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June 17 1998

Mini-symposium on Population Dynamics: Wednesday June 17 1998
Vrije Universiteit zaal P046 gebouw W&N, De Boelelaan 1087, 1081 HV Amsterdam
Route to the VU from Station Zuid/WTC.
express tram 51 (1 minute), alight: at the VU tram stop
tram 5 (1 minute), alight: at the VU tram stop
bus 23 (3 minutes), alight: at De Boelelaan in front of the VU Main Building
bus 63 (3 minutes), alight: at Buitenveldertselaan VU Main Building

It is a 10 minute walk to the VU from Station Zuid/WTC
A map of the VU-campus can be found on:

13:30 - 14:15
Sergio Rinaldi (Politecnico di Milano, Italy)
Optimality versus Chaos in Ecosystems
14:15 - 14:45
Martin Boer (Vrije Universiteit)
The dynamics of tri-trophic food chains
14:45 - 15:00
15:00 - 15:30
Peter de Ruiter (Department of Environmental Studies, Utrecht and DLO-Institute for Agrobiology and Soil Fertility (AB-DLO), Haren)
Energetics, Patterns of Interaction Strengths and Stability in Real Ecosystems
15:30 - 16:00
Anje-Margriet Neutel (DLO-Institute for Agrobiology and Soil Fertility (AB-DLO), Haren)
Diagonal dominance as a condition on ecosystem organisation
16:00 - 16:30
Hugo van den Berg (Vrije Universiteit)
Another Look at the Community Matrix

Summary of the talk of S. Rinaldi: Optimality versus Chaos in Ecosystems

Simple nonlinear models of ecosystems can be chaotic for realistic values of their environmental and demographic parameters. Although only a few laboratory and field data confirm this possibility, many others indicate that a great number of populations are not too far from being chaotic. This is consistent with some studies in evolution, natural selection, self-organization, and thermodynamics, which support the idea that biological systems should tend toward the edge of chaos. Thus, the following question naturally arises: do ecosystems enjoy special properties when they are at the edge of chaos? A sharp answer to this question can be given for a simple but important class of ecosystems, namely tritrophic food chains. To this aim, a comparative analysis of three different models was performed, focusing on the role played by the nutrient available to the bottom of the ecosystem. The study was carried out through bifurcation analysis and simulation and showed that top predator mean abundance is maximum at the edge of chaos, but can collapse shortly after that maximum. The consequence in the case of exploited resources, is that maximization of the mean yield through enrichment pushes ecosystems toward the edge of chaos, where dramatic collapses are highly probable.

Reference papers:
O. De Feo and S. Rinaldi. "Yield and dynamics of tri-trophic food chains", American Naturalist, 150, 328-345, 1997.
A. Gragnani, O. De Feo, and S. Rinaldi. "Food chains in the chemostat: relationships between mean yield and complex dynamics", Bulletin of Mathematical Biology, 1, 1-16, 1998.
S. Rinaldi, O. De Feo, and A. Gragnani. "Optimality and chaos of tri-trophic food chains", VII International Congress of Ecology (INTECOL), Florence, Italy, July 19-25,1998.

Summary of the talk of Martin Boer: The dynamics of tri-trophic food chains

Continuous time models of bi-trophic food chains have only two basic patterns: approach to an equilibrium or to a limit cycle. In contrast, models of tri-trophic food chains exhibit complex dynamical phenomena, including chaotic behavior. The appearance of these phenomena in tri-trophic food chains suggests that complex dynamics may be common in natural food webs. Therefore, it may be a good idea to gain a better understanding of the dynamics of a simple tri-trophic food chain, before an attempt is made to analyse more complicated models of food chains and food webs.

In this talk, the asymptotic behavior of a tri-trophic food chain model composed by a logistic growing prey, a Holling type II intermediate predator, and a Holling type II top predator is described. Transversal homoclinic and heteroclinic orbits to a limit cycle of saddle type are presented. It will be shown that these orbits and their bifurcations are important to gain a better understanding of the asymptotic behavior of tri-trophic food chains.

