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General strong stabilisation criteria for food chain models

Explore the application of bifurcation analysis in theoretical ecology, focusing on the stabilisation mechanisms in predator-prey interactions and population dynamics.

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General strong stabilisation criteria for food chain models

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  1. General strong stabilisation criteria for food chain models George van Voorn, Thilo Gross, Bob Kooi, Ulrike Feudel, Bas Kooijman http://www.bio.vu.nl/thb/ george.van.voorn@falw.vu.nl Wageningen, October 28, 2005 10.45-11.00 h

  2. Overview What is theoretical ecology? What is bifurcation analysis? How do we use bifurcation analysis in theoretical ecology? Mechanisms studied in our work Results of application Discussion

  3. Theoretical ecology • Theoretical ecology • Study predator-prey interactions • Population dynamics predator prey

  4. Theoretical ecology • Theoretical ecology • Study predator-prey interactions • Population dynamics Food web models Using mathematics Y predator X prey

  5. Toolkit: bifurcation analysis Dynamical systems, generated by ODE’s dX/dt = rX - dY/dt = - dY Parameter variation can lead to qualitative differences in system behaviour

  6. Y Stable equilibrium Fixed K: Y(t), t  ∞ K Unstable equilibrium KTC = The value of K at which the predator invades, K being an “enrichment” parameter Predator invasion criteria • Different types of analysis of food web models • Asymptotic behaviour (t  ∞) • Parameter variation  bifurcation analysis Predator invasion: transcritical bifurcation KTC

  7. Stable period solution Stable equilibrium Unstable equilibrium Predator-prey cycle criteria Predator-prey cycles: Hopf bifurcation Y Y K < KH K > KH X X For 2D predator-prey systems we can give the values of KH and KTC symbolically For larger dimensional systems we need numerical analysis

  8. Lotka-Volterra Ecological modelling For study predator-prey interactions use of several models Most basic: Lotka-Volterra Realistic?! Y a*X*Y X

  9. Resource competition Step up Prey compete for resources Logistic growth model Consumption by prey is limited by competition

  10. Saturated interactions Step up Predators need time to handle prey Holling type-IIfunctional response Rosenzweig-MacArthur Do we have all the basic features?!

  11. Predator interactions Another step up Predators also interact with each other  Intraspecific interference Beddington-DeAngelis

  12. Destabilisation Extinction Continued persistence One-parameter analysis One-parameter bifurcation analysis RM vs. BD  Classical RM TI = 0 Beddington-DeAngelis TI = 0.04 KTC (RM) = KTC (BD), KH (RM) ≠ KH (BD), where K = enrichment parameter Intraspecific predator interactions  Stabilising effect

  13. Multi-parameter analysis Weakly stabilising vs. strongly stabilising mechanisms: The limits for K  ∞ are equal; shift of value KH  Weakly stabilising Different asymptotes  Strongly stabilising

  14. Discussion Results: Interference effects: for TI > TI~ no destabilisation, for any amount of enrichment General application: Multi-parameter asymptotic behaviour  Stability criteria Other mechanisms have the same effect (not shown), e.g. cannibalism, inedible prey, …  Broader application range G.A.K. van Voorn, T. Gross, B.W. Kooi, U. Feudel and S.A.L.M. Kooijman (2005). Strongly stabilized predator–prey models through intraspecific interactions. Theoretical population biology (submitted)

  15. Future work Different interaction function  different stability properties Application approach to large-scale food webs

  16. Thank you for your attention! Thanks to: Thilo Gross, Bob Kooi, Ulrike Feudel, Bas Kooijman, João Rodriguez and Hans Metz and http://www.bio.vu.nl/thb/ george.van.voorn@falw.vu.nl

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