Population Dynamics
Focus on births (B) & deaths (D)
B = bNt , where b = per capita rate
(births per individual per time)
D = dNt
N = bNt – dNt = (b-d)Nt
Exponential Growth
• Density-independent growth models
Discrete birth intervals (Birth Pulse)
vs.
Continuous breeding (Birth Flow)
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1
2
0
t
N
=
N

t
0
1
0
0
=
2
,
N
=
2
0
>1
<1
=1
N
8
0
6
0
4
0
Nt = N0 t
2
0
0
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T
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Geometric Growth
• When generations do not overlap, growth can be
modeled geometrically.
Nt = Noλt
–
–
–
–
Nt = Number of individuals at time t.
No = Initial number of individuals.
λ = Geometric rate of increase.
t = Number of time intervals or generations.
Exponential Growth
Birth Pulse Population (Geometric Growth)
e.g., woodchucks
(10 individuals to 20 indivuals)
N0 = 10, N1 = 20
N1 =  N0 ,
where  = growth multiplier = finite rate
of increase
 > 1 increase
 < 1 decrease
 = 1 stable population
Exponential Growth
Birth Pulse Population
N2 = 40 = N1 
N2 = (N0  ) = N0 2
Nt = N0  t
Nt+1 = Nt 
Exponential Growth
• Density-independent growth models
Discrete birth intervals (Birth Pulse)
vs.
Continuous breeding (Birth Flow)
Exponential Growth
• Continuous population growth in an unlimited
environment can be modeled exponentially.
dN / dt = rN
• Appropriate for populations with overlapping
generations.
– As population size (N) increases, rate of population
increase (dN/dt) gets larger.
Exponential Growth
• For an exponentially growing population, size
at any time can be calculated as:
Nt = Noert
•
•
•
•
•
Nt = Number individuals at time t.
N0 = Initial number of individuals.
e = Base of natural logarithms = 2.718281828459
r = Per capita rate of increase.
t = Number of time intervals.
Exponential Population Growth
Exponential Population Growth
ln(N t)
l
n
(
N
)
=
l
n
(
N
)
+
r
t
t
0
r
=
e
p
lo
s
Nt = N0ert
Difference
Eqn
Note: λ = er
l
n
(
N
)
0
T
i
m
e
Exponential growth and change over time
N = N0ert
Number (N)
Slope (dN/dt)
dN/dt = rN
Time (t)
Slope = (change in N) / (change in time)
= dN / dt
Number (N)
ON THE MEANING OF r
rm - intrinsic rate of increase – unlimited
resourses
rmax – absolute maximal rm
r
- also called rc = observed
r>0
r<0
r=0
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r
m
r
<
0
m
x
r
>
0
m
r
>
>
0
m
x
r
m
a
x
Intrinsic Rates of Increase
• On average, small organisms have higher
rates of per capita increase and more
variable populations than large organisms.
Growth of a Whale Population
• Pacific gray whale (Eschrichtius robustus)
divided into Western and Eastern Pacific
subpopulations.
– Rice and Wolman estimated average annual
mortality rate of 0.089 and calculated annual
birth rate of 0.13.
0.13 - 0.089 = 0.041
– Gray Whale population growing at 4.1% per yr.
Growth of a Whale Population
• Reilly et.al. used annual migration counts
from 1967-1980 to obtain 2.5% growth rate.
– Thus from 1967-1980, pattern of growth in
California gray whale population fit the
exponential model:
Nt = Noe0.025t
• What values of λ allow
• What values of r allow
– Population Growth
– Population Growth
• λ > 1.0
– Stable Population Size
• λ = 1.0
– Population Decline
• λ < 1.0
• r>0
– Stable Population Size
• r=0
– Population Decline
• r<0
Logistic Population Growth
• As resources are depleted, population growth rate
slows and eventually stops
• Sigmoid (S-shaped) population growth curve
– Carrying capacity (K): number of individuals of a
population the environment can support
• Finite amount of resources can only support a finite number
of individuals
Logistic Population Growth
Logistic Population Growth
dN / dt = rN
dN/dt = rN(1-N/K)
• r = per capita rate of increase
• When N nears K, the right side of the equation nears
zero
– As population size increases, logistic growth rate becomes a
small fraction of growth rate
• Highest when N=K/2
• N/K = Environmental resistance
Exponential & Logistic Growth
(J & S Curve)
Logistic Growth
Actual Growth
Populations Fluctuate
Limits to Population Growth
• Environment limits population growth by
altering birth and death rates
– Density-dependent factors
• Disease, Resource competition
– Density-independent factors
• Natural disasters
Galapagos Finch Population
Growth
Logistic Population Model
A.
