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Predator-Prey models Intrinsic dynamics & Stability

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Patterns and processes

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Extrinsic drivers of fluctuation The environment can exert pressures on the organisms – Press perturbations – Pulse perturbations Affect growth rates or mortality rates The organisms lag behind

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Fluctuations in biodiversity a,b The green and black plots show the number of known marine animal genera versus time. The trend line (blue) is a third-order polynomial fitted to the data. c, As b, with the trend subtracted and a 62-Myr sine wave superimposed. d, The detrended data after subtraction of the 62-Myr cycle and with a 140-Myr sine wave superimposed. Rohde & Muller 2005, Nature 434, 208-210

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Intrinsic patterns in simple models Simple difference equation models Time progresses in a discrete, step-wise manner Births and deaths described by r Adjust r so that more births occur below K and more deaths above K

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Growth rate around K Simple linear effect on r At K, r=0 Below K, r>0 Above K, r<0 Pushes N towards K

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Complex Behaviour of this equation 8 2 1 Chaos

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Multiple equilibria

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Continuous time population model Very similar to discrete equation Births occur instantaneously and N grows at all times N tends towards K as positive and negative growth rates around it push it back to K

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Can’t recreate complex dynamics

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Time lag is needed Growth rate is now a function of the population at some point in the past (T) Can be hard now to reach K as growth takes time Lemmings populations Solving these in the computer a little more tricky

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Predator-prey models Predator intake rates Type 3 functional response a = encounter rate Th = handling time See Chapter 11 of Ted Case’s book An Illustrated Guide to Theoretical Ecology

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Prey dynamics a = encounter rate Th = handling time K = prey carrying capacity R = prey density C = predator density

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Predator dynamics a = encounter rate Th = handling time k = prey to predator conversion efficiency w = mortality rate R = prey density C = predator density

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Stable populations a=0.002, r=0.5, d=0.1, k=0.5, K=100, Th=2

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Oscillatory dynamics a=0.002, r=0.5, d=0.1, k=0.5, K=100, Th=3

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Boom and Bust a=0.002, r=0.5, d=0.1, k=0.5, K=100, Th=4

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Further Reading An illustrated guide to theoretical ecology by Ted J Case, Oxford University Press 2000. – Chapters 5,6,11,12,13

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Something to think about…. So far we have not discussed stochasticity (random processes) Parameters in all these models might fluctuate either according to some seasonal pattern, or might be entirely random Stochasticity can be a powerful driver of non- equilibrium behaviour (or can have little influence)

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Next tutorial Shaw et al 2004. The Shape of Red Grouse Cycles. Journal of Animal Ecology 73, 767-776 http://dx.doi.org/10.1111/j.0021- 8790.2004.00853.x http://dx.doi.org/10.1111/j.0021- 8790.2004.00853.x Cattadori et al. 2005. Parasites and climate synchronize red grouse populations. Nature 433, 737-741. http://dx.doi.org/10.1038/nature03276 (see also the supplementary material) http://dx.doi.org/10.1038/nature03276

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