Evolutionary emergence of size-structured food webs Nicolas Loeuille* and Michel Loreau Laboratoire d���Ecologie, Unite �� Mixte de Recherche 7625, Ecole Normale Supe ��rieure, 46 Rue d���Ulm, F-75230 Paris, Cedex 5, France Edited by Simon A. Levin, Princeton University, Princeton, NJ, and approved March 10, 2005 (received for review November 12, 2004) Explaining the structure of terrestrial and aquatic food webs remains one of the most important challenges of ecological theory. Most existing models use emergent properties of food webs, such as diversity and connectance as parameters, to determine other food-web descriptors. Lower-level processes, in particular adapta- tion (whether by behavioral, developmental, or evolutionary mechanisms), are usually not considered. Here, we show that complex, realistic food webs may emerge by evolution from a single ancestor based on very simple ecological and evolutionary rules. In our model, adaptation acts on body size, whose impact on the metabolism and interactions of organisms is well established. Based on parameters defined at the organism scale, the model predicts emergent properties at the food-web scale. Variations of two key parameters (width of consumption niche and competition intensity) allow very different food-web structures and function- ings to emerge, which are similar to those observed in some of the best-documented food webs. complex adaptive system evolutionary assembly macroevolution food- web structure Amodel lthough prevailing food-web models, such as the cascade (1, 2), the niche model (3), and the more recent nested-hierarchy model (4), are able to describe food-web struc- ture satisfactorily, they fail to provide clear mechanisms explain- ing how this structure emerges. There are two reasons for this shortcoming. First, these models are parameterized by using some emergent properties of observed food webs (usually di- versity and connectance), although these properties result from lower-level processes. Second, they consider only binary food webs in which species and trophic links are either present or absent, but are not quantified. The dynamical aspects of food webs, linked to population dynamics and adaptation processes, although ubiquitous in ecosystems (5), are absent from these theoretical studies. Consideration of adaptive processes may provide new ap- proaches to understand the structure and functioning of food webs. A few recent pioneering studies (6���8) have suggested that complex food webs may emerge from evolutionary assembly processes. These models consider variations of a large set of traits that determine the strength of predator���prey interactions. Considering a large number of traits is appealing because many characteristics are likely to have a role in trophic interactions. However, because these traits are numerous and are not explic- itly identified, it is not possible to understand the evolutionary dynamics in terms of selective pressure or to test model predic- tions by using empirical data. Here, we propose a simpler approach in which body size is the single biological trait subject to evolution. This simple approach allows us to make testable predictions on how evolution shapes ecosystem structure and functioning. Model and Methods The reasons for using body size as a key trait are numerous. The trophic cascade model (1, 2) is based on a hierarchy among species, and body size is a good candidate to explain this hierarchy (9, 10). Body mass, M, is also tightly linked to individual metabolism, B, by the allometric relation B Ma. The exponent a in this relation is usually 0.75 (11���13), or 0.25 if metabolism is measured per unit mass (mass-specific metabolic rate) (13). Because of this allometric relation, it is possible to correlate body size and a number of life-history traits of organ- isms, thereby making a link between organismic and community scales (13, 14). We modeled the population dynamics of species i with biomass Ni and body size xi by dNi dt Ni f xi j 0 i 1 xi xj Nj m xi j 1 n xi xj Nj j i 1 n xj xi Nj , [1] where the xi are ranked by increasing values, f(xi) is the produc- tion efficiency of species i, and m(xi) is its mass-specific mortality rate. Because these two parameters are related directly to mass-specific metabolic rate, they are assumed to depend on body size (13): f(xi) f0xi 0.25 and m(xi) m0xi 0.25. The function (xi xj) describes the consumption rate exerted by predator i on prey j. It is assumed to be a Gaussian function with standard deviation s and a maximum value when the body sizes of the predator and the prey are separated by a distance d as follows: xi xj 0 s 2 exp xi xj d 2 s2 , with xi xj (Fig. 1). The choice of this type of function is based on the idea that, for a predator of a given size, energy gains should increase with its prey body size, whereas the probability of such successful attacks should decrease with the prey body size. As a result, body size should then be optimum at an intermediate value. This type of relationship between interaction strength and body size is supported by empirical data (13, 15, 16). Exploitation competition among individuals of similar size is implicit in the function because they consume the same kind of resources. Also, these individuals of similar body size may also hamper, interfere physically, or even harm each other while competing for the resource. This interaction is called ������interfer- ence competition.������ The function ( xi xj ) corresponds to the interference-competition rate and is defined as follows: xi xj 0 if xi xj 0 if xi xj . The value of was chosen to be small (0.25), which means that competition is assumed to occur mainly within species or with closely related types. Interference competition among species with similar body sizes has been shown in some groups (17) the mechanisms involved may be interference (18, 19). Habitat partitioning leads to similar effects (20). Our qualitative results This paper was submitted directly (Track II) to the PNAS office. *To whom correspondence should be sent at the present address: Section of Integrative Biology, Patterson Building, Room 141, University of Texas, Austin, TX 78712. E-mail: nicolas.loeuille@normalesup.org. �� 2005 by The National Academy of Sciences of the USA www.pnas.org cgi doi 10.1073 pnas.0408424102 PNAS April 19, 2005 vol. 102 no. 16 5761���5766 ECOLOGY

