**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:
http://www.vu.nl/eng1/Algemeen/route.html

**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**- Tea

**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.

References:

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: kooi@bio.vu.nl

Fri May 15 12:26:03 MET DST 1998