Transcript Population Growth and Regulation

Population Growth
and Regulation
BIOL400
31 August 2015
Population
 Individuals
of a single species sharing time
and space
 Ecologists must define limits of
populations they study

Almost no population is closed to immigration
and emigration
Exponential Population Growth
 Equations:


Nt = N0ert
dN/dt = rN
 Model



terms:
r = per-individual rate of change (= b – d)
= intrinsic capacity for increase, given the
environmental conditions
N = population size, at time t
e = 2.718
Fig. 8.13 p. 131
Exponential Population Growth
 Assumptions


of the Model:
Constant per-capita rate of increase,
regardless of how high N gets
Continuous breeding
Geometric Population Growth

Model modified for discrete annual breeding
 Nt = N0t
  = er


 is the annual rate of increase in N
Example: N0 = 1000,  = 1.10
•
•
•
•
N1 = 1100
N3 = 1331
N5 = 1611
N25 = 10,834
N2 = 1210
N4 = 1464
N10 = 2594
N100 = 13,780,612
Q: Can you spot the oversimplification of nature here?
Fig. 8.10 p. 129
Fig. 9.1 p. 144
R0 = per-generation multiplicative rate of increase
Logistic Population Growth
 Equations:

Nt = K/(1 + ea-rt)
• K = karrying kapacity of the environment
• a positions curve relative to origin

dN/dt = Nr[(K-N)/K]
 Assumption:
Growth rate will slow as N
approaches K
Fig. 9.4 p. 146
Table p. 148
As N increases, per-capita rate of increase declines, but the
absolute rate of increase always peaks at ½ K
Data from Populations
in the Field
Fig. 9.8 p. 150

Cormorants in Lake
Huron
 Low numbers due to
toxins
 Increase is not
strongly sigmoid
Fig. 9.9 p. 150

Ibex in Switzerland
 Reintroduced after
elimination via
hunting
 Roughly sigmoid
(=logistic) but with big
decline in 1960s
Fig. 9.10 p. 151

Whooping cranes of
single remaining wild
population

15 in 1941, now over
200

r increased in 1950s
 Every mid-decade,
there is a mini-crash

Apparently related to
predation cycles
Fig. 9.15 p. 154

Cladocerans


Predominant lake
zooplankton
No constant K; big
swings seasonally
Can We Improve Our Models?
 1)
Theta logistic model
 2) Time-lag logistic model
 3) Stochastic models
 4) Population projection matrices
Theta Logistic Model
New term, , defines
curve relating growth
rate to N
 dN/dt = Nr[(K-N)/K]

Fig. 9.12 p. 152
Fig. 9.13 p. 152
Time-Lag Models
 Logistic
model in which population growth
rate depends not on present N, but on N
one (or more) time periods prior
 Assumes population’s demographic
response to density may be delayed
Fig. 9.14 p. 153

With time lag, stable
ups and downs may
occur
Fig. 11.14 p. 170

Water fleas show
stable approach to K
at 18C

Time lag effect occurs
at 25C

Daphnia store energy
to use when food
resources collapse
Stochastic Models

Predict a range of
possible population
projections, with
calculation of the
probability of each
Fig. 9.17 p.
Population Projection Matrices
 Use
matrix algebra to project population
growth, based on fecundity and agespecific survivorship
• Fig. 9.18A p. 157
 Application:
Determining whether
changes in one aspect or another of the
life history of an organism have the greater
impact on r (calculate “elasticity” of each
life-history parameter)
Fig. 9.19 p. 159
HANDOUT—Biek et al. 2002
Survivorship in a Population

Three types of
curves are
recognized
following Pearl
(1928)
 Examination of
the survivorship
of various
species shows
that most have a
mixed pattern
Fig. 8.6 p. 124
Fig. 8.8 p. 126
Life Table

Used to project population growth
 Can be used to determine R0, from which r or 
can be calculated
 1) Vertical (=Static): useful if there is long-term
stability in age-specific mortality and fecundity
 2) Cohort: data taken from a population
followed over time (ideally, a cohort followed
until all have died)


Observing year-year survivorship, or
Collecting data on age at death
Table 8.5 p. 128
Table 8.3 p. 122