The geometry of evolved community matrix spectra

Modelling complex evolved food webs

Our interest here is to develop a conceptual comparison between the eigenvalue spectrum of a complex, evolved food web and a random matrix analog. We therefore focus on the widely-used generalised Lotka–Volterra equations for consumer-resource interactions. For simplicity, we further restrict to a single basic nutrient source, and require that species feeding on the basic nutrient source are never omnivorous³¹, e.g., plants do not consume other plants. The original Lotka–Volterra equations^32,33 describe spatially and temporally homogeneous, consumer-resource relations. The generalised Lotka–Volterra equations^34,35,36 can be used to describe the dynamics of larger, more complex food webs, and encode the dynamics of primary producers as

where (S_i), (iin {1,dots ,n_1}), denote the population densities of primary producers in units of biomass, normalised to the system carrying capacity and (n_1) denotes the total number of primary producers, (k_i>0) denote the growth rates of the corresponding primary producer (S_i), that is, the maximal reproduction rate at unlimited nutrient availability. We use (k_i=k) for all primary producers. The negative sum on the species (S_j) encodes logistic growth by accounting for nutrient depletion by all primary producers. For all other species, (S_k), (kin {n_1+1,dots ,n}) with n the total number of species in the food web, the equations read

Here, (S_k) is again measured in units of normalised biomass. In Eqs. (1) and (2), (alpha _j>0) is the decay rate of a species (S_j), representing death not caused by consumption through other species. (eta _{ki}ge 0) is the link-specific interaction strength between consumer (S_k) and resource (S_i). On the RHS of either equation, note the final term representing the diminishing effects experienced by each resource species, which is caused by consumption. This term is mirrored by the first term in Eq. (2), which describes the strengthening effect on the consumer side. The coefficients (beta _{ki}le 1) encode link-specific consumption efficiency—that is, potentially incomplete use of energy removed from a resource species by its consumer. (beta _{ki}=1) would describe perfect consumption efficiency whereas in real food webs this value is estimated to lie considerably lower³⁷. In our simulations we use (beta _{ki}=beta) for all interactions present.

Equations (1) and (2) describe a simplified food web structure where consumption is modelled by the simple Holling type-I response³⁸, where consumer resource fluxes scale proportional to the product of consumer and resource biomass density and there are no saturation effects. Moreover, Eqs. (1) and (2) assume that the food web is rigid in that species are incapable of adapting their consumption behaviour to changes within the food web, such as a decreasing population of resources or competition from an invasive species³⁹. Yet, these equations allow for a coherent description of the energy fluxes between species and constitute an established framework for complex consumer-resource relations to evolve.

To evolve food webs we simulate Eqs. (1) and (2) numerically. New species are added successively to an existing food web. We assume that invasion attempts occur on a slow timescale, such that equilibrium can be reached before the subsequent invasion attempt, though occasionally, the food web does not converge to its equilibrium state. After each invasion attempt the steady state species vector (mathbf {S^*}) is computed. In case of feasibility the eigenvalues of the community matrix are evaluated in order to determine the linear stability of the steady states. If feasibility is not obtained, that is, if (mathbf {S^*}) contain negative populations, Eqs. (1) and (2) are integrated numerically until extinctions occur and feasibility of the remaining species is reached (Details: “Materials and methods”). Examples of several invasion attempts are shown in Fig. 2.

Loops profoundly impact food web evolution

To make sure our results do not depend on the details of the invasion process we allow for several qualitatively distinct evolutionary processes: (i) treelike food webs, where each consumer has a single resource; (ii) non-omnivorous food webs with loops; (iii) omnivorous food webs. Loops are known to be relevant for sustained limit cycles and chaotic attractors, thus widening the range of dynamical properties. Indeed, we find treelike food webs to stand out in that fitness, measured by species decay rates, indefinitely increases in the evolutionary process (Fig. 1a, dotted red line), a finding consistent with the recent literature³⁰. This indefinite fitness improvement hinges on the absence of network loops: a given primary producer can only be replaced by an invading primary producer of greater intrinsic fitness, that is, lower decay rate.

Allowing for network loops, evolved food web do not show indefinite fitness improvement (Fig. 1b,c) and mean species richness somewhat decreases (Fig. 1, insets). All histograms show a systematic difference in odd and even species richness, with food webs of odd species richness being the most frequent. This tendency is most pronounced for treelike food webs. We interpret this as a manifestation of the requirement of non-overlapping pairing²⁸. Treelike food webs are feasible and stable if every species in the food web can be coincidentally paired with a connected species or nutrient that is not part of another pairing. In food webs of even species richness the nutrient is never included in such a pairing. Food webs consisting of several smaller trees that are connected through the nutrient source are therefore only feasible if every tree satisfies this requirement individually. On the contrary, the nutrient is always included in a pairing in food webs of odd species richness, and therefore odd food webs are more likely to be feasible. To a lesser extent this tendency is also found in the histograms representing food webs with network loops. We interpret this as resulting from the fact that 40-60% of the food webs from simulations allowing network loops are in fact treelike.

