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This study explores the impact of intraspecific interactions on food chain stability through model analysis using the Rosenzweig-MacArthur and Mass-balance models. The research focuses on the functional responses, destabilization mechanisms, and bifurcation analyses in ecological systems. It highlights how mutual interference influences searching efficiency and predator-prey dynamics. The paper discusses intraspecific interference effects on trophic interactions and stable equilibria in food web models. Overall, it sheds light on the complexities of predator-prey relationships influenced by intraspecific competition.
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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 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)
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.
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?!
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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