Metabolic adjustment enhances food web stability

Understanding ecosystem stability is one of the greatest challenges of ecology. Over several decades, it has been shown that allometric scaling of biological rates and feeding interactions provide stability to complex food web models. Moreover, introducing adaptive responses of organisms to environmental changes (e.g. like adaptive foraging that enables organisms to adapt their diets depending on resources abundance) improved species per-sistence in food webs. Here, we introduce the concept of metabolic adjustment, i.e. the ability of species to slow down their metabolic rates when facing starvation and to increase it in time of plenty. We study the reactions of such a model to nutrient enrichment and the adjustment speed of metabolic rates. We found that increasing nutrient enrichment leads to a paradox of enrichment (increase in biomasses and oscillation amplitudes and ultimately extinction of species) but metabolic adjustment stabilises the system by dampening the oscillations. Metabolic adjustment also increases the average biomass of the top predator in a tri-trophic food chain. In complex food webs, metabolic adjustment has a stabilising effect as it promotes species survival by creating a large diversity of metabolic rates. However, this stabilising effect is mitigated in enriched ecosystems. Phenotypic plasticity of organisms must be considered in food web models to better understand the response of organisms to their environment. As metabolic rate is central in describing biological rates, we must pay attention to its variations to fully understand the population dynamics of natural communities.

Introduction predicted the food web structures of natural communities. The trophic interactions across 94 species are set according to the algorithm detailed by Williams and Martinez (2000) with 95 an expected connectance equal to 0.15. The basal species described by Williams Here N is equal to 6, that means the biggest species is one million times larger than the 101 smallest ones.
These equations describe changes in relative, biomass densities of primary producers (2a) 106 and consumer species (2b). In these equations B i is the biomass of species i, r i is the 107 mass-specific maximum growth rate of primary producers, G i is the logistic growth rate 108 of primary producers (Equation (3)), x i is i's mass-specific metabolic rate, y i is the maxi-109 mum consumption rate of consumers relative to their metabolic rate, e ji is j's assimilation 110 efficiency when consuming population i and F ij describes the realised fraction of i's maxi-111 mum rate of consumption achieved when consuming j (equation (4)). Primary producers 112 growth rate is modelled by a logistic growth with a shared carrying capacity K which 113 ensures a comparable primary production among food webs, regardless the number of 114 primary producers (equation 3).
The consumption rate of prey depends on a Holling type II functional response with 116 predator interference (Equation (4)). The preference of consumers for their prey ω ij are 117 set to 1/p i with p i the number of consumer i's prey as we have no a priori information on 118 preferences. Thus, all consumption rates are only driven by consumer body masses and 119 prey biomass densities. ω ij are recalculated after each extinction to follow the changes of 120 the number of prey p i .
Here B 0 is the half-saturation density of i and c the predator interference.

122
Basically, mass specific biological rates (biomass production, metabolic rate and maximum 123 consumption rate) follow the negative-quarter power-law relationship with species body Thus, the loss due to respiration and the gain due to consumption both directly depend 130 on the metabolic rate (Equation (2b)).
With X the metabolic adjustment coefficient representing the speed of the adjustment.

144
The higher X is, the faster the response of species to modifications of their growth rate is.

145
The metabolic rate is bounded by 1 and 0.001 to ensure a minimum metabolic rate and to The model is coded in C + + and the simulations performed with the GSL ODE solver.

151
The simple tri-trophic food chain only contains a primary producer, a herbivore and a 152 carnivore. Their body masses are respectively set to 1, 10 2 and 10 4 . For the complex food of parameters is tested for 100 different food webs.

160
The first system we consider is a simple tri-trophic food chain containing a primary pro-163 ducer, a herbivore and a carnivore. The effects of the resource availability on species dy-164 namics are represented by bifurcation diagrams (Fig.1). The food chain without metabolic 165 adjustment (X = 0) displays large biomass oscillations whose amplitude increases with the rate increases with carrying capacity K until it oscillates for K > 7 (Fig.1B).

179
The tri-trophic food chain has fixed points along a gradient in metabolic adjustment co-180 efficients for a carrying capacity K = 2 (Fig.2), except for X = 0 (origin of the x-axis 181 corresponding to the situation described in Fig.1A). Increasing the metabolic adjustment 182 coefficient increases the biomass of the herbivore and of the carnivore while it decreases 183 the biomass of the primary producer. However, we observe an increase in the primary 184 producer biomass and a decrease in the herbivore biomass for the low values of X. The 185 metabolic rate of the herbivore is maximum for X > 0 and the metabolic rate of the 186 carnivore first sharply increases with the increasing metabolic adjustment coefficient X 187 and then it decreases (Fig.2B). The response is similar for K = 5 and X < 4 but for 188 X ≥ 4 the system oscillates ( Fig.2A), yet it is not chaotic (Supplementary material Ap-189 pendix A, Fig.A4B). Increasing the metabolic adjustment coefficient does not increase 190 the amplitude of biomass oscillations, it even decreases them for the primary producer.

191
The biomass of the carnivore increases with X, the amplitude of the oscillations of its We can identify two groups of species in complex food webs: 'slow species' with a low 209 biomass (< 10 −2 ) and a low metabolic rate (< 10 −2.5 ) and 'fast species' with a high 210 biomass (> 10 −2 ) and a high metabolic rate (> 10 −2.5 ) ( Fig.3B and 3C). Increasing 211 the carrying capacity K does not seem to change the repartition of species in these 212 two categories (Fig.3B) while more species are in an intermediate category (low biomass 213 and high metabolic rate) at low values of metabolic adjustment coefficient X (Fig.3C).

214
This difference is confirmed in Fig.3D where three groups of species can be identified for for top consumers whose loss rate only depends on metabolic rate and not on predation.

260
In our tri-trophic food chain, carnivores have a highly variable metabolic rate while the 261 herbivore's metabolic rate always stays at the upper limit of metabolic rate range. This

262
can be attributed to a trophic cascade: the carnivore controls the herbivore population 263 and the primary producer thrives. Thus, the herbivore always has plenty of resources, 264 and increasing the metabolic rate increases more the ingestion rate and the growth rate    Figure 1: Bifurcation diagrams of the tri-trophic food-chain containing a primary producers (green), a herbivores (blue) and a carnivores (red). The bifurcation is performed along gradients in the carrying capacity K for A) biomass density and B) metabolic rate for a metabolic adjustment coefficient X = 0 or X = 2.
Figure 2: Bifurcation diagrams of the tri-trophic food-chain containing a primary producers (green), a herbivores (blue) and a carnivores (red). The bifurcation is performed along gradients in the metabolic adjustment coefficient X for A) biomass density and B) metabolic rate for a carrying capacity K = 1 or K = 2. Each point represents one species and 100 food webs are tested for each combination of K and X. D) Distribution of the average metabolic rate of each species along a metabolic adjustment coefficient gradient (K = 1.5). The domains a, b and c represent respectively species with minimum or low metabolic rate, species with intermediate metabolic rate and species with maximum metabolic rate.