Less is worse than none: ineffective adaptive foraging can destabilise food webs

Consumers can potentially adjust their diet in response to changing resource abundances, thereby achieving better foraging payoffs. Although previous work has explored how such adaptive foraging scales up to determine the structure and dynamics of food webs, consumers may not be able to perform perfect diet adjustment due to sensory or cognitive limitations. Whether the effectiveness of consumers’ diet adjustment alters food-web consequences remains unclear. Here, we study how adaptive foraging, specifically the effectiveness (i.e. rate) with which consumers adjust their diet, influences the structure, dynamics, and overall species persistence in synthetic food webs. We model metabolically-constrained optimal foraging as the mechanistic basis of adaptive diet adjustment and ensuing population dynamics within food webs. We compare food-web dynamical outcomes among simulations sharing initial states but differing in the effectiveness of diet adjustment. We show that adaptive diet adjustment generally makes food-web structure resilient to species loss. Effective diet adjustment that maintains optimal foraging in the face of changing resource abundances facilitates species persistence in the community, particularly reducing the extinction of top consumers. However, a greater proportion of intermediate consumers goes extinct as optimal foraging becomes less-effective and, unexpectedly, slow diet adjustment leads to higher extinction rates than no diet adjustment at all. Therefore, food-web responses cannot be predicted from species’ responses in isolation, as even less-effective adaptive foraging benefits individual species (better than non-adaptive) but can harm species’ persistence in the food web as a whole (worse than non-adaptive). Whether adaptive foraging helps or harms species coexistence has been contradictory in literature Our finding that it can stabilise or destabilise the food web depending on how effectively it is performed help reconcile this conflict. Inspired by our simulations, we deduce that there may exist a positive association between consumers’ body size and adaptive-foraging effectiveness in the real world. We also infer that such effectiveness may be higher when consumers cognise complete information about their resources, or when trophic interactions are driven more by general traits than by specific trait-matching. We thereby suggest testable hypotheses on species persistence and food-web structure for future research, in both theoretical and empirical systems.

Introduction the same amount of energy to any consumer species. The consumer's energy intake rate can thus be 120 evaluated by its total (i.e., sum over its diet) biomass consumption rate. 121 In a food web, where multiple species are feeding on each other, the total biomass consumption 122 rate of the j th consumer species is 124 where D j is the set of k species that constitutes its diet, and for the i th resource species in its diet, 125 a ij is its per-capita search rate (m 2 s −1 as we model 2D foraging), A ij its attack success probability, 126 h ij its per-capita handling time (s) of a successful foraging attack, and F ij quantifies its time cost for 127 engaging in an unsuccessful attack attempt (sensu Meire & Ervynck 1986), expressed as a proportion 128 (here assumed to be a constant) of h ij . n i is the i th resource species' numerical abundance (here, 2D 129 density, in m −2 ) and m i its body mass (kg; mean body mass ignoring intraspecific variation). For 130 detailed parameterisation, see Supplementary Information section S1. 131 To identify a consumer's optimal diet, one needs to find the specific set D that maximises its 132 resultant C with given resource abundances. In response to changing resource abundances, the diet 133 adjustment predicted by OF-either including new resources into, or excluding existing resources 134 from, the current OF diet-must always follow the profitability order of the consumer's potential to the j th consumer species is therefore Once resource profitabilities have been calculated, the consumer's realised optimal diet can then be 141 derived by identifying the highest C generated with eqn (1) from its narrowest (i.e., eating only 142 the most profitable species) to broadest (i.e., eating all species) possible diets following a decreasing The population dynamics of species in each focal food web were modelled using a generalised Lotka- 148 Volterra model, where the species' numerical abundances described by ordinary differential equations 149 as (in matrix notation) 150ṅ = N (r + Bn) .
(2) 151 Here, r and n are vectors of species' intrinsic growth rates (r) and numerical abundances (n), respec- species' intrinsic growth rate and intraspecific interference strength both scale with its body mass.

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The per-capita intrinsic growth rate of the i th species scales as

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where m i is its body mass (kg). The per-capita intraspecific interference strength of the i th species 160 (b ii , the diagonal elements of B) scales as

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where b 0 a positive scaling constant (for values, see further below). The negative intrinsic growth rates 163 represent consumers' natural mortality rates, while the negative interference strengths represent the 164 typically negative effects of increasing density on species' population growth.

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The per-capita trophic interaction strengths between the j th consumer species and its i th resource 166 species (coupled b ij and b ji , the off-diagonal of B) can then be derived from biomass consumption 167 rates by accounting for the fraction of population numerical loss and gain contributed by per-capita 168 consumer and resource, respectively. That is,

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The first equation is the numerical loss from the population due to being consumed, and the second 171 the numerical increase from consuming others. The scalar is the consumer's efficiency of converting 172 consumed biomass to its own biomass (here assumed to be a constant, see SI section S1).

