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Predator-Prey models

Predator-Prey models. Intrinsic dynamics & Stability. Patterns and processes. 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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Predator-Prey models

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  1. Predator-Prey models Intrinsic dynamics & Stability

  2. Patterns and processes

  3. 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

  4. Fluctuations in biodiversity • a,bThe 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

  5. 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

  6. 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

  7. Complex Behaviour of this equation 2 1 8 Chaos

  8. Multiple equilibria

  9. 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

  10. Can’t recreate complex dynamics

  11. 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

  12. 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

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

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

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

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

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

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

  19. 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)

  20. 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 • 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)

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