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,age- structure and abundance models see and more on age-structure 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

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

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10 Distribution of individual fitness w for two bird species

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18 The Green function

19 Expected time to extinction

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25 The expected time to extinction is often very large

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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.

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37 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

38 Population size Total reproductive value Age-structured population, no density regulation. (The total reproductive value serves as a filter)

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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:

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45 Poisson Gamma negative binomial

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51 10 minutes’ break

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58 Homogeneous models with speciations and extinctions

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60 Homogeneous models with speciations and extinctions Inhomogeneous Poisson process Homogeneous Poisson process

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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

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69 Neutral model, demographic noise only Model with environmental noise Typical lifetime of species

70 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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78 For more details see Engen and Sæther (2013 Evolution, in press)

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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].[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)

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96 We can now investigate the evolutionary component due to harvesting for different harvesting strategies: Constant harvesting Proportional harvesting Threshold harvesting

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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