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Simulation of food-web dynamics 174 Building a synthetic species pool, we simulated 100,000 species whose body masses (kg) were randomly

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where m i is its body mass, and n 0 a positive scaling constant (for value, see further below). Synthetic

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We wired all synthetic communities into "initial webs" for subsequent simulation of population 186 dynamics. An initial web could be set at a steady state where all species coexist with positive 187 abundances, and all consumers' current diets are exactly their optimal diets derived (eqn (1)) with 188 such species abundances. However, an steady state cannot be analytically solved a priori in a Type-

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II system with its dynamic abundance-dependent interaction strengths, and a fully feasible (i.e., all 190 species coexist) equilibrium is highly improbable with all metabolic constraints (i.e., mass-scaling 191 rules) we have set. Therefore, we approximated such a steady state, and ensured that all generated 192 initial webs are comparable, by constraining each initial web to have the following properties:

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• The species abundances are as close as possible to an "analytical" dynamical equilibrium of the

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• All consumers feed on their optimal diets based on the given values of species abundances and all 199 other parameters, i.e., the initial web is itself an OF web. This ensures that, across replicates, any

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OF diet adjustment taking place during simulation is triggered by species abundance fluctuation 201 after the simulation begins, not by any preset difference among initial webs.

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• The web has a ∼ 0.1 connectance, which is comparable to observed values in empirical food

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With the assigned interference strengths (eqn (4)), as well as the known trophic interaction strengths 216 (eqn (5) with above-given n 0 and h 0 combinations) and intrinsic growth rates (eqn (3)), these OF to their abundances (both in log 10 scale), was identified and taken as the reference equilibrium. The 227 specific parameter set of n 0 , h 0 , and b 0 leading to this reference equilibrium, the mass-scaled species 228 abundances with this n 0 (not the reference equilibrium abundances), and the corresponding OF web 229 wired at their combination (i.e., the initial web) together presented the "initial state" for subsequent 230 food-web dynamical simulation. 231 We then simulated the population dynamics from the initial state by numerically solving eqn (2) 232 using the deSolve R package (Soetaert et al. 2010) and its ode function. Each simulation was run for 233 10 8 time units (10 3 as an integration time step). During the simulation, any species whose abundance 234 (density) fell below 10 −9 was considered extinct and its abundance set to 0. From the same shared 235 initial state, food-web dynamics were allowed to unfold based on each of five foraging schemes (so five  (1)). Note that as species abundances at the initial state are close, yet never 242 equal, to those at the reference analytical equilibrium, population dynamics with abundance 243 fluctuations will still occur in this scheme even with fixed interaction strengths. This is a "null", 244 control scheme that eliminates both the abundance dependence of parameters and the consumers' 245 OF diet adjustment.

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• Type.II: the trophic interaction strengths are dynamically updated with real-time species abun-247 dances following eqn (5), but the consumers are still not adjusting their diets. This can be 248 seen as another null scheme, simulating dynamics with Type-II functional responses but without 249 consumers' OF diet adjustment.

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• OF.fast: the trophic interaction strengths are as in the Type.II scheme. The optimal diet of 251 each consumer is re-calculated every 10 4 time units (10 time steps) based on species abundances 252 at that juncture using eqn (1). Consumers are accordingly allowed to adjust their diet by one 253 resource, following the resources' profitability order towards achieving their currently optimal 254 diet. That is, a consumer may broaden or narrow its diet by one resource, or keep the same diet, 255 depending on which act makes its diet closer to the re-calculated optimal one. The population 256 dynamics simulation is resumed after the diet adjustment with an updated food-web topology 257 and corresponding interaction matrix (B).

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• OF.mid: the same as the OF.fast scheme, but the re-derivation of optimal diets and consumers' 259 diet adjustment happens every 10 5 time units (100 time steps).

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• OF.slow: the same as the OF.fast scheme, but the re-derivation of optimal diets and consumers' 261 diet adjustment happens every 10 6 time units (1000 time steps). 262 We expect the comparison of the Type.I and Type.II schemes to reveal the influence of resource

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In terms of both species persistence and food-web structure, our simulation generated clear scheme- abundances, extinctions happened relatively frequently (Fig. 1) was not effective enough, as the OF.slow scheme had even more cumulative extinctions than the non-291 diet-adjusting Type.II scheme (Fig. 1). Notably, as we judged species extinction using an abundance 292 threshold (Methods), in the Type.II scheme there were occasionally a few consumers having all their 293 initial resources go extinct without yet having their own abundances drop below the threshold by the 294 end of the simulation. These "resourceless" consumers were actually doomed to go extinct without 295 the ability to establish new links to resources, but were technically not recorded as extinctions-296 the number of extinctions in the Type.II scheme was thus slightly underestimated. Therefore, more 297 precisely speaking, the OF.slow scheme may not necessarily have led to more extinctions (Fig. S1), 298 but certainly accumulated extinctions faster (i.e., higher extinction rates) than the Type.II scheme.

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Ineffective OF diet adjustment, though by definition being still adaptive (in terms of gaining more 300 energy) to individual consumers, was surprisingly harmful to the overall species persistence in food 301 webs.

