We investigate in detail the model of a trophic web proposed by Amaral and Meyer [Phys. Rev. Lett. 82, 652 (1999)]. We focused on small-size systems that are relevant for real biological food webs and for which the fluctuations are playing an important role. We show, using Monte Carlo simulations, that such webs can be non-viable, leading to extinction of all species in small and/or weakly coupled systems. Estimations of the extinction times and survival chances are also given. We show that before the extinction the fraction of highly-connected species ("omnivores") is increasing. Viable food webs exhibit a pyramidal structure, where the density of occupied niches is higher at lower trophic levels, and moreover the occupations of adjacent levels are closely correlated. We also demonstrate that the distribution of the lengths of food chains has an exponential character and changes weakly with the parameters of the model. On the contrary, the distribution of avalanche sizes of the extinct species depends strongly on the connectedness of the web. For rather loosely connected systems we recover the power-law type of behavior with the same exponent as found in earlier studies, while for densely-connected webs the distribution is not of a power-law type.
Deep Dive into Extinction risk and structure of a food web model.
We investigate in detail the model of a trophic web proposed by Amaral and Meyer [Phys. Rev. Lett. 82, 652 (1999)]. We focused on small-size systems that are relevant for real biological food webs and for which the fluctuations are playing an important role. We show, using Monte Carlo simulations, that such webs can be non-viable, leading to extinction of all species in small and/or weakly coupled systems. Estimations of the extinction times and survival chances are also given. We show that before the extinction the fraction of highly-connected species (“omnivores”) is increasing. Viable food webs exhibit a pyramidal structure, where the density of occupied niches is higher at lower trophic levels, and moreover the occupations of adjacent levels are closely correlated. We also demonstrate that the distribution of the lengths of food chains has an exponential character and changes weakly with the parameters of the model. On the contrary, the distribution of avalanche sizes of the extinc
Food webs describe the resources and trophic relationships among species within an ecosystem. The first semiquantitative descriptions of food webs were given by biologists at the end of the nineteen century [1,2]. Later on prey-predator relationship between species were defined in terms of oriented graphs with hierarchical or layered structures [3]. The problem of describing such food webs was then taken over by mathematicians and physicists, and different modeling levels and types of models have been proposed.
A first group of models is constituted by the so-called static models in which the links between different species are assigned once and for all, according to different scenarios (random, scale-free or small-world graphs [4,5], for example). Some properties of these food webs were analyzed and compared with available biological data, and the comparison usually turned out to be quite poor.
The second group of models contains the so-called dynamic food web models. The novelty consists in recognizing that the links between the species are generally not arbitrary and quenched, but emerge as the result of some intrinsic biological dynamics. There are then many possibilities to model the evolutionary dynamics [6]. The simplest one concerns two-layered systems with preypredator Lotka-Volterra type of dynamics (for a short review, see [7]). A very large body of work has been devoted to the study of population dynamics equations for more than two species [8,9]. In such cases, the links among the species can be modified according to the evolutionary dynamics. One important issue is the control of the robustness of such models when the complexity of the system is increased. Moreover, at a more refined level of description, the Lotka-Volterra mean-field dynamics can be replaced by individual-based models [10,11] taking into account the particularities of the interacting individuals and thus offering the possibility to include the stochastic fluctuations. These dynamic food webs models allow therefore to treat on an equal footing both the micro-and the macro-evolution of an ecosystem [12,13].
The richness of the models mentioned above has its own drawbacks. Indeed, the number of control parameters defining the models is usually quite large; moreover the dynamics is nonlinear. Thus, it is often impossible to get a global picture of the properties of the system. Accordingly, it is desirable to study some models which are as simple as possible, in order to clarify the relative importance of the various ingredients, while being able to capture the generic properties expected for food webs. Several proposals have been made along this line in the past years, see e.g. [14,15]. In particular, Amaral and Meyer [16] proposed such a “minimal” model whose numerical solution leads to a power-law distribution of extinction-avalanche sizes, in good agreement with available data from fossile record. It was shown later that this model is self-organized critical [17] and that the power law can be obtained analytically. Furthermore, taxonomic effects have been added to the model [18], but without significant effects.
In this work we are revisiting the Amaral -Meyer model (AM hereafter) with the aim of investigating several of its properties which are relevant for real food webs and which have not been addressed in the previous works. The paper is organized as follows. In Sec. II, the model is described and several technical details concerning the Monte-Carlo simulations, as well as the values of the control parameters, are given. Section III contains the main results. First, the dependence of the survival chance and of the average extinction time on the number of niches N and on the maximum number of feeding species k is studied. The problem of extinction due to stochastic effects is also discussed. Then the question of the pyramidal structure of the food web is approached. Time correlations between the occupied niches at different levels are investigated. The time evolution of the ratio of omnivores is also computed, both for viable and non-viable food webs. The distribution of food-tree sizes as a function of the values of N and k is found to exhibit different regimes. Finally, the problem of avalanches of species extinctions is revisited. In contradiction with previous results, it is found that strong deviations from simple power laws for the size distribution of these avalanches can be observed for large values of k. Some of our predictions are compared with real biological data and are found to be in good agreement. Conclusions are relegated to Sec. IV.
The AM food web model consists of L trophic levels, each of them containing the same number N of niches, which can be either empty or occupied by a single species. Each species from level l = 2, 3, …, L feeds on at most k (k 1) species that are randomly selected from the level below, (l -1) (see Fig. 1). Therefore a species from level l is a predator for some species at
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