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/[email protected]

Wageningen, October 28, 200510.45-11.00 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 ecologyStudy predator-prey interactionsPopulation dynamics

Theoretical ecology

prey

predator

Theoretical ecologyStudy predator-prey interactionsPopulation 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 equilibriumFixed K: Y(t), t ∞

Unstable equilibrium

Different types of analysis of food web models

Asymptotic behaviour (t ∞) Parameter variation

KTC

KTC = The value of K at which the predator invades,K being an “enrichment” parameter

bifurcation analysis

Predator-prey cycle criteriaPredator-prey cycles: Hopf bifurcation

For 2D predator-prey systems we can give the values of KH and KTC symbolicallyFor larger dimensional systems we need numerical analysis

Stable period solution

K < KH K > KH

Unstable equilibriumStable equilibrium

Y

X

Y

X

Ecological modellingFor study predator-prey interactions use of several models

Most basic: Lotka-Volterra

Realistic?!

X

Y

Lotka-Volterra

a*X*Y

Step upPrey compete for resources

Logistic growth model

Consumption by prey is limited by competition

Resource competition

Step upPredators need time to handle prey

Holling type-II functional response

Rosenzweig-MacArthur

Do we have all the basic features?!

Saturated interactions

Another step upPredators also interact with each other Intraspecific interference

Beddington-DeAngelis

Predator interactions

One-parameter analysis

Destabilisation Extinction Continued persistence

Classical RMTI = 0

Beddington-DeAngelisTI = 0.04

One-parameter bifurcation analysis RM vs. BD

KTC (RM) = KTC (BD), KH (RM) ≠ KH (BD),where K = enrichment parameterIntraspecific predator interactions Stabilising effect

Multi-parameter analysis

Weakly stabilising vs. strongly stabilising mechanisms:The limits for K ∞ are equal; shift of value KH Weakly stabilisingDifferent asymptotes Strongly stabilising

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

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 Metzand

http://www.bio.vu.nl/thb/[email protected]