Variability in space

Source-sink structure
 (arithmetic)
Source-sink structure
with the rescue effect
In time
 (geometric)
G < A
G declines with
increasing variance
 (arith & geom)
Increase the number of
subpopulations increases
the growth rate (to a point),
and slows the time to
extinction
Temporal variability reduces population growth rates
Cure – populations decoupled with respect to variability,
but coupled with respect to sharing individuals
Overview of
population growth:
New Concepts:
discrete
continuous
density
independent
Geometric
X
Exponential
X
density
dependent
Discrete
Logistic
Logistic
- Stability
- DI (non-regulating)
vs.
DD (regulating) growth
- equilibrium
Variability in growth
(1) Individual variation in births and deaths
(2)
X Environmental (extrinsic variability)
(3) Intrinsic variability
8000
7000
BUT, most populations
appear more regulated
than this…..
6000
5000
4000
Series1
3000
2000
1000
0
And
0
20
40
60
80
100
THERE ARE LIMITS
TO GROWTH!!!!
e.g., Australian sheep
Limits are manifested
in (-) density dependence
in population vital rates:
mortality/survivorship
reproduction
Density dependence often affects
more than a single component of
those rates:
At higher densities, song sparrows:
(a) smaller % reproductive males
(b) fewer young fledged/female
(c) lower juvenile survivorship
How do populations grow?
Logistic Growth
K
N
time
1 dN
N dt 0
K
dN = rN (K-N)
dt
K
population
1 dN = r (K-N)
N dt
K
per capita
N
K = Carrying capacity: the
maximum density of
individuals that the
environment can support
1 dN = r (K-N)
N dt
K
If N = 0
= r (K-(0))
K
=r K
K
=r
1 dN = r (K-N)
N dt
K
N
If N = 0
Exponential
growth-like
time
= r (K-(0))
K
=r K
K
=r
That’s Exponential Growth
1 dN = r (K-N)
N dt
K
If N = K
= r (K-(K))
K
=r 0
K
=0
1 dN = r (K-N)
N dt
K
Zero
growth
K
If N = K
N
time
= r (K-(K))
K
=r 0
K
=0
That’s Zero Growth
1 dN = r (K-N)
N dt
K
K
N
Put the two together
time
LOGISTIC GROWTH
r
+
growth
1 dN
N dt 0
K
1 dN = r (K-N)
N dt
K
N
growth
=r
=r
(
(
K _ N
K
K
)
)
1 _ 1K N
= r _ Kr N
Y= b + mX
2nd Simplest expression of population growth:
2 parameters: r = intrinsic growth rate and
K = carrying capacity
Per capita growth rate is (-) density dependent
Second Law of Ecology: There are limits to growth
K
N
EQ stability regulation
Log. 


time
Exp.
N
time
Severe
drought
Rinderpest
innoculation
Rainfall
Total food
per capita food
So what about
Density-dependence?
Proportion of animals
Live wildebeest
Lion/hyena killed
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
To download a version of Populus:
http//www.cbs.umn.edu/populus/download/download.html
Density
Discrete Logistic Growth
500
600
400
500
400
300
300
200
r=0.2
100
0
0
200
10
20
r=1.0
100
30
0
0
10
20
30
time
600
600
500
500
400
400
Damped
oscillations
300
300
200
r=1.8
100
0
0
10
20
30
200
2-point
limit cycle
100
r=2.0
0
0
10
20
30
900
1000
800
900
800
700
700
600
600
500
500
400
400
300
300
200
200
100
r=2.2
0
0
10
20
100
30
0
0
3000
1500
Chaos
r=2.8
1000
2000
500
1000
0
0
10
20
30
40
50
60
70
0
0
r=2.5
4-pt cycle
10
20
extinction
10
20
30
30
r=4.0
40
50
60
70
Chaos – “unpredictable” population
dynamics incurred through very high
growth rate and time lags between
growth and negative feedback.