Discrete equation
Nt = 2, R = 0.15,
K = 450
Logistic Population Growth
- Built in time lag = 1
- Nt+1 depends on Nt
N(t)
500.0
400.0
300.0
200.0
100.0
0.0
0
20
40
60
Time
 Nt 
N t 1  N t  r N t 1  K 


80
100
I.
B.
Logistic Population Model
Density Dependence
I.
Logistic Population Model
C. Assumptions
• No immigration or emigration
• No age or stage structure to influence births
and deaths
• No genetic structure to influence births and
deaths
• No time lags in continuous model
I.
Logistic Population Model
C. Assumptions
Per Capita Growth Rate
(N(t+1) - N(t))/N(t)
• Linear relationship of
per capita growth rate
and population size
(linear DD)
0.2000
0.1500
0.1000
0.0500
0.0000
K
0.0
100.0
200.0 300.0
N(t)
400.0 500.0
I.
Logistic Population Model
C. Assumptions
• Linear relationship of per capita growth rate
and population size (linear DD)
• Constant carrying capacity – availability
of resources is constant in time and space
– Reality?
I.
Logistic Population Model
Discrete equation
Nt = 2, r = 1.9,
K = 450
N(t)
Logistic Population Growth
600.0
500.0
400.0
300.0
200.0
100.0
0.0
Damped
Oscillations
r <2.0
0
20
40
60
Time
80
100
I.
Logistic Population Model
Discrete equation
N(t)
Logistic Population Growth
Nt = 2, r = 2.5,
K = 450
600.0
500.0
400.0
300.0
200.0
100.0
0.0
0
20
40
60
Time
80
100
Stable Limit
Cycles
2.0 < r < 2.57
* K = midpoint
I.
Logistic Population Model
Discrete equation
Nt = 2, r = 2.9,
K = 450
Logistic Population Growth
800.0
N(t)
600.0
400.0
200.0
0.0
0
20
40
60
Time
80
100
Chaos
r > 2.57
• Not random
change
• Due to DD
feedback and time
lag in model
Underpopulation or Allee Effect
• Opposite type of DD
– population size down and population growth down
b=d
b=d
d
Vital rate
b<d
r<0
b
N*
N
K
I.
Review of Logistic Population
Model
D. Deterministic vs. Stochastic
Models
Random R0 and K
Logistic Growth Model
Nt = 1, r = 2,
K = 100
120.000
100.000
N(t)
80.000
* Parameters set
deterministic
behavior same
60.000
40.000
20.000
0.000
0
20
40
60
Time (t)
80
100
120
I.
Review of Logistic Population
Model
D. Deterministic vs. Stochastic
Random R0 and K
Models
Logistic Growth Model
Nt = 1, r = 0.15,
SD = 0.1;
K = 100, SD = 20
120.000
100.000
N(t)
80.000
60.000
40.000
20.000
0.000
0
20
40
60
Time (t)
80
100
120
* Stochastic
model, r and K
change at
random each
time step
I.
Review of Logistic Population
Model
D. Deterministic vs. Stochastic
Random R0 and K
Models
Logistic Growth Model
Nt = 1, r = 0.15,
SD = 0.1;
K = 100, SD = 20
120.000
100.000
N(t)
80.000
60.000
40.000
* Stochastic
model
20.000
0.000
0
20
40
60
Time (t)
80
100
120
I.
Review of Logistic Population
Model
D. Deterministic vs. Stochastic
Random R0 and K
Models
Logistic Growth Model
Nt = 1, r = 0.15,
SD = 0.1;
K = 100, SD = 20
120.000
100.000
N(t)
80.000
60.000
40.000
* Stochastic
model
20.000
0.000
0
20
40
60
Time (t)
80
100
120
II.
