Emergence of complexity in evolving niche-model food webs

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Abstract

We have analysed mechanisms that promote the emergence of complex structures in evolving model food webs. The niche model is used to determine predator–prey relationships. Complexity is measured by species richness as well as trophic level structure and link density. Adaptive dynamics that allow predators to concentrate on the prey species they are best adapted to lead to a strong increase in species number but have only a small effect on the number and relative occupancy of trophic levels. The density of active links also remains small but a high number of potential links allows the network to adjust to changes in the species composition (emergence and extinction of species). Incorporating effects of body size on individual metabolism leads to a more complex trophic level structure: both the maximum and the average trophic level increase. So does the density of active links. Taking body size effects into consideration does not have a measurable influence on species richness. If species are allowed to adjust their foraging behaviour, the complexity of the evolving networks can also be influenced by the size of the external resources. The larger the resources, the larger and more complex is the food web it can sustain. Body size effects and increasing resources do not change size and the simple structure of the evolving networks if adaptive foraging is prohibited. This leads to the conclusion that in the framework of the niche model adaptive foraging is a necessary but not sufficient condition for the emergence of complex networks. It is found that despite the stabilising effect of foraging adaptation the system displays elements of self-organised critical behaviour.

Introduction

Diversity and complexity are ubiquitous in ecological systems. Especially in food web theory it is a long standing question, how complex systems arise and how their apparent stability can be explained on theoretical grounds (McCann, 2000). In this paper we will concentrate on the question how large and complex ecological communities are generated through coevolution of the species establishing the food web. The structure of natural food webs is known to be far from random. Several simple models exist that describe food web structure in part very well. They all introduce a more (cascade model, Cohen et al., 1990, Solow and Beet, 1998) or less (niche model, Williams and Martinez, 2000, nested hierarchy model, Cattin et al., 2004) strict hierarchy of the species that is inspired by the empirical result that species in a food web tend to be grouped into distinct trophic levels. The models are completely static, i.e. they neither consider dynamical aspects of the nodes (population dynamics) nor those of the network as a whole. The restrictions they impose on the link structure of the food webs constrain the possible processes that may have led to these structures. However, these models do not provide explicit mechanisms for such processes that could explain how the complicated structures they describe emerge or how these structures can persist.

In order to understand the structural properties of food webs one has to take into account that food webs are subject to dynamics on different time scales. On an ecological time scale, predator–prey interactions and competition for resources determine the population dynamics of each species. The composition of food webs changes on a larger, evolutionary time scale due to the appearance of new species (through speciation or immigration) and extinction of species. On a third, behavioural time scale the linkage pattern (feeding relationships) in the food webs may change due to adaptive behaviour of the species. Species number and connectance, typically used to parameterise the static models mentioned above, are therefore emerging from lower level mechanisms.

Inspired by the idea of self-organised criticality (SOC) in ecological systems, a variety of models has been developed that explicitly include the evolving character of food webs (Bak and Sneppen, 1993, Solé and Manrubia, 1996, Amaral and Meyer, 1999, Slanina and Kotrla, 1999). They focus on statistical properties of extinction events and species lifetime distributions. Since they are (deliberately) very simple, they do not tell us anything about biologically reasonable network structures. More meaningful for this task are assembly models (Drake, 1990, Morton and Law, 1997). They use an external species pool as source of immigration to the food web under consideration. The networks they produce have a more realistic structure and the models incorporate population dynamics to determine which species dies out and which persist. However, we think these models suffer from the fact that the species pool is limited, and that the species have not coevolved but are preselected according to their ecological function (resource, consumer, predator).

