1 demographic and environmental stochasticity in population processes 1) population dynamics 2)...
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Demographic and environmental stochasticity in population processes
1) Population dynamics2) Community dynamics3) Selection in fluctuating environments
Steinar Engen, Centre for Biodiversity Dynamics, Dept. of Mathematical Science, NTNU, Trondheim, Norway
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For more detailed lectures on diffusion, extinction, dynamics ,age-structure 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
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Single species population dynamics
Environmental and demographic stochasticityDiffusion theory
Extinction Some harvesting statistics
Age structure
Steinar Engen, Centre for Biodiversity Dynamics, Department of Mathematical Sciences
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Population fluctuations for some species
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Distribution of individualfitness wfor two bird species
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The Green function
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Expected time to extinction
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The expected time to extinction is often very large
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The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time
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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.
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The total reproductive value V of the population growth exactly exponential, and lnV has exactly linear growth.
The age distribution approaches the stable age distribution
R. A. Fisher
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Population size
Total reproductive value
Age-structured population, no density regulation.
(The total reproductive value serves as a filter)
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Community dynamics History The infinite allele model – neutral models Independent species dynamics Comparison of two dynamic models giving Fisher’s log series model
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The discovery of abundance patterns by Corbet and Williams in 1942:
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Poisson Gamma negative binomial
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10 minutes’ break
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Homogeneous models with speciations and extinctions
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Homogeneous models with speciations and extinctions
Inhomogeneous Poisson process
Homogeneous Poisson process
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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
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Neutral model, demographic noise only
Model with environmental noise
Typical lifetime of species
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Stochastic components of selection
Decomposition of the Robertson-Price equation,covariance formula for selection in an age-structured population
Density-dependent selection, r- and K-selection in a stochastic environment
Evolutionary effects of different non-selective harvesting strategies in a fluctuating population
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For more details see Engen and Sæther (2013 Evolution, in press)
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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)
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What about non-selective harvesting?
What are the effects?
With a given mean annual yield, are there differences between harvesting strategies?
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This result can be generalized to a sexual population with multinormally distributed z and constant G-matrix(Engen et al. 2012)
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We can now investigate the evolutionary component due to harvesting for different harvesting strategies:
Constant harvesting
Proportional harvesting
Threshold harvesting
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Demographic and environmental stochasticity in population processes
1) Population dynamics2) Community dynamics3) Selection in fluctuating environments
Steinar Engen, Centre for Biodiversity Dynamics, Dept. of Mathematical Science, NTNU, Trondheim, Norway