The effects of intraspecific interactions on the stability of a simple food chain

George van Voorn, Thilo Gross, Bob Kooi, Ulrike Feudel, Bas Kooijman

Dresden, July 18-22 2005

http://www.bio.vu.nl/thb/george.van.voorn@falw.vu.nl

Van Voorn et al.

Overview

Introduction•Stability in food chain models – several mechanisms•Functional responses•Intraspecific interference between predators•Models: Rosenzweig-MacArthur and Mass-balance

Model analysis•Asymptotic behaviour in food chain models (bifurcations)•Stability criteria (RM)•Numerical results (MB)

Discussion•Other functional responses (literature search)

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Food chain stability

A few highlights regarding food chain stability:

•Destabilisation through nutrient enrichment ‘Paradox of enrichment’Rosenzweig, M.L. (1971). Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science, 171:385-387.

•Maintenance costs for living cellsNisbet, R.M., Cunningham, A., and Gurney, W.S.C. (1983). Endogenous metabolism and the stability of microbial prey-predator systems. Biotechnology and bioengineering, 25:301-306.

•Ecosystem nutrient recyclingDeAngelis, D.L. (1992). Dynamics of Nutrient Cycling and Food Webs. Chapman & Hall.

•Properties of functional form of interaction functionGross, T., Ebenhöh, W. and Feudel, U. (2005). Enrichment and foodchain stability: the impact of different forms of predator-prey interaction. Journal of Theoretical Biology, 227:349-358.

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Trophic interaction functions

Laboratory experiments on predator-prey systems Wiedenmann, R.N. & O’Neil, R.J. (1991). Laboratory measurements of the functional response of Podisus maculiventris (Say) (Heteroptera: Pentatomidae). Environmental Entomology, 20:610-614.

resemblance Holling type II FR (Holling, 1959), but:•1 predator•No other organisms, only preyField tests: significantly lower attack rates

Searching efficiency of predators < with increasing numbersHassell, M.P. (1971). Mutual interference between searching insect parasites. Journal of Animal Ecology, 40:473-486.

Predators hampered by other factors than handling time?!

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Intraspecific interference

where = searching time [m t/V]

= handling time [t]

= interacting time [t]

Mutual interference through intraspecific interactions

Beddington-DeAngelis functional response (BD-FR)Beddington, J.R. (1975). Mutual interference between parasites or predators and its effect on searching efficiency. Journal of Animal Ecology, 44:331-340.DeAngelis, D.L., Goldstein, R.A. and O’Neill, R.V. (1975). A model for trophic interaction. Ecology, 56:881-892.

Time scale separation Kooi, B.W., Poggiale, J.C., Auger, P. and Kooijman, S.A.L.M. (2002). Aggregation methods in food chains with nutrient recycling. Ecological modelling, 157:69-86.

If kSI = 0 Holling type II FR

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Food web models

Classical Rosenzweig-MacArthur Mathematically more tractable• Logistic growth prey• Linear mortality

Mass-balanced chemostat model

recycling

maintenance

explicit nutrient dynamics

F(X,Y) is replaced by either Holling type II-FR or BD-FR

recycling ofmaintenance products

products

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Predator invasion criteria

Y

K

Predator invasion: transcritical bifurcation

Stable equilibriumFixed K: Y(t), t ∞

Unstable equilibrium

Analysis of food web modelsAsymptotic behaviour bifurcation analysis

KTC

KTC = The value of K at which the predator invades(RM: can be expressed algebraically)

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Predator-prey cycle criteria

Predator-prey cycles: Hopf-bifurcation

The value of KH above which cycling occurs can also be calculated algebraically for 2D predator-prey systems

Unstable equilibrium Stable period solution

K < KH K > KH

Stable equilibrium

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Results: 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)Intraspecific predator interactions Stabilising effect

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Hopf surface

TranscriticalsurfaceClassical paradox of enrichment

Results: multi-parameter analysisMulti-parameter bifurcation analysis RM vs. BD

=

TI = 0

<

TI > 0

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Multi-parameter asymptotic behaviour

For the RM-model:

With BD-FR:

The limits for K ∞ are equal There is always a Hopf-bifurcation There is always destabilisation through nutrient enrichmentWeakly stabilising: shift of value KH

There is a parameter region with no Hopf-bifurcation There is possible avoidance of POEStrongly stabilising: different asymptotes

<

Multi-parameter asymptotic behaviour Stability criteria

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MB with Holling type II

Recycling: weakly destabilising

Recycling Mass balanced model

Same asymptoteswith and without

recycling

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MB with BD functional response

Different asymptotic bifurcations

Always stable

MB with BD-FR (also) strongly stabilising

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Maintenanceψ = proportional to maintenance

Same asymptotes ψ = 0.25

Same asymptotes ψ = 0.05

Maintenance: weakly stabilising

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Discussion (1)Conclusions:

Definition stability Grimm, V. and Wissel, C. (1997). Babel, or the ecological stability discussions: an inventory and analysis of terminology and a guide for avoiding confusion. Oecologia, 109:323-334.Rinaldi, S. and Gragnari, A. (2004). Destabilizing factors in slow-fast systems. Ecological modelling, 180:445-460.

For nutrient enrichment well-defined criteria for strong andweak stabilisation is possible

Bifurcation analysis yields:

•Recycling weakly destabilising•Maintenance weakly stabilising•Intraspecific interactions strongly stabilising but: Other strongly stabilising mechanisms?!

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Strong stabilisation: inedible prey

TC

H

Predators (can) waste time on inedible preyKretzschmar, M., Nisbet, R.M. and McCauley, E. (1993). A predator-prey model for zooplankton grazing on competing algal populations. Theoretical Population Biology, 44:32-66.

Functional response for predator also depends on inedible prey non-prey dependent term alters occurrence of Hopf

Interaction edible preyand inedible prey

No interaction inedible prey,only with edible prey

No difference

Differentasymptotes

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Strong stabilisation: inducible defences

Inducible defences: predation leads to prey that invests energy in defence more time lost on handlingVos, M., Kooi, B.W., DeAngelis, D.L. and Mooij, W.M. (2004). Inducible defences and the paradox of enrichment. Oikos, 105:471-480.

Occurrence of Hopf altered by inducible defences limit Hopf ≠ limit TC (other FR)

TC: no prey with defences

H: prey defensible, more time/prey

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Strong stabilisation: cannibalismCannibalism: predators feed partially on other predators Alternative food sourceKohlmeier, C. and Ebenhöh, W. (1995). The stabilizing role of cannibalism in a predator-prey system. Bulletin of Mathematical Biology, 57:401-411.

Measure of cannibalism

H

η > η* never destabilisation

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Discussion (2)Intraspecific interactions strongly stabilising and:Literature search shows many more mechanisms lead to functional responses not solely depending on prey-density Strongly stabilising effects

RM: mathematically more tractableGross, T., Ebenhöh, W. and Feudel, U. (2005). Enrichment and foodchain stability: the impact of different forms of predator-prey interaction. Journal of Theoretical Biology, 227:349-358.

symbolic bifurcation analysisMB: numerical bifurcation analysis

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The effects of intraspecific interactions on the stability of a simple food chain

Thanks to:Thilo Gross, Bob Kooi, Ulrike Feudel, Bas Kooijman,

João Rodriguez

http://www.bio.vu.nl/thb/george.van.voorn@falw.vu.nl

The end