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Food-web structure also varied through time. As no diet adjustment was allowed in the Type.I 303 and Type.II schemes, all their structural changes were purely driven by the loss of nodes and links 304 via species extinctions during the dynamics. Accordingly, the Type.I scheme saw very little structural 305 change throughout, while the structure in the Type.II scheme first changed rapidly then became near-306 constant (Fig. 2), largely reflecting the patterns of cumulative species extinction (Fig. 1). In general, 307 food webs in the Type.II scheme became less connected, less nested, and less modular over time.

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Also, their number of top consumers slightly decreased, while the mean resource-consumer body-size 309 ratio slightly increased (Fig. 2). Conversely, food webs in the OF schemes showed different structural 310 change trajectories. Generally, these food webs became slightly more connected, equally nested, less 311 modular, had fewer top consumers (even fewer than the in the Type.II schemes) and a slightly smaller 312 mean resource-consumer body-size ratio by the end of dynamical simulation (Fig. 2). Notably, unlike 313 in the Type.I and Type.II schemes, here in the OF schemes the structural changes were driven by 314 both species extinction and trophic link rewiring due to consumers' diet adjustment. Therefore, food-315 web structures did not change toward their final states monotonically, but instead had fluctuations 316 throughout the simulation (Fig. 2). Looking at the ending phase of simulation, in most structural 317 measures, the OF schemes tended to be closer to the shared initial state than the Type.II scheme, 318 and the OF.fast tended to be the closest (Fig. 2). These findings suggested that, while adaptive diet 319 adjustment is itself a mechanism that can alter food-web structure, an effective diet adjustment can, 320 in the long run, maintain some structural properties even though the size of the food web may have 321 changed due to species extinctions.

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A closer look at species identities within food webs (Fig. 3) further revealed the qualitative dif-323 ference in extinctions (i.e. "who went extinct") among the schemes. This also helped us to better 324 understand the causal links between diet change, extinctions, and changes in food-web structure. In 325 the Type.II scheme, the initial extinctions could occur on consumer species of any body size (Fig. 3), 326 but most large consumers went extinct eventually, leaving some intermediate-and small-sized con-327 sumers in the food web by the end of simulation. Although species were not adjusting their diet 328 in the Type.II scheme, due to the extinction of large consumers, some intermediate-or small-sized consumers were freed from predation and themselves became new top consumers (Fig. 3). As a result 330 of such trophic role transitions, the decrease in number of top consumers did not appear to be severe 331 ( Fig. 2) even though the "initial" top consumers had mostly gone extinct. In contrast, the OF schemes 332 immediately generated species role transitions through rewiring rather than extinction (Fig. 3). Early 333 in the dynamics, the large consumers broadened their diet toward smaller species (SI Fig. S2), mak-334 ing the web more connected and more nested while less modular. Meanwhile, the smallest few top 335 consumers became the resources of larger ones, thus their roles turned into intermediate consumers 336 and caused a decrease in the number of top consumers (Fig. 2, 3). Over time, species extinction 337 was largely suppressed in the OF.fast schemes, where those that went extinct were mostly large-sized tively species perform such diet adjustments is a crucial factor determining whether adaptive foraging 347 benefits or harms the overall species persistence in food webs.

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Our simulations showed that abundance-dependent trophic interaction strengths following Type-II 349 functional responses, in comparison to fixed strengths, exacerbated species extinctions (Fig. 1). This the feedback required for species persistence (Fig. 4), and we therefore observed fewer cumulative 362 extinctions in the OF.fast and OF.mid schemes than in the Type.II scheme (Fig. 1).

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Interestingly, we found that OF diet adjustment increases species persistence only when the ad- to extinction (Fig. 3). Although the OF diet adjustment allows the very few largest consumers to per-392 sist. The remaining large consumers besides these few still eventually go extinct, even with relatively 393 effective diet adjustment (Fig. 3). Notably, again echoing Rooney et al.     Figure 4: A consumer's biomass intake rate and the predation pressure it imposes on percapita more-profitable resource. This is illustrated in a one-consumer-two-resource system. A and B are the two resource species, where A is the more-profitable one. The black dashed lines indicate the scenario with optimal foraging diet adjustment. The biomass consumption rate (top panels) follows a multi-species Type-II function, and the per-capita predation pressure on A (bottom panels) is quantified by A's contribution to the consumer's biomass intake rate divided by A's numerical abundance. The consumer that forages without diet adjustment but feeds "single-mindedly" on A (red lines) leads to a monotonically increasing predation pressure on A with its decreasing abundance (bottom panels), which gives the typical destabilising effect of a Type-II functional response on A's population. In contrast, making a diet adjustment to include B when A is rare (black dashed lines) creates a sudden fall of predation pressure at the threshold abundance of A (indicated by arrows, bottom panels). This leads to a lower predation pressure at the low-abundance end of A than in the previous scenario, and therefore mitigates partly the destabilising effect. Comparing the left with right panels reveals that such mitigation is more effective when the less-profitable resource (B) is abundant.