Models of food web evolution

The two major types of food web models are those that are static and phenomenological, and those that are dynamic. The first type includes models such as the cascade, niche and nested hierarchy models (I have talked about these models elsewhere). The static models use a set of rules to reproduce statistical patterns of food web topologies, and have proven valuable to food web research but they have their limitations. One criticism is that using these models forces us to use what are called “trophic food webs” where trophically similar species are collapsed into a single node. I think that the future of food web research lies in models that explicitly incorporate the dynamics of predator prey interactions. Within this class of models are two basic methods of building food webs. The two methods are using community assembly or evolution based algorithms. In this post I am going to briefly go over a number of models that are utilize evolution based algorithms to introduce new species to the community. The following is modified from an appendix I wrote for my dissertation proposal, with links to arXiv preprints (when available) or journals.

There are five models of food webs based on evolution that have been proposed, as well as several modifications of some of these (Loeuille & Loreau 2010). These models are not designed to be realistic representations of biological evolution. Instead they should be thought of as theoretical approximations of how coevolution may act to drive the construction of community interactions (Christensen et al. 2002). The models I will describe here are Webworld (Caldarelli et al. 1998; Drossel et al. 2001, 2004), Recursive Evolutionary Branching (Ito & Ikegami 2006), Matching (Rossberg et al. 2006), Tangled Nature (Christensen et al. 2002; Anderson & Jensen 2005), and Body Size Evolution (Loeuille & Loreau 2005, 2010).

In the Webworld model species are defined by a particular set of abstract traits, which could in theory represent either morphological or behavioral characteristics (Caldarelli et al. 1998). Each species is assigned L traits out of a pool of K traits. It is important to note once again that these traits are abstract and only represented by integers 1… K. A KxK matrix (m) is defined such that its elements mab represent the score of a particular trait a against trait b. Predator-prey relationships are thus assigned by summing the score of species i against species j, where a positive score means i is adapted to be a predator of j and 0 means no interaction (Drossel et al. 2001). Competition for resources is hierarchically based on predator scores for a given species, meaning that a higher score indicates a better competitor. Speciation in the Webworld model occurs probabilistically as a function of population size. When a speciation event occurs, the new species is generated by randomly replacing 1 of the L traits of the chosen species (constrained such that the event must generate a unique set of species traits).  Other than the choice of L and K, the model has only three parameters to be set; the trophic efficiency of consumers, the total amount of external resources, and a parameter that determines the strength of competition (Caldarelli et al. 1998). This model does not consider individual variation, genetics, the mechanisms of speciation, and there is no distinction between the modes of reproduction (Caldarelli et al. 1998). It has, however, been used to consider the implications of varying the form of the functional response in predator-prey equations such that more realistic population dynamics can be incorporated (Drossel et al. 2001, 2004).

I suppose the model proposed by Ito and Ikegami (2006) would be termed the Recursive Evolutionary Branching (REB) model although they do not explicitly name it. This model is essentially a continuous version of the Webworld model described above (Loeuille & Loreau 2010). Unlike the Webworld model, however, in the REB model species are given two sets of traits; one for defining its location as a resource (resource traits), and one for defining the set of resources it uses (utilization traits). Ito and Ikegami (2006) assumed a single dimensional resource space and a two dimensional phenotype space to reduce the complexity of the model. In other words, given a 1 dimensional resource space each of the k species has a distribution describing it as a resource and a distribution describing the resources it utilizes. The distributions were described by a delta function. Evolutionary dynamics in the REB model are described as a diffusion process where most species are assumed to have a large number of individuals and the magnitude of a mutation is assumed to be small.

The Matching model (Rossberg et al. 2006) combines parts of both the Webworld and REB models. Like the Webworld model the Matching model represents species as a set of traits, and like the REB model there is a set of traits describing it as a resource (vulnerability) and as a predator (foraging). Additionally unlike either model these traits are represented as binary character strings with a defined length n. The strength of an interaction is then defined as the number of predator foraging traits that match prey vulnerability traits, so long as it exceeds some threshold value (defining the presence of a link).  Additionally, species are given a parameter which defines the size of a species, and they can only consume prey that are below a size threshold defined as their lambda more than their size (Rossberg et al. 2006). Lambda is a loopiness parameter, determining the probability of a trophic loop occurring. Unlike the previously described models the Matching model does not explicitly incorporate population dynamics, instead defining extinction, speciation, and invasion probability with specified parameters. New species invading the system are given their two binary trait sets and size randomly, while speciation events cause foraging and vulnerability traits to flip with a given probability and a random value (from a standard normal distribution) is added to the size parameter. This model assumes that evolution in all directions is equally likely.

I will only briefly outline the Tangled Nature model, as I do not anticipate using it in my research (and it gets confusing). The model is unique in that it may incorporate multiple interaction types, rather than being restricted to trophic interactions (Loeuille & Loreau 2010). Additionally it is an individual based model, rather than focusing on populations or species as the base unit. Individuals are defined as a binary string of “alleles” of length L, which are allowed to mutate with a defined probability. The model assumes that the interactions between species are determined by the interaction between the L loci. The strength of the interaction is given by a matrix with terms drawn from the uniform distribution along an interval of [-c,c] with a given probability.

While the first four models described typically utilize multiple traits to describe the interactions between organisms, the Body Size Evolution model uses only a single trait (body size, as the name implies). Loeiulle and Loreau (2005) developed a model of community evolution where each species is given a characteristic body size. This body size allows demography to be determined based on allometric relationships, trophic interactions to be described by defining an optimum feeding size range, and competition based on similar body sizes to be modeled. Population dynamics are modeled using differential equations with allometrically determined production efficiency (growth rate) and mortality, as well as functions describing consumption and competition based on body size differences between species. Basal species in the model consume inorganic nutrient, which is also modeled with a differential equation describing its rate of change as a function of input, output, and nutrient recycling. Species are modified at a rate (usually 10-6) per unit biomass at each time step. If a given population has a mutation event a new daughter population is created with a body size that is drawn from a uniform distribution along the interval [0.8x, 1.2x] where x is the parent body size.

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