Why do loops counteract indefinite fitness improvement? This can be seen as a manifestation of relative, rather than absolute, fitness, where a species can consume two resources and thereby can help eliminate even primary producers of high intrinsic fitness (Fig. 1b,c). An example of this is illustrated in Fig. 2f), where the intrinsically fittest producer is a node in a food web loop, and is driven to extinction during the invasion of a producer with lower intrinsic fitness.

The evolution of intrinsic fitness in Fig. 1 implies that allowing for interaction loops makes resident species more vulnerable to extinction during invasions, because parameters that characterise high intrinsic fitness before an invasion might characterise low intrinsic fitness during the invasion. This is supported by the cumulative distribution of resident times (Fig. S1a), where residence times in food webs with network loops fall off faster than the residence times in treelike food webs. In Fig. S1b we observe that in accordance with this, the distribution of extinction event size falls off faster for treelike food webs (Fig. S1b), where the extinction event size is measured relative to the total number of species (species richness) in the food web. Fig. S1b therefore implies that interaction loops make food webs less robust to invasions, as invasive species tend to create larger extinction events here than in treelike food webs. Finally, we find invasive species to have higher success rates when invading food webs with interaction loops, and the success rate is found to increase with (beta). In simulations with (beta =0.75) we observe 11.5%, 27.2% and 29.8% for treelike, non-omnivorous, and omnivorous food webs with loops, respectively. The implications of this are twofold. On one hand, it is easier to assemble feasible food webs when multiple resources and omnivory are allowed. On the other hand, these food webs are more susceptible to invasions and their resident species are more vulnerable. If a food web contains two-resource species, removal of one of the two resources of a species (S_i) by an invader can already lead to a cascading extinction of S, as exemplified by Fig. S2.

Robustly bi-modal eigenvalue spectra

We now turn to the eigenvalue spectra of the evolved complex food webs, which we present as two-dimensional histograms in the complex plane (Fig. 3). Each simulation conducts (10^5) invasion attempts, yet the number of unique feasible food webs is considerably lower, that is, approximately equal to the aforementioned rates of successful invasions. Furthermore, the number of unique feasibly food webs drastically decreases with species richness. While the data shown represent relatively small networks, we find that key spectral features are very systematic as function of species richness. A generic feature is that spectra typically have many eigenvalues with small negative real parts. Further, the real parts scatter more and more closely at small negative values, as species richness increases beyond two. All spectra contain a considerable fraction of purely real eigenvalues, typically making up 15–30% of a spectrum.

The origin of purely real eigenvalues

The first column in Fig. 3 represents food webs with species richness two. These simple food webs only have one feasible configuration, namely that of one primary producer and one consumer. Any differences between spectra in the left column are therefore purely statistical. These food webs can be considered as isolated interactions between a consumer and its resource, hence the analytical eigenvalues of this food web can provide some insight on the dynamics underlying the eigenvalue spectra. From the analytical eigenvalues we obtain that an eigenvalue is purely real if the inequality

is fulfilled (Details: Sec. S3). Here, (alpha _1) and (alpha _2) are the decay rates of the resource and the consumer, respectively, and (beta eta) is short for (beta _{21}eta _{21}), the “consumption rate” of the consumer. (gamma) can be interpreted as the inverse intrinsic fitness of the food web.

From feasibility, we have the additional requirement of (gamma < beta eta), hence, the consumer’s “consumption rate” is bounded also from below. As k decreases, the lower and upper boundaries on (beta eta) approach one-another until they are equal for (k=0). A food web with low producer growth rate is therefore likely to have complex eigenvalues. In the opposite limit, when (krightarrow infty), or equivalently (alpha _1 rightarrow 0), we see that (gamma) reduces to (alpha _2). In the first limit Eq. (3) reduces to (beta eta le infty) which will always be satisfied and all eigenvalues are therefore purely real in this limit. This corresponds to a food web where the consumer has infinite access to resources and there is no stress or constraints on the web that could cause oscillations. In the limit where (alpha _1 rightarrow 0), the eigenvalues pick up an imaginary component when (beta eta) is large compared to (alpha _2) and k. This occurs when the consumer population has a large intrinsic growth rate, thus heavily exploiting its resource.

Overall, purely real eigenvalues characterise food webs where consumption of the resource is moderate compared to the intrinsic fitness of the resource. This corresponds to an over-damped limit where the consumer does not consume enough to cause any significant displacement of the resource population, hence a perturbation of the consumer population will not spread to its resource. For higher species richness the Jacobian quickly becomes too complicated to be solved analytically. Even so, we expect the dynamics between a consumer and its resources to be conceptually analogous, namely that “sustainable over-consumption” yields oscillating densities and complex eigenvalues.

