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

Van Voorn et al. The effects of intraspecific interactions on the stability of a simple food chain. George van Voorn , Thilo Gross, Bob Kooi, Ulrike Feudel, Bas Kooijman. http://www.bio.vu.nl/thb/ george.van.voorn@falw.vu.nl. Dresden, July 18-22 2005. Van Voorn et al. Overview. Overview

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

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  1. Van Voorn et al. The effects of intraspecific interactions on the stability of a simple food chain George van Voorn, Thilo Gross, Bob Kooi, Ulrike Feudel, Bas Kooijman http://www.bio.vu.nl/thb/ george.van.voorn@falw.vu.nl Dresden, July 18-22 2005

  2. Van Voorn et al. Overview Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5 • 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)

  3. Van Voorn et al. Food chain stability Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5 • 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 cells • Nisbet, 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 recycling • DeAngelis, D.L. (1992). Dynamics of Nutrient Cycling and Food Webs. Chapman & Hall. • Properties of functional form of interaction function • Gross, 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.

  4. Van Voorn et al. Trophic interaction functions Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5 • 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 prey • Field tests: significantly lower attack rates • Searching efficiency of predators < with increasing numbers • Hassell, M.P. (1971). Mutual interference between searching insect parasites. Journal of Animal Ecology, 40:473-486. •  Predators hampered by other factors than handling time?!

  5. Van Voorn et al. = searching time [m t/V] If kSI = 0  Holling type II FR = handling time [t] = interacting time [t] Intraspecific interference Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5 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. where

  6. Van Voorn et al. explicit nutrient dynamics recycling of maintenance products products recycling maintenance Food web models Mass-balanced chemostat model Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5 • Classical Rosenzweig-MacArthur •  Mathematically more tractable • Logistic growth prey • Linear mortality F(X,Y) is replaced by either Holling type II-FR or BD-FR

  7. Van Voorn et al. Y Stable equilibrium Fixed K: Y(t), t  ∞ K Unstable equilibrium KTC = The value of K at which the predator invades (RM: can be expressed algebraically) Predator invasion criteria Analysis of food web models Asymptotic behaviour  bifurcation analysis Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5 Predator invasion: transcritical bifurcation KTC

  8. Van Voorn et al. K < KH K > KH Stable period solution Stable equilibrium Unstable equilibrium Predator-prey cycle criteria Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5 Predator-prey cycles: Hopf-bifurcation The value of KH above which cycling occurs can also be calculated algebraically for 2D predator-prey systems

  9. Van Voorn et al. Destabilisation Extinction Continued persistence Results: one-parameter analysis One-parameter bifurcation analysis RM vs. BD  Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5 Classical RM TI = 0 Beddington-DeAngelis TI = 0.04 KTC (RM) = KTC (BD), KH (RM) ≠ KH (BD) Intraspecific predator interactions  Stabilising effect

  10. Van Voorn et al. Results: multi-parameter analysis Hopf surface TI = 0 TI > 0 = < Transcritical surface Classical paradox of enrichment Multi-parameter bifurcation analysis RM vs. BD  Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5

  11. Van Voorn et al. < Multi-parameter asymptotic behaviour Multi-parameter asymptotic behaviour  Stability criteria Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5 For the RM-model: The limits for K  ∞ are equal  There is always a Hopf-bifurcation  There is always destabilisation through nutrient enrichment Weakly stabilising: shift of value KH With BD-FR:  There is a parameter region with no Hopf-bifurcation  There is possible avoidance of POE Strongly stabilising: different asymptotes

  12. Van Voorn et al. Same asymptotes with and without recycling Recycling: weakly destabilising MB with Holling type II Recycling  Mass balanced model Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5

  13. Van Voorn et al. Different asymptotic bifurcations Always stable MB with BD functional response Intraspecific interactions Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5 MB with BD-FR  (also) strongly stabilising

  14. Van Voorn et al. Same asymptotes ψ = 0.05 Same asymptotes ψ = 0.25 Maintenance ψ= proportional to maintenance Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5 Maintenance: weakly stabilising

  15. Van Voorn et al. 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 and • weak stabilisation is possible • Bifurcation analysis yields: • Recycling weakly destabilising • Maintenance weakly stabilising • Intraspecific interactions strongly stabilising • but: • Other strongly stabilising mechanisms?! Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5

  16. Van Voorn et al. H No difference Different asymptotes TC Strong stabilisation: inedible prey Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5 Predators (can) waste time on inedible prey Kretzschmar, 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. No interaction inedible prey, only with edible prey Interaction edible prey and inedible prey Functional response for predator also depends on inedible prey  non-prey dependent term  alters occurrence of Hopf

  17. Van Voorn et al. TC: no prey with defences H: prey defensible, more time/prey Strong stabilisation: inducible defences Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5 Inducible defences: predation leads to prey that invests energy in defence  more time lost on handling Vos, 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)

  18. Van Voorn et al. Measure of cannibalism H η > η* never destabilisation Strong stabilisation: cannibalism Cannibalism: predators feed partially on other predators  Alternative food source Kohlmeier, C. and Ebenhöh, W. (1995). The stabilizing role of cannibalism in a predator-prey system. Bulletin of Mathematical Biology, 57:401-411. Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5

  19. Van Voorn et al. 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 Overview Intro 1 Intro 2 Intro 3 Intro 4 Intro 5 Intro 6 Results 1 Results 2 Results 3 Results 4 Results 5 Results 6 Discuss 1 Discuss 2 Discuss 3 Discuss 4 Discuss 5 RM: mathematically more tractable Gross, 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 analysis MB: numerical bifurcation analysis

  20. The end 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

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