Models pretend that nature is
simple, predictable and unchanging:
25.0
20.0
15.0
r = 0.3
r = 0.5
10.0
r = 0.8
5.0
0.0
0
5
10
15
20
25
30
The real world is messy.
Saether et al. 2002
Does this “messiness” matter to populations and
communities?
(“messiness” = heterogeneity, variability)
Causes of variability
Patterns of variability
How does variability matter to populations?
Causes of variability
Environmental
weather related: temperature, precipitation, light,…
site related: soil properties, slope, vegetation cover,…
biological: the number of competitors, predators,…
Demographic stochasticity
Random variability in the growth rates of small
populations arising chance events in individual
survival and reproduction. Can lead to chance
extinction in small populations.
Chaotic dynamics
Random-like variability from strong and non-linear
interactions between variables.
Environmental variability
Patterns in space or time
spatial variation: when sites differ
temporal variation: when conditions change over time
Deterministic or random variability
deterministic: predictable variation that follows precise rules
random: has only statistical properties: mean, variance, etc.
Types of deterministic variability
periodic (in time or space)
progressive change (spatial gradient, temporal trend)
Patterns in time:
40
30
Average monthly
temperature in Tucson
Precipitation in Moab, Utah
300
250
Winter
Summer
200
20
150
100
10
50
0
1 2 3 4 5 6 7 8 9 1011 12
Month of the year
0
1930
1940
1950
1960
1970
Year
1980
1990
2000
Patterns in time:
Patterns in time:
Patterns in space:
Patterns in space:
Demographic stochasticity:
High per-capita variability in growth rate due to
random fluctuation in individual survivorship, and
fecundity.
Small populations can go extinct, even though
average growth rates are positive.
Excel Worksheets:
• Demographic Stochasticity
Demographic stochasticity
N
50
Exponential model
with
N0 = 20
45
40
35
30
b = 0.01 per year
d = 0.002 per year
25
20
15
10
5
29
36
43
50
57
64
71
78
85
92
99
97
22
91
15
16
14
N0 = 5
12
10
8
6
4
2
85
79
73
67
61
43
37
31
25
19
13
7
0
1
N
8
55
1
49
0
A documented extirpation
Three-spined stickleback
(Gasterosteus aculeatus)
The population
disappeared from a
pond in Wales in 2000
(Wootton 2010)
Chaotic fluctuations:
Some entirely deterministic processes can produce
fluctuations indistinguishable from random fluctuations.
This was first discovered by the meteorologist Edward
Lorenz who worked the prediction of weather patterns.
Edward N. Lorenz
(1917-2008)
The Butterfly Effect
Whether or not a hurricane forms may
depend on whether or not a butterfly flapped
its wings several weeks before
Chaotic dynamics have extreme sensitivity
to initial conditions.
The Butterfly Effect
(Slingo and Palmer 2011)
The Discrete Logistic Model:
N t 1  N t  rmax
Nt 

N t 1 

K 

Equilibrium defined as Nt+1 = Nt
N*,1 = 0
N*,2 = K
Excel Worksheets:
• Discrete logistic equation
The existence of the various forms and
changing patterns of variability drives
evolution and creates and maintains
biodiversity.
A prairie
Six species of Solidago (golden rod) are found in old fields and prairies.
nemoralis
canadensis
missouriensis
gigantea
speciosa
graminifolia
Frequency of occurrence
species
Species usually occupy only a subset
of an environments range of conditions.
Oklahoma Prairie
Hour of day
Species are active only during a fraction of the total
available time.
Summary:
There are various sources and patterns of variability in nature
producing spatial and temporal heterogeneity.
Forms of patterns: random, cyclic, or persistent trends (both in
space and time).
A special case of variability is called: demographic stochasticity: the
random fluctuations of small populations (which can have large and
important consequences) .
Chaos is a special case of variability that is seemingly random but
actually caused by deterministic processes.
Much of ecology is concerned with understanding species’
responses to variability in its various forms in evolutionary and
ecological time.