Population Ecology: Growth & Regulation

Photo of introduced (exotic) rabbits at “plague proportions” in Australia from Wikimedia Commons.

Life Cycle Diagram

0 1 2 30.3 0.8 0

2 4

seed 1 to 2 yr old adult

2 to 3 yr old adult

seedling

age

survival

fecundity

Life Table (a.k.a. Actuarial Table)

Cain, Bowman & Hacker (2014), Table 10.3

Demographic rates often vary withage, size or stage

Life Table (a.k.a. Actuarial Table)

Cohort Life Table Fates of individuals in a cohort are followed from birth to death

Static Life TableSurvival & 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

Cain, Bowman & Hacker (2014), Table 10.3

lx = Survivorship; proportion surviving from birth (age 0) to age xFx = Age-specific fecundity; average number of offspring

produced by a female at age x

Life Table (a.k.a. Actuarial Table)

Population growth from t0 (beginning population size) to t1 (one year later)

Cain, Bowman & Hacker (2014), Table 10.4

F1 = 2, so 6 x 2 = 12F2 = 4, so 24 x 4 = 96

108 offspring

Life Table (a.k.a. Actuarial Table)

Population growth from t0 (beginning population size) to t1 (one year later)

Cain, Bowman & Hacker (2014), Table 10.4

Nt+1

Nt

Population growth rate = = = 1.38138

100=

Life Table (a.k.a. Actuarial Table)

Cain, Bowman & Hacker (2014), Fig. 10.8 B

If age-specific survival & fecundity remain constant, the population settles into a stable age distribution and population growth rate

11 = 1.32

1 = 1.38

12 = 1.32

13 = 1.32

etc. = 1.32

Leslie Matrix

Age-structured matrix model (L) of population growth parameters

Example of a Leslie matrix from Wikimedia Commons

Age structure at t+1 Age structure at t

Dominant Eigenvalue of L = Dominant Eigenvector of L = stable age distribution

Age-specific survival & fecundity

Lefkovitch Matrix

Stage-structured matrix model (L) of population growth parameters

Example of a Lefkovitch matrix adapted from Leslie matrix from Wikimedia Commons

Stage structure at t+1 Stage structure at t

Dominant Eigenvalue of L = Dominant Eigenvector of L = stable stage distribution

Stage-specific survival & fecundity

Population Age Structure

Age structure for China in 2014 from Wikimedia Commons; China implemented a “one-child policy” in 1960s

Useful for predicting population growth

Survivorship Curves

Cain, Bowman & Hacker (2014), Fig. 10.5

Which is most likely to characterize an

r-selected species?

K-selected species?

Geometric growth when reprod. occurs at regular time intervals

Exponential Growth

Cain, Bowman & Hacker (2014), Fig. 10.10

Nt+1 = Nt

Population grows by a constant proportion

in each time step

Nt = tN0

=Geometric population

growth rate or

Per capita finite rate of increase

Exponential Growth

Cain, Bowman & Hacker (2014), Fig. 10.10

Exponential growth when reproduction occurs “continuously”

Reproducing is not synchronous in discrete

time periods

dN

dt = rN

N(t) = N(0)ert

r =Exponential growth

rateor

Per capita intrinsic rate of increase

Exponential Growth

Cain, Bowman & Hacker (2014), Fig. 10.11

= er

Nt = tN0

N(t) = N(0)ert = ertN(0)

Geometric

Exponential

r = ln()

Exponential decline /

decay

Constant population

size

Exponentialgrowth

Peter Turchin

The Fundamental Law of Population Ecology

Original idea from Turchin (2001) Oikos

“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.

Laws of Thermodynamics

Image of Carnot engine from Wikimedia Commons

Earth

Bio-geo-chemical

processes

1st Law of Thermodynamics Law of Conservation of EnergyRelated to Law of Conservation of Mass

E=mc2

Sun

Limited Scope for Population Increase

Quote from Cain, Bowman & Hacker (2014), pg. 227

“No population can increase in size forever.”

< 10100 < Number of particles in the universe

Limits to Exponential Growth

Cain, Bowman & Hacker (2014), Fig. 10.14

Density independent

Density dependent

Density-independent factors can limit population size

Limits to Exponential Growth

Cain, Bowman & Hacker (2014), Fig. 10.14

Density independent

Density dependent

Density-independent factors can limit population size

Density-dependent factors can regulate

population size

Logistic Growth

Cain, Bowman & Hacker (2014), Fig. 10.18

K = Carrying Capacity

r = Intrinsic Rate of Increase

r- vs. K-selection

Cain, Bowman & Hacker (2014), Fig. 10.18

K = Carrying Capacity

r = Intrinsic Rate of Increase