Download 1 - NTNU

1
Demographic and environmental
stochasticity in population processes
1) Population dynamics
2) Community dynamics
3) Selection in fluctuating environments
Steinar Engen, Centre for Biodiversity Dynamics, Dept. of
Mathematical Science, NTNU, Trondheim, Norway
2
For more detailed lectures on diffusion, extinction, dynamics ,agestructure and abundance models see
www.math.ntnu.no/~steinaen/lovund_2012
and more on age-structure
www.math.ntnu.no/~steinaen/ISEC_2012
and/or Lande, Engen and Sæther 2003, Stochastic Population
Dynamics in Ecology and Conservation, Oxford University Press.
These lectures are based on co-authored papers with Russell Lande
and Bernt-Erik Sæther during the last 20 years. For a list see
my home page
www.math.ntnu.no/~steinaen
3
Single species population
dynamics
Environmental and demographic stochasticity
Diffusion theory
Extinction
Some harvesting statistics
Age structure
Steinar Engen, Centre for Biodiversity Dynamics,
Department of Mathematical Sciences
4
Population
fluctuations
for some
species
5
6
7
8
9
10
Distribution of
individual
fitness w
for two bird
species
11
12
13
14
15
16
17
18
The Green function
19
Expected time to extinction
20
21
22
23
24
25
The expected time
to extinction is often
very large
26
27
28
29
30
31
32
The rate of increase in fitness of any
organism at any time is equal to its genetic
variance in fitness at that time
33
The rate of increase in fitness of any
organism at any time is equal to its genetic
variance in fitness at that time
The rigour of its demonstration requires that the terms
employed should be used stricktly as defined; the ease
of its interpretation may be increased by appropriate
conventions of measurement. For example, the ratio p:q
should stricktly be evaluated at any instant by
enumeration, not necessarily of the census population,
but of all individuals having reproductive value, weighted
according to the reproductive value of each.
34
35
36
37
The total reproductive value V
of the population growth
exactly exponential, and lnV
has exactly linear growth.
R. A. Fisher
The age distribution
approaches the stable age
distribution
38
Age-structured population, no density regulation.
(The total reproductive value serves as a filter)
Population size
Total reproductive
value
39
40
41
42
Community dynamics
History
The infinite allele model – neutral models
Independent species dynamics
Comparison of two dynamic models giving
Fisher’s log series model
43
The discovery of abundance patterns by
Corbet and Williams in 1942:
44
45
Poisson
Gamma
negative binomial
46
47
48
49
50
51
10 minutes’ break
52
53
54
55
56
57
58
Homogeneous models with speciations and extinctions
59
60
Homogeneous models with speciations and extinctions
Inhomogeneous
Poisson process
Homogeneous Poisson process
61
62
63
64
65
66
67
So what is the difference between the two types
of model?
Answer:
Temporal fluctuations of abundant (observable)
species are very different.
Neutral model with demographic noise only:
extremely small fluctuations and species
turnover rates
Gamma model with environmental noise: More
realistic fluctuations and turnover rates
68
69
Neutral model,
demographic noise only
Typical lifetime of species
Model with environmental
noise
70
Stochastic components of selection
Decomposition of the Robertson-Price equation,
covariance formula for selection in an agestructured population
Density-dependent selection, r- and K-selection in
a stochastic environment
Evolutionary effects of different non-selective
harvesting strategies in a fluctuating population
71
72
73
74
75
76
77
78
For more details see
Engen and Sæther (2013
Evolution, in press)
79
80
Figure 1. Observed decline in mean carapace length
of rock lobsters captured in the fishery at two locations
off the coast of Western Australia from 1972 to 2005 [10].
Only animals with a carapace length of greater than
76 mm (dotted line) can be legally harvested. This decline
apparently is partially an evolutionary response to
extremely high annual exploitation rates of adults (∼75%),
combined with a required minimum carapace length
of 76 mm in harvested individuals.
(Allendorf et al. 2008)
81
What about non-selective
harvesting?
What are the effects?
With a given mean annual
yield, are there differences
between harvesting strategies?
82
This result can be generalized to a
sexual population with multinormally
distributed z and constant G-matrix
(Engen et al. 2012)
83
84
85
86
87
88
89
90
91
92
93
94
95
96
We can now investigate the evolutionary
component due to harvesting for different
harvesting strategies:
Constant harvesting
Proportional harvesting
Threshold harvesting
97
98
99
10
0
10
1
10
2
Demographic and environmental
stochasticity in population processes
1) Population dynamics
2) Community dynamics
3) Selection in fluctuating environments
Steinar Engen, Centre for Biodiversity Dynamics, Dept. of
Mathematical Science, NTNU, Trondheim, Norway