There are only few studies that build food webs by stochastic speciation events and apply population dynamics to determine each species’ fate. These are the Webworld model (Drossel et al., 2001, Drossel et al., 2004), the model studied by Lässig et al. (2001), and the more recently published model proposed by Loeuille and Loreau, 2005, Loeuille and Loreau, 2006. In the model from Lässig et al., species are grouped into distinct trophic levels and they prey only on the species on the next lower level. The network structure is changed by stochastic redirection of links. Computer simulations leading to complex network structures have not been published for this model. Evolution in the Webworld model is based on variations of a large number of traits that can be either switched on or off but whose biological meanings are not specified. In contrast to that, Loeuille and Loreau employ only one continuous trait with clear biological meaning (body size) to characterise a species and to let evolution act on. This allows them to order species on a one-dimensional axis just as in the static models discussed at the beginning. Computer simulations of both models successfully produce realistic food webs with statistical properties that are comparable to empirical food webs. It is found that in the Webworld model, complex food webs only arise if adaptive dynamics are applied.1

The issue of this paper is to use another model and to test more systematically conditions and mechanisms that are essential for the emergence of complexity in food webs with an evolving structure. Complexity is measured by species number, number of links per species or connectance (total number of links in the network divided by total number of possible links) and the length of food chains. For the last point we will discuss the trophic level structure of the emerging food webs, i.e. the number of species on each trophic level. In general, the trophic level of a species is defined as its mean distance from the external resources via chains of predator–prey links. Clearly, a food web with many trophic levels can be regarded as being more complex than one consisting only of basal species (species with no prey except for external resources).

We will focus on the impacts of foraging adaptation and of different metabolic rates of predators and prey that are caused by differences in body size. To be able to distinguish the effects on the food webs, the model will be refined gradually. As framework for our simulations we use the niche model. In its static setup (Williams and Martinez, 2000), it is known to produce networks of very realistic structure (it does equally well than the nested hierarchy model (Stouffer et al., 2005), but is easier to extend to a model with non-static structure). Furthermore, the niche model has the advantage that it constrains the network structure only very weakly, i.e. only very few configurations of species are excluded a priori. We believe this to be an advantage over the model proposed by Loeuille and Loreau (2005), which essentially prohibits looping and cannibalism and imprints an effective cascade-like structure on the food web to be built up.

The outline of this paper is as follows. In Section 2, the underlying model and the population dynamics are introduced. Section 3 describes the algorithm of the computer simulations. Results of these computer simulations with and without adaptive foraging are presented at the beginning of Section 4. In the following subsections, refinements of the basic model are introduced and their impact on the food web structure is discussed. Particular attention is paid to food webs resulting from simulations that include body size effects on metabolism since these networks display the highest degree of complexity. The section also includes a comparison of the results obtained with the three stages of the model discussed so far. The article is concluded by a discussion of our results (Section 5).

Section snippets

The model

The rules by which trophic links between species are assigned are taken from the niche model proposed by Williams and Martinez (2000). The species in the network are ordered on a one-dimensional niche axis. A predator species i preys on all species j, whose niche values nj (the position on the niche axis) fall in a region on the niche axis with width niri,0<ri1 and centre cini (see Fig. 1). We will call the parameter ri the relative width of the feeding range ci±(niri)/2, i.e. the part of the

Computer simulations

The simulations start with an initial network consisting only of a single species feeding on external resources. The population dynamics (1) are computed until they reach their stationary behaviour. Then a new species is introduced to the network by a speciation process, and the population dynamics are computed again. The new species may add stably to the network or cause the extinction of one or more existing species (including itself). This process of species introduction and computation of

Results

To address our central question of what are the mechanisms that promote the emergence of complexity in model food webs, the following measures of food web complexity are investigated. The first one is the species richness S (number of species in the network) that is reached during the simulations. Next, the links between the species are considered by computing connectance C=L/S2 and link density L/S. Here, L is the number of links (feeding relationships) in the network. The last, and most

Conclusions

In this paper, we have used the well-studied niche model as a framework to construct a dynamical and adaptive model of evolving food webs. While the speciation process adds a stochastic element to the evolution of the networks, extinctions do not occur at random as in many simpler evolutionary food web models (Bak and Sneppen, 1993, Solé and Manrubia, 1996, Amaral and Meyer, 1999, Slanina and Kotrla, 1999, Rossberg et al., 2007), but follow deterministically from the population dynamics. We

Acknowledgements

Discussions with Satoshi Uchida are gratefully acknowledged. We also thank Ulrich Brose for the advice to include body-size effects in the dynamics of the model. Fig. 3, Fig. 5, Fig. 7 were built from the data by Thomas Schösser.

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