Summary of the talk of Peter C. de Ruiter, Anje-Margriet Neutel, John C. Moore: Energetics, Patterns of Interaction Strengths and Stability in Real Ecosystems

Central in our understanding of the structure and stability of ecosystems are the strengths of the trophic interactions among the populations constituting community food webs. For seven real soil food webs, we constructed material flow descriptions of the food webs. Feeding rates were calculated from the observed population sizes, death rates and energy conversion From these food web energetics, the strengths of the trophic interactions could be estimated, following the principles of May and using standard Lotka-Volterra equations. A distinction was made between the per capita effects of predators on their prey (negative (ij)) and the per capita effects of prey on their predators (positive (ji)). Both types of per capita effects were patterned along trophic position: the absolute value of the negative (ij) decreased with trophic position, and the positive (ji) increased with trophic position. In other words: The patterning consisted of the simultaneous occurrence of strong "top down" effects at lower trophic levels and strong "bottom up" effects at higher trophic levels. This patterning of the interaction strengths was found to be an important factor in the stability of the webs. This appeared from a comparative analysis in which the stability of community matrix representations of the seven food webs including the estimated patterns of interaction strengths (lifelike matrices) was compared with that of matrices with similar structures but without the pattern, that is (i) theoretical matrices in which interaction strength was sampled from proposed theoretical intervals, (ii) disturbed matrices in which the lifelike patterns of interaction strength were disturbed by randomly permuting the non-zero pairs of elements, and (iii) test matrices, in which the values of the parameters (population sizes, specific death rates and energy conversion efficiencies) used to calculate the feeding rates and interaction strengths were not based on observations but were randomly chosen. This comparison showed that the lifelike matrices were more likely to be stable than their theoretical, disturbed or test counterparts. The comparison with the theoretical and disturbed matrices showed that including the estimated values of interaction strength enhanced stability, and that this enhancement could not be attributed to the occurrence of particular ranges of element values nor to the overall strength of the trophic interactions relative to the strength of intra-group interference, hence resulted from the way in which the element values were arranged in a specific pattern. Furthermore, the comparison with the test matrices showed that the high likelihood of stability of the lifelike matrices was not due to an artifact of the (equilibrium) assumptions underlying the equations we used to calculate the feeding rates and interaction strengths, but was connected to the field and laboratory data. As the interaction strengths were directly derived from the energetic organisation of the food webs, the results show that energetics and community structure govern ecosystem stability through imposing stabilising patterns of the interaction strength. However, It is not yet clear how mathematically the patterning relates to stability.

Summary of the talk of Anje-Margriet Neutel: Diagonal dominance as a condition on ecosystem organisation

Diagonal dominance has long been known to guarantee stability in community matrices. This matrix property is mainly interpreted as intraspecies regulation dominating over the interactions among species. In the pursuit of stability constraints on ecosystem structure much attention therefore has been given to intraspecies regulation. However, it has recently been argued that certain general trophic structures in natural communities provide for system stability through patterns in relatively strong species interactions, under weak self-regulation (De Ruiter et al. 1995, Neutel et al. in prep.). Here, we show that, despite relatively weak intraspecific interaction, such patterns in fact enable systems to meet the criterion of diagonal dominance. The results indicate that the criterion of diagonal dominance could help in identifying general principles ecosystem organisation.

De Ruiter, P.C., A.M. Neutel, and J.C. Moore. 1995. Energetics, patterns of interaction strengths, and stability in real ecosystems. Science 269:1257-1260.
Neutel, A.M., C. Kaldeway, and P.C. de Ruiter. How pyramids of biomass make complex structures stable. in prep.

Summary of the talk of Hugo van den Berg: Another Look at the Community Matrix

It is a well-known fact that the community matrix ``summarizes the outcomes of all possible press [permanent perturbation] experiments,'' as Peter Yodzis once put it. I discuss how the community matrix can be viewed as an internal sensitivity matrix, which combines with an external sensitivity matrix to give the overall sensitivity matrix of the community. The formalism need not be restricted to trophic fluxes: indeed, mass transfers between abiotic and biotic components, as well as ageing and death may be treated in the same manner. Of course, there is no such thing as the community matrix for an ecosystem. It all depends on how the distinct components are chosen, as well as on the time scale for which the mass transfers are considered stationary. The overall sensitivity matrix depends not only on these two choices, but also on the parametrization of the external influences. One may establish an overall sensitivity matrix by direct manipulation of the community, or, alternatively, through the modelling and study of the individual mass transfer interactions.

Information: Bob Kooi, Theoretical Biology, Vrije Universiteit Amsterdam. E-mail:

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Next: June 21-261998 Up: Calendar Previous: June 2-61998

Bob Kooi
Fri May 15 12:26:03 MET DST 1998