GAKvanVoorn Wageningen2005

General strong stabilisation criteria for food chain models : 

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

Slide2: 

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

Slide3: 

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

Slide4: 

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

Toolkit: bifurcation analysis: 

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: 

Predator invasion criteria Predator invasion: transcritical bifurcation Different types of analysis of food web models Asymptotic behaviour (t  ∞) Parameter variation KTC  bifurcation analysis

Predator-prey cycle criteria: 

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

Lotka-Volterra: 

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

Resource competition: 

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

Saturated interactions: 

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

Predator interactions: 

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

One-parameter analysis: 

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

Multi-parameter analysis: 

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: 

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)

Future work: 

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

Thank you for your attention!: 

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