do not hinge on this assumption of interference competition among similar-sized types because we also considered the case in which this competition is absent ( 0 0). Last, N0 is the amount of inorganic nutrient whose trait x0 0 is expressed for mathematical convenience but does not evolve. Its dynamics is defined as follows: dN0 dt I eN0 v i 1 n m xi Ni i 1 n j 1 n xi xj NiNj i 1 n j 0 i 1 1 f xi xi xj NiNj i 1 n xi NiN0 [2] where I is the input of inorganic nutrient, e is its output rate, and v is the percentage of remaining nutrient within the system during the recycling process. Numerical simulations were performed by using the Runge��� Kutta method in FORTRAN 90. Food webs emerge progressively from a single ancestor by mutation���selection processes. For each population, the mutation rate was 10 6 per unit mass at each time step. If a mutation occurs in a population x, a new population is created whose trait is drawn randomly in the interval [0.8x, 1.2x]. The initial biomass of the mutant is 10 20, which is also the threshold biomass below which any population is assumed to go extinct. The initial population has a trait x d and consumes inorganic nutrient. The simulations were run during 108 time steps. Their computation time varied from a few hours to several weeks depending on the total diversity reached (varying from 1 to 200 species). The total time for the 201 simulations exceeded 1 year. Of these simulations, we focus here on the results of the 36 simulations that varied the two parameters that proved to be critical in determining food-web structure [i.e., niche width (nw 0.5, 1, 2, 3, 4, and 5) and competition intensity ( 0 0, 0.1, 0.2, 0.3, 0.4, and 0.5)]. Niche width is defined as nw s2 d, a relative measure of the variance of body sizes consumed by each species. Trophic position is determined recursively from the bottom to the top of the food web. The trophic position of a target species is defined as the average trophic position of the species it consumes weighted by the proportion of nutrient these represent in the diet of the target species, plus one. For details about the method, see ref. 21. Results As shown in Fig. 2, the structure of the food web gradually stabilizes after an initial period of strong diversification. It is mainly determined by niche width and, to a lesser extent, by competition intensity. If niche width and competition intensity are small, species are packed in distinct trophic levels. If niche width or competition intensity are large, the structure of the food web is somewhat blurred, with no distinct trophic levels. The community then appears as a continuum of species homoge- neously spaced along an axis of body size or trophic position. Given the link between body size and predation (Fig. 1), trophic position is strongly correlated with body size. Thus, the patterns displayed by Fig. 2 would be identical if body size, instead of trophic position, were plotted against time. The observed patterns (Fig. 2) can be explained by the way that the food web structures itself in a bottom-up manner. If niches are narrow, then only those species whose size is close to d are able to consume the basal resource efficiently enough to survive. Smaller or larger species appear transiently but are not favored in the long run. Similarly, only species that are packed at size 2d can consume species with size d efficiently. There- fore, they have a selective advantage over species with interme- diate body size. More generally, species from trophic level i evolve to a body size close to id. Thus, when niche width is small, the food web is strongly structured in groups of different sizes corresponding to distinct trophic levels (Fig. 2 A and B). If niches are broad, the advantage (in terms of resource consumption) of species with sizes that are multiples of d is less important compared with species with intermediate sizes. This advantage may then be offset by other processes, such as the dependence of mean generation time [which is the inverse of mortality rate m(xi)] on body size and interference competition. Although predation tends to pack species in distinct trophic levels, interference competition tends to have the opposite effect of homogenizing the body-size distribution. This homogeniza- tion occurs because species compete only if their body sizes are similar enough. When competition is intense, it has a more important role than predation does, and the structure of the food web is less distinct. This effect is particularly visible for nw 0.5 or 1 (Fig. 2 G and H). When niche width is large (nw 2), the trophic structure is blurred in all three cases. Our results show that complex food webs may emerge from simple ecological and evolutionary rules in a system that initially contains a single species consuming an inorganic nutrient. The variety of emerging structures is comparable with empirical observations of real ecosystems. Food webs with distinct trophic levels are commonly found in freshwater ecosystems (22���24), whereas food webs with a more continuous trophic structure may be more common in soil terrestrial or marine ecosystems (25). Although interaction strength is a continuous property that emerges from the coevolutionary dynamics, most empirical data and food-web models (1, 3) are binary (presence or absence of species and trophic links). Therefore, we transformed the food webs obtained here into binary food webs to make comparisons with these previous studies. Each type was then regarded as a species, and a consumption link was considered to be present between any two species if their interaction strength exceeded the threshold value of 0.15. The food web was then comparable with the classical binary webs studied extensively in the ecolog- ical literature. Variations in the threshold value may affect estimates of connectance, the proportion of omnivores, food- Fig. 1. Trophic interactions. Interaction strength, as measured by function , between two species as a function of the difference between their body sizes is shown. The vertical line indicates a target species. Species whose traits are larger are its potential predators, and species whose traits are smaller are its potential prey. Mutations acting on the body-size trait of the target species will give birth to types whose trophic interactions are slightly different. 5762 www.pnas.org cgi doi 10.1073 pnas.0408424102 Loeuille and Loreau