General strong stabilisation criteria for food chain models George van Voorn, Thilo Gross, Bob Kooi, Ulrike Feudel, Bas Kooijman Wageningen, October 28, h

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 Overview

Theoretical ecology Study predator-prey interactions  Population dynamics Theoretical ecology prey predator

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

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

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

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

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

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

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

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

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

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

Discussion Results: Interference effects: for T I > T I ~  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)

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

Thank you for your attention! Thanks to: Thilo Gross, Bob Kooi, Ulrike Feudel, Bas Kooijman, João Rodriguez and Hans Metz and