 2, so 6 x 2 = 12
F2 = 4, so 24 x 4 = 96
Cain, Bowman & Hacker (2014), Table 10.4
108 offspring
Life Table (a.k.a. Actuarial Table)
Population growth from t0 (beginning population size)
to t1 (one year later)
Population growth rate =  =
Cain, Bowman & Hacker (2014), Table 10.4
Nt+1
138
=
Nt
= 1.38
100
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
13 = 1.32
etc. = 1.32
Cain, Bowman & Hacker (2014), Fig. 10.8 B
Leslie Matrix
Age-structured matrix model (L) of population growth parameters
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
Lefkovitch Matrix
Stage-structured matrix model (L) of population growth parameters
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
Population Age Structure
Useful for predicting population growth
Age structure for China in 2014 from Wikimedia Commons; China implemented a “one-child policy” in 1960s
Survivorship Curves
Which is most likely to
characterize an
r-selected species?
K-selected species?
Cain, Bowman & Hacker (2014), Fig. 10.5
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
Cain, Bowman & Hacker (2014), Fig. 10.10
Exponential Growth
Exponential growth when reproduction occurs “continuously”
Reproducing is not
synchronous in discrete
time periods
dN
= rN
dt
N(t) = N(0)ert
r=
Exponential growth
rate
or
Per capita intrinsic
rate of increase
Cain, Bowman & Hacker (2014), Fig. 10.10
Exponential Growth
Geometric
Nt = tN0
Exponential N(t) = N(0)ert = ertN(0)
 = er
r = ln()
Exponential
decline /
decay
Cain, Bowman & Hacker (2014), Fig. 10.11
Constant
population
size
Exponential
growth
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 necessary for survival & reproduction
continues expanding indefinitely as the population expands.
Original idea from Turchin (2001) Oikos
Laws of Thermodynamics
1st Law of Thermodynamics  Law of Conservation of Energy
Related to Law of Conservation of Mass
E=mc2
Sun
Earth
Bio-geochemical
processes
Image of Carnot engine from Wikimedia Commons
Limited Scope for Population Increase
“No population can increase in size forever.”
Number of particles
in the universe
Quote from Cain, Bowman & Hacker (2014), pg. 227
< 10 < 
100
Limits to Exponential Growth
Density independent
Density-independent
factors can limit
population size
Density dependent
Cain, Bowman & Hacker (2014), Fig. 10.14
Limits to Exponential Growth
Density independent
Density-independent
factors can limit
population size
Density dependent
Density-dependent
factors can regulate
population size
Cain, Bowman & Hacker (2014), Fig. 10.14
Logistic Growth
r = Intrinsic Rate
of Increase
K = Carrying
Capacity
Cain, Bowman & Hacker (2014), Fig. 10.18
r- vs. K-selection
r = Intrinsic Rate
of Increase
K = Carrying
Capacity
Cain, Bowman & Hacker (2014), Fig. 10.18