Existence and construction of large stable food webs

Ecological diversity is ubiquitous despite the restrictions imposed by competitive exclusion and apparent competition. To explain the observed richness of species in a given habitat, food-web theory has explored nonlinear functional responses, self-interaction, or spatial structure and dispersal—model ingredients that have proven to promote stability and diversity. We return instead here to classical Lotka-Volterra equations, where species-species interaction is characterized by a simple product and spatial restrictions are ignored. We quantify how this idealization imposes constraints on coexistence and diversity for many species. To this end, we introduce the concept of free and controlled species and use this to demonstrate how stable food webs can be constructed by the sequential addition of species. The resulting food webs can reach dozens of species and generally yield nonrandom degree distributions in accordance with the constraints imposed through the assembly process. Our model thus serves as a formal starting point for the study of sustainable interaction patterns between species.

Figure 1

Definitions. (a) Simple food web consisting of four trophic levels. We define the nutrient source to be trophic level zero. The trophic level of any species is the average trophic level of its resources plus 1. Colored circles mark species, light blue ovals mark pairings of species, and (T) marks top species, i.e., species without consumers. Side branches are marked by “SB.” A branching species (or branching point) is marked by “B.” $Q_c={\alpha }^{\prime \prime }/\beta$ (b) An example of a pairing, consisting of a “free” consumer and a “controlled” resource species. (c) Definition of ${\alpha }_{\text{eff}}$ for a controlled species: ${\alpha }_{\text{eff}}$ equals the sum of the decay coefficient of the species itself and the steady-state density of its controlled consumer(s). (d) Illustration of the condition [Eq. (10)] for feasibility at branching points, showing the comparison between ${\tilde{\alpha }}_{\text{eff},f}^{\left(l\right)}$ and ${\alpha }_{\text{eff},c}^{\left(l\right)}$ for two branches formed by a free and controlled species, respectively.

Figure 2

Closure at the basal level. (a) Case of free nutrients: All basal species are controlled. (b) Case of controlled nutrients: One basal species (density $Q_f^{\left(1\right)})$ is free. Dashed lines in both panels symbolize links to possible controlled consumers.

Figure 3

Assembly of a sustainable model food web. (a) Sequential additions, starting from a single basal species. Colors from blue to orange indicate early and late additions, respectively. The area of nodes is proportional to steady-state species densities $Q_i$. Lines indicate consumer-resource links. The latest addition is marked by a white point in each panel. Number of addition marked below the webs. Note that after each addition, a new nonoverlapping pairing (Fig. 1) can be found. (b) Steady-state concentrations $Q_i$ vs species richness for the web in (a), in correspondingly chosen colors. Gray line: trajectory of a particular species, addition 15; compare (a). (c) Average concentration $\langle Q\rangle$. The smooth line indicates the inverse of species richness. Note the logarithmic vertical axes in panels (b) and (c). (d) Total biomass (black) and Lyapunov function (orange) averaged over multiple realizations of food webs as a function of species richness. Thin lines indicate the 10th and 90th biomass percentiles, respectively. (i) $\beta =1$; (ii) $\beta =1/2$. (e) Diversity $D\equiv exp\left(-{\sum }_i{q}_i\text{ln}{q}_i\right)$ per species for $\beta =1$ (i) and $\beta =1/2$ (ii), respectively.

Figure 4

Total species richness obtained in assembly of treelike food webs. In our simulations, a single nutrient source was taken to provide the carrying capacity unity to trophic level one in the food web. Species richness depends on the minimal allowed spontaneous decay rate ${\alpha }_{\text{min}}$; two examples are shown for ${\alpha }_{\text{min}}=0.01$ (black) and ${\alpha }_{\text{min}}=0.02$ (blue). The parameter $\beta$ quantifies consumption efficiency, i.e., the proportion of consumed biomass available to the consumer in terms of population growth.

Figure 5

Distribution of species richness according to tropic levels. (a) Assembly prediction with $\beta =1$ and ${\alpha }_{\text{min}}=0.01$. (b) Assembly prediction with $\beta =0.2$ and ${\alpha }_{\text{min}}=0.01$. (c) Empirical data from seven different food webs, compiled by Dunne et al. [38]. The original data are as follows: Carpinteria Salt Marsh, Estero de Punta Banda, Bahia Falsa [39, 40]; Flensburg Fjord [41], Sylt Tidal Basin [42]; Otago Harbor [43]; and Ythan Estuary [44]. The trophic level of a given species was determined as the average path length to the nutrient level when following predation links downward toward the nutrient level.

Figure 6

Degree distributions. Blue and black curves mark actual and randomized degree distributions, respectively. Panels from (a) to (e) correspond to trophic levels 1 to 5, respectively. Left and right panels correspond to $\beta =1$ and $1/2$, respectively. Each degree distribution is normalized to the total number of links for each trophic level, i.e., for trophic level $l$ the total number of links is the species richness at trophic level $l+1$. Note that in the panel (e) for $\beta =1$, both curves coincide as there are very few links and the randomization gave identical results to the original distribution.

Figure 7

Food web showing species population densities. Species densities (circles) and feeding links (arrows). Circle areas are proportional to species densities. Colors from blue to red indicate earlier and later additions of species.