Environmental Stochasticity
A. Defined
• Unpredictable change in environment occurring in
time & space
• Random “good” or “bad” years in terms of
changes in r and/or K
• Random variation in environmental conditions in
separate populations
• Catastrophes = extreme form of environmental
variation such as floods, fires, droughts
• High variability can lead to dramatic fluctuations
in populations, perhaps leading to extinction
II.
Environmental Stochasticity
A. Defined
• Unpredictable change in environment occurring in
time & space
• Random “good” or “bad” years in terms of
changes in r and/or K
• Random variation in environmental conditions in
separate populations
• Catastrophes = extreme form of environmental
variation such as floods, fires, droughts
• High variability can lead to dramatic fluctuations
in populations, perhaps leading to extinction
II.
Environmental Stochasticity
A. Defined
• Unpredictable change in environment occurring in
time & space
• Random “good” or “bad” years in terms of
changes in r and/or K
• Random variation in environmental conditions in
separate populations
• Catastrophes = extreme form of environmental
variation such as floods, fires, droughts
• High variability can lead to dramatic fluctuations
in populations, perhaps leading to extinction
II.
Environmental Stochasiticity
B. Examples – variable fecundity
Relation Dec-Apr
rainfall and number
of juvenile
California quail per
adult (Botsford et
al. 1988 in
Akcakaya et al.
1999)
II.
Environmental Stochasiticity
B. Examples - variable
survivorship
Relation total
rainfall pre-nesting
and proportion of
Scrub Jay nests to
fledge (Woolfenden
and Fitzpatrik 1984
in Akcakaya et al.
1999)
II.
Environmental Stochasiticity
B. Examples – variable rate of
increase
6
5
Muskox
population on
Nunivak Island,
1947-1964
(Akcakaya et al.
1999)
4
3
2
1
0
0.925 0.975 1.025 1.075 1.125 1.175 1.225 1.275
Growth Rate
Number of Animals
II.
Environmental Stochasiticity
- Abundance
Example
of random
of Wildebeest
in the K
Serengeti
1600000
1400000
1200000
1000000
800000
600000
400000
200000
0
1954
1959
1964
1969
1974
1979
1984
1989
Years
• Serengeti wildebeest data set – recovering from Rinderpest
outbreak
– Fluctuations around K possibly related to rainfall
Exponential vs. Logistic
dN
 N
 rN 1  
dt
 K
dN
 rN
dt
No DD
All populations same
DD
All populations same
No Spatial component
Space Is the Final Frontier in
Ecology
• History of ecology = largely nonspatial
e.g.,
*competitors mixed perfectly with prey
*homogeneous ecosystems with
uniform distributions of resources
• But ecology = fundamentally spatial
– ecology = interaction of organisms with their
[spatial] environment
Incorporating Space
Metapopulation: a population of
subpopulations linked by dispersal of
organisms
Two processes = extinction & recolonization
• subpopulations separated by unsuitable
habitat (“oceanic island-like”)
• subpopulations can differ in population
size & distance between
Metapopulation Model
(Look familiar?)
dp
 cp 1  p   ep
dt
p = habitat patch (subpopulation)
c = colonization
e = extinction
Metapopulation Model
(Look familiar?)
dp
 cp 1  p   ep
dt
dp
 (m  e ) p[1  ( p /(1  e / m ))]
dt
dN
 (b  d ) N [1  ( N / K )]
dt
Rescue Effect
Another Population Model
Source-sink Dynamics: grouping of
multiple subpopulations, some are sinks
& some are sources
Source Population = births > deaths = net
exporter
Sink Population = births < deaths
>1
<1
<1
Metapopulations
• Definition of Population?
• Groups of populations within which there is a
significant amount of movement of individuals via
dispersal
Classic
Metapopulation
Metapopulation Con’t
• This kind of population structure
applies when there are “groups” of
populations occupying habitat that
occurs in discrete patches (patchy).
• These patches are separated by areas
of inhospitable habitat, but connected
by routes for dispersal.
• Populations fluctuate
independently of each other
The probability of dispersal from
one patch to another depends on:
• Distance between patches
• Nature of habitat corridors linking the
patches
• Ability of the species to disperse (vagility or
mobility) – dependent on body size
Who Cares?
Why bother discussing these models?
Metapopulations & Source-sink Populatons
highlight the importance of:
• habitat & landscape fragmentation
• connectivity between isolated
populations
• genetic diversity