The effects of intraspecific interactions on the stability of a simple food chain George van Voorn, Thilo Gross, Bob Kooi, Ulrike Feudel, Bas Kooijman.

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Presentation transcript:

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 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) 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.

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: 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: 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: 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.

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: 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:  Predators hampered by other factors than handling time?! 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.

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: DeAngelis, D.L., Goldstein, R.A. and O’Neill, R.V. (1975). A model for trophic interaction. Ecology, 56: 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: If k SI = 0  Holling type II FR 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.

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 of maintenance products products 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.

Predator invasion criteria Y K Predator invasion: transcritical bifurcation Stable equilibrium Fixed K: Y(t), t  ∞ Unstable equilibrium Analysis of food web models Asymptotic behaviour  bifurcation analysis K TC K TC = The value of K at which the predator invades (RM: can be expressed algebraically) 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.

Predator-prey cycle criteria Predator-prey cycles: Hopf-bifurcation The value of K H above which cycling occurs can also be calculated algebraically for 2D predator-prey systems Unstable equilibriumStable period solution K < K H K > K H Stable equilibrium 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.

Results: one-parameter analysis Destabilisation Extinction Continued persistence Classical RM T I = 0 Beddington-DeAngelis T I = 0.04 One-parameter bifurcation analysis RM vs. BD  K TC (RM) = K TC (BD), K H (RM) ≠ K H (BD) Intraspecific predator interactions  Stabilising effect 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.

Hopf surface Transcritical surface Classical paradox of enrichment Results: multi-parameter analysis Multi-parameter bifurcation analysis RM vs. BD  = T I = 0 < T I > 0 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 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 enrichment Weakly stabilising: shift of value K H  There is a parameter region with no Hopf-bifurcation  There is possible avoidance of POE Strongly stabilising: different asymptotes < 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 Van Voorn et al.

MB with Holling type II Recycling: weakly destabilising Recycling  Mass balanced model Same asymptotes with and without recycling 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.

MB with BD functional response Different asymptotic bifurcations Always stable MB with BD-FR  (also) strongly stabilising 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 Van Voorn et al.

Maintenance ψ = proportional to maintenance Same asymptotes ψ = 0.25 Same asymptotes ψ = 0.05 Maintenance: weakly stabilising 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 (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: Rinaldi, S. and Gragnari, A. (2004). Destabilizing factors in slow-fast systems. Ecological modelling, 180: 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.

Strong stabilisation: inedible prey TC H 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: Functional response for predator also depends on inedible prey  non-prey dependent term  alters occurrence of Hopf Interaction edible prey and inedible prey No interaction inedible prey, only with edible prey No difference Different asymptotes 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.

Strong stabilisation: inducible defences 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: Occurrence of Hopf altered by inducible defences  limit Hopf ≠ limit TC (other FR) TC: no prey with defences H: prey defensible, more time/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 Van Voorn et al.

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: Measure of cannibalism H η > η *  never destabilisation 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 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: symbolic bifurcation analysis MB: numerical 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 Van Voorn et al.

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 The end