1500
Chaos
r=2.8
1000
500
10
20
30
40
50
60
70
1500
time
Density
0
0
Extrinsic
variability
1000
500
0
0
10
20
time
30
40
K=1000; r=3.0
Density
3000
2000
1000
0
0
10
20
30
time
40
50
Islands < 1.0 ha support
too few shrews to persist
Population culled by 25%
Density
K=1000; r=3.0
3000
3000
2000
2000
1000
1000
0
0
10
20
30
time
40
50
0
0
10
20
30
time
40
50
Extrinsic
variability
Population culled by 25%
1500
120
Density
100
1000
80
60
500
40
20
0
0
10
20
30
40
0
0
10
20
30
40
Variability comes in 2 flavors: Extrinsic and Intrinsic
Recognizing the type of variability is important
because different types require different solutions.
Intrinsic –  growth rate or population size
Extrinsic –  migration, # populations, population size
Overview of
population growth:
New Concepts:
discrete
continuous
density
independent
Geometric
X
Exponential
X
density
dependent
Discrete
X
Logistic
Logistic
X
- Stability
- DI (non-regulating)
vs.
DD (regulating) growth
- equilibrium
Variability in growth
(1) Individual variation in births and deaths
(2)
X Environmental (extrinsic variability)
(3)
X Intrinsic variability
REVIEW
- Cure: Dispersal from sources can
- Populations consist of sources ( > 1)
and sinks (<1), the latter doom to extinction…….. Rescue sinks
- Cure: Many populations that share
- Populations have good years and bad years
and temporal variation is bad …………………………… individuals (dispersal)
- Cull the population or otherwise
- Populations can grow chaotically by overand under-shooting Carrying capacity…………………. reduce its growth
- Recognize and keep density above
- Populations with an Allee Effect can decline
to extinction if N is too low……………………………….. the critical density
2 Models of growth
Exponential – all populations have the capacity
to growth exponentially, but
N
Growth has no limits and is density independent
time
1 dN = r
N dt
1 dN
N dt
N
Logistic – recognizes limits to growth (Carrying
K
capacity) and incorporates the negative effect
individuals have on their growth rate
N
(- Density Dependence)
time
1 dN
= r (K-N)
N dt
K
r
1 dN
N dt 0
K
N
Stable EQ @ K
One other variation is the ALLEE
EFFECT where individuals also have
+ Density Dependence at low density
- DD
Individuals inhibit
their growth
1 dN
N dt
0
N
K
+ DD
e.g., social behavior
safety in numbers
Important Concepts we have touch upon under Population Growth
- Life Tables: Understanding how patterns of age-specific survivorship
and maternity has consequences for population growth and can be
manipulated to achieve a management goal
- Variability: In space, populations exist as sources ( > 1) and sinks
( < 1), the latter of which must receive migrants to persist (Rescue Effect)
In time, environmental variation is an anathema to population growth, but
it too has a cure: increase the number of populations, migration,
- Intrinsic Variability: Appreciate the difference between external and
internal variation arising from time lags and delayed density dependence.
Its cure is radically different than for external variation – and requires
culling population size or otherwise reducing the growth rate.
Important Concepts we have touch upon under Population Growth
- EQ, stability, and Pop. regulation: Attainable only under
(-) density dependence. Negative feedback is Universal
- Domains of Attraction: Specifically, under the Allee Effect,
population extinction is an “attractant” below some critical density
-----------------------------------------------------------------------The concept of the limits to growth is manifested in the
Carrying Capacity
Species Social Behavior is manifested in the Allee Effect
But otherwise, we have incorporated the biology of species
as phenomena and have not appreciated the actual details
But we will……
Where’s the Biology?
Wildebeest populations growth
competition for grass occurs
Individuals are energy stressed
Lions kill off weak individuals
1 dN = r (K-N)
N dt
K
??Energy/stress??
The Phenomenological Approach
THE GOOD: Modeling the phenomena allows us to look past
the details … we don’t need separate models for
every organism
THE BAD: We only get a superficial understanding ….
when the details matter we’re left scratching our heads
This tradeoff between DETAIL and GENERALITY
Is pervasive throughout science