The set of smallest and largest real-valued eigenvalues is obtained when (beta eta) is only slightly larger than (gamma), hence barely satisfying the criterion of feasibility. The eigenvalues then reduce to (lambda _{pm } = -frac{k-alpha _1}{2} pm frac{k-alpha _1}{2}). (lambda _+) is always zero, that is, food webs of species richness two are always stable, and with our choice of parameters (lambda _- ge -0.95). We observe approximately the same range of real values in all numerical spectra of any species richness, thereby implying that the choice of parameters might be more important for the spectrum width than the structure of the food web.

The overall shape is qualitatively similar for all food web structures (see: Fig. 3). Importantly, omnivorous spectra are the only ones to contain also eigenvalues with positive real part, that is, unstable eigenvalues. These food webs do therefore not converge to their equilibrium state after an invasion, but are displaying periodic or chaotic dynamics (Details: “Materials and methods”). The unstable eigenvalues are all barely larger than zero, hence hardly visible in Fig. 3. Interestingly, non-omnivorous food webs with network loops exhibit the same species richness and approximate connectivity as the omnivorous food webs, yet they do not yield unstable eigenvalues. The differences between treelike food webs and food webs with network loops discussed earlier must therefore be unrelated to the stability of the food webs, thus emphasising the difference between stability to perturbation of a given food web and its robustness to invasions. For omnivorous food webs the fraction of unstable eigenvalues increases with species richness and decreases with (beta). Intuitively, it seems reasonable that there is a relation between instability and low consumption efficiency. A species with a low consumption efficiency has to compensate by consuming more biomass, thereby putting more stress on its resources. Only for (beta =1) are there no unstable omnivorous eigenvalues.

We now compare the evolved spectra (Fig. 3) to their random counterparts (Fig. 4). The diagonal entries represent self regulation of each species and are set to (d = -1). Off-diagonal entries are drawn from ({mathcal {N}}(0, 1)) with probability p(N), and are otherwise 0.

where N is “species richness”, that is, the number of rows (or columns) of the matrix. This corresponds to the implemented connectivity in the simulation allowing network loops and omnivory, that is, the connectivity of omnivorous food webs given no extinctions occur (Details: Sec. S5). As predicted by spectral theory of random matrices, the spectra are centred around d on the real axis and approach a circular geometry as the size of the matrix increases. Already for (N=2) does the spectrum contain unstable eigenvalues. The fraction of unstable eigenvalues increases with N as the circle radius increases. Also for random spectra do we observe a large fraction of purely real eigenvalues. We attribute this to the small size of the matrices, being much smaller than the infinity limit for which the law was derived⁴⁰.

Finally, we study the real-part frequency distributions of eigenvalues of all four types (treelike, non-omnivorous, omnivorous and random). The frequency distributions for species richness 2–9 can be seen in Fig. 5, where each distribution consists of data from various values of (beta) (see Table 1). In order to facilitate comparison of the functional form of the frequency distributions, rather than the range, the frequency distributions are scaled to be bounded by (-1) on the real axis, that is, we divide each data point by ((|min {x}|)^{-1}) where x is the data points of the distribution. Frequency distributions representing the evolved food webs follow approximately the same curve for a given species richness, and are distinctively different from the random matrices. As also seen in Fig. 3 omnivorous distributions are the only to extended to positive values for species richness greater than two.

Once again, we observe quantitative differences between food webs with odd and even species richness: For odd species richness the distribution is bi-modal with a global maximum near (x=0) and a secondary maximum near the lower limit, that is (x=-1). For even species richness, the distribution is initially less strongly peaked. Yet, as species richness increases, a sharp peak emerges around (x = 0). The distribution thus becomes more similar to that of the food webs with an odd number of species.

The intermediate part of the spectrum is increasingly depleted of eigenvalues at higher species richness. Comparing Fig. 5 with Fig. 3 we see that the left part of all distributions consists of purely real eigenvalues, whereas it is mostly complex eigenvalues that make up the global maximum near (x=0). This implies that perturbations can be divided into two main groups: perturbations from which the food web quickly returns to the respective steady state, and perturbations that induce oscillations from which the food web takes very long to recover. The peak consisting of purely real eigenvalues near (x=-1) does not change notably with species richness, indicating that, independent of species richness, food webs are robust to certain perturbations. In accordance with this we observe that food webs of all species richness usually return quickly to their steady states after an unsuccessful invasive species goes extinct. The main peak (near (x=0)) becomes both higher and narrower with increasing species richness, that is, the food webs become quasi-stable. In larger food webs there are more species that can be disturbed by a perturbation, which might prolong the effect of the perturbation, that is, push eigenvalues towards zero on the real axis. Overall, we thus find that the histogram of complex food webs becomes strongly bi-modal as food webs consisting of many species are approached in an evolutionary process, whereas random matrix spectra are consistently uni-modal. In Sec. S8–S9 we consider the robustness of the results in Fig. 5 by varying the parameter distributions and implementing Holling type-II response, respectively.

Source: Ecology - nature.com