Emergence and maintenance of biodiversity in an evolutionary food-web model

Abstract

Ecological communities emerge as a consequence of gradual evolution, speciation, and immigration. In this study, we explore how these processes and the structure of the evolved food webs are affected by species-level properties. Using a model of biodiversity formation that is based on body size as the evolving trait and incorporates gradual evolution and adaptive radiation, we investigate how conditions for initial diversification relate to the eventual diversity of a food web. We also study how trophic interactions, interference competition, and energy availability affect a food web’s maximum trophic level and contrast this with conditions for high diversity. We find that there is not always a positive relationship between conditions that promote initial diversification and eventual diversity, and that the most diverse food webs often do not have the highest trophic levels.

A Appendix

A.1 Evolutionary dynamics

To increase computational efficiency, we have implemented the evolutionary process such that for any community of coexisting morphs we draw the next successfully invading mutant trait value and the time at which the invasion occurs.

A.1.1 Determining the rate of successful mutations

We first determine the probability p _i that a mutant in morph i is successful. Recalling that we assume a mutant’s trait value to be normally distributed with variance $\sigma_m^2$ around its parent’s trait value, and denoting by p _r (m) the probability that a mutant with trait value m successfully invades a community characterized by the trait values r = (r ₁, ..., r _n ) of the resident morph, p _i is given by

$$ p_i = \frac{1}{\sigma_m} \int_{-\infty}^{\infty} \phi \left( \frac{m - r_i}{\sigma_m} \right) p_r(m)\,dm, $$

where ϕ is the probability density function of the standard normal distribution. In general, this integral would have to be evaluated numerically, but for small values of σ _m , a second-order Taylor approximation of p _r (m) around r _i can be used. With τB _i denoting the rate at which mutations occur in morph i, the rate at which successful mutations occurs in morph i is s _i  = τB _i p _i .

A.1.2 Drawing the next successful mutant

The morph i in which the next successful mutant occurs is drawn with probability s _i /S, with S = ∑  _i s _i . We then draw the mutant’s trait value m from the probability density function

$$f(m) = \frac{1}{\sigma_m p_i} \phi \left( \frac{m - r_i}{\sigma_m} \right) p_r(m).$$

To do this, we use the rejection method described in Press et al. (1992). After choosing a sufficiently large maximal value D of the deviation |m − r _i |, and after determining an upper bound F _m so that F(m) > fm for all m, we draw a uniformly distributed bivariate random deviate (m, y) with |m − r _i | < D and y < F _m . If y < f(m), we take m as the new mutant trait value; otherwise, the bivariate random draw is repeated. Alternatively, we can draw m from the normal distribution σ((m − r _i )/σ _m )/σ _m and y from the uniform distribution over the unit interval. If y < p _r (m), we take m as the new mutant trait value; otherwise, the bivariate random draw is repeated.

Finally, since the arrival of successful mutants follows a Poisson process with intensity S, the waiting time between such events is drawn from an exponential distribution with parameter S (see e.g. Grimmett and Stirzaker 1992, pp 248).

A.1.3 Implementing the oligomorphic stochastic model

The evolutionary dynamics of the oligomorphic stochastic model (Ito and Dieckmann 2007) are obtained by assuming a time-scale separation between population dynamics and evolutionary dynamics and by making a simplifying assumption for the conditions under which a mutant trait value replaces the resident trait value from which it arises. Specifically, we implemented a variant of the oligomorphic stochastic model that considers a community viable if the corresponding system of Lotka-Volterra equations has an interior rest point (see Hofbauer and Sigmund (1998) for a thorough account of the relationship between Lotka-Volterra dynamics and the existence of interior rest points).

It is worth noting that the method described by Law and Morton (1996) for species-assembly models based on Lotka-Volterra dynamics in particular, the use of permanence to determine a community’s viability can possibly be adapted to evolutionary models based on Lotka-Volterra dynamics, which would enable an improvement over the algorithm described here.

A.2 Definition of trophic levels

Although the concept of trophic level is frequently encountered in the literature, it is not always easy to define. Most of the proposed methods (see Yodzis 1989 for examples) are either difficult to apply to food webs containing cycles and loops, or expensive to calculate in practice. Furthermore, if trophic levels are defined as taking integer values only, the implied abrupt jumps in trophic level will make it difficult to track the dynamics of evolving food webs.

Following Odum and Heald (1975) and Adams et al. (1983), we define a morphs’ real-valued fractional trophic level as 1 plus the average trophic level of its prey. Denoting by t _i the fractional trophic level of morph i = 0,...,n (with t ₀ = 1 corresponding to the basal autotrophic resource) and by v _ij the proportion of energy entering morph i from morph j (with ∑  _j v _ij  = 1), the fractional trophic levels t _i are defined as the solutions of the linear equations

$$ t_i = 1 + \sum\limits_{j=0}^{n} v_{ij} t_j. $$