Topological structure and interaction strengths in model food webs

Abstract

We report the results of carrying out a large number of simulations on a coevolutionary model of multispecies communities. A wide range of parameter values were investigated which allowed a rather complete picture of the change in behaviour of the model as these parameters were varied to be built up. Our main interest was in the nature of the community food webs constructed via the simulations. We identify the range of parameter values which give rise to realistic food webs and give arguments which allow some of the structure which is found to be understood in an intuitive way. Since the webs are evolved according to the rules of the model, the strengths of the predator–prey links are not determined a priori, and emerge from the process of constructing the web. We measure the distribution of these link strengths, and find that there are a large number of weak links, in agreement with recent suggestions. We also review some of the data on food webs available in the literature, and make some tentative comparisons with our results. The difficulties of making such comparisons and the possible future developments of the model are also briefly discussed.

Introduction

Food webs, which represent the links between predators and prey in an ecological community, are complex networks which present several problems to the modeller. Firstly, can the range and nature of webs be specified a priori, perhaps using some generic biological principles? Secondly, can a dynamics describing the change in population sizes of species in the web be defined on the network? Thirdly, given that the number of links in a food web of a reasonable size will be of the order of several hundred, is it possible in practice, or even desirable, to give values to all of these link strengths?

The formidable difficulties associated with answering these questions led most early researchers to adopt approaches to the modelling of food webs which either bypassed some of these questions entirely, or implemented them in a very simple way. For example, May (1973) assumed the network to be a random graph, the interactions to be of randomly chosen strength and linearised the dynamics near a fixed point. Other food web modellers ignored the dynamics completely and simply gave rules to construct static graphs Cohen et al., 1990, Williams and Martinez, 2000. Even in the recent flurry of interest concerning network structure and topology (Albert and Barabási, 2002), most modellers have concentrated on specifying the nature of the network, rather than defining the dynamics on the network. The problem with this approach is that there is no reason why network structures which are appealing or which are found in social or other networks (small world, scale invariance) should apply to food webs. A recent study indeed shows that this is the case (Dunne et al., 2002a).

This suggests a more sophisticated approach to the modelling of food webs should be adopted. A clue as to how we might move forward is that it is clear that the structure of the network depends on the dynamics of the network, and cannot be divorced from it. It is therefore not appropriate to specify the web and then define dynamics—the two are interdependent. For example, a predator–prey link between any two species will disappear if either of the species becomes extinct and this will depend on the nature of the population dynamics that is chosen to govern their interaction. It is also clear that the dynamics on the network will be strongly influenced by the nature of the network itself. This strongly suggests that we cannot separate the population dynamics on the network from the dynamics which changes the network structure, which will occur on much longer time scales.

These comments address the first two of the questions posed at the beginning of this section, but there still remains the difficulty of knowing how to generate the several hundred quantities which specify the parameters in the model dynamics at any given time. The solution to this problem which we favour is to assemble the food web from one (or very few) species, so that the parameters are determined by the dynamics of web assembly. We have already explained that this dynamics is inextricably linked to the population dynamics. Starting with only one species simply amounts to giving an initial condition to the dynamics. In this way we determine those food web structures that can actually be reached rather than simply those that are possible.

This approach combines the reductionist and holistic treatments of ecosystems. We specify the rules that govern the dynamics of the interacting species in our communities but we then let the system itself parameterise those rules, and thus the ecosystem structure is an emergent property of the system. In this way we avoid having to deal with the intractable complexity of real ecosystems whilst still simulating the complexity of behaviour they exhibit (Jørgensen et al., 1992).

A model based on this philosophy was developed by some of us a few years ago and has been under study since then. An original version of the model (Caldarelli et al., 1998) was superseded by a later version with more realistic population dynamics (Drossel et al., 2001), and reviews which discuss the model specifically (Quince et al., 2002) and in a more general context (Drossel and McKane, 2003) are available. Our aim in this paper is to present a more extensive set of results from the model, emphasising aspects that were not stressed in previous investigations. A prime example is the distribution of link strengths, which is a topic which has been discussed extensively in the last year or so (Berlow et al., 2004) and which is an emergent attribute in our model, and consequently a fundamental test of the whole approach.

We begin by outlining the model in Section 2 and then, in order to provide some intuition on how a particular web is built up, we describe the time evolution of a single web in Section 3. The structure of the model food webs which are dynamically constructed through simulation of the model are explored in Section 4 and compared with data in Section 5. The distribution of link strengths in the model is explored in Section 6 and a discussion of our broad conclusions, as well as possible future avenues of investigation, is given in Section 7.