Flexible diets, fragile routes

The question

Are complex ecosystems more stable than simple ones, or less? Random-network models, starting with May (1972), predict less: adding species and feeding links makes a community less likely to be stable. Yet real ecosystems are very complex, and they persist.

Kondoh (2003) proposed an explanation. Real predators do not eat from a fixed menu; they shift their foraging towards whichever prey currently gives the best return. In his simulations, when predators adapted in this way, more complex food webs were more likely to keep all their species.

What we did

We built hundreds of random food webs on a computer, with 12 to 500 species, and ran each one two ways: predators that split their effort evenly and never change, and predators that adapt. Then we looked at where each community ends up, and how it gets there.

We also ran them with and without an extinction line, a rule that removes any species whose numbers drop below a tiny value. Kondoh used one, as do most food-web models.

The model in two equations

The model is Kondoh's. Each species i has a population density N_i, which changes as

Read the bracket left to right. A species grows at its own rate r_i, is limited by crowding (−s_iN_i), gains from what it eats and loses to what eats it. Here r_i is the species' net growth rate on its own (births minus deaths), and s_i sets how strongly it limits itself, so on its own it settles at r_i/s_i. In Kondoh's setup every r_i is positive and s_i = 1. In the gain term, f_ij is the rate at which predator i captures prey j, a_ij is the share of its foraging effort spent on j, and e is the fraction of eaten food converted into new growth. The loss term is the same interaction seen from the prey's side, summed over its predators k.

The second equation is the adaptation rule:

Here e f_ij N_j is the return per unit of effort that predator i gets from prey j, and the sum after it is the predator's average return over its whole diet. Effort on a prey grows if it returns more than average and shrinks if it returns less, at a rate set by G. This is the replicator equation of evolutionary game theory (Hofbauer & Sigmund 1998), applied to each predator's diet. A fixed diet is a_ij = 1/k_i, where k_i is the number of prey species that i can eat.

Each web has one end state

Start an adaptive food web from any populations, any diets and any adaptation speed, and it reaches the same end state: the same species, at the same densities. The end state is set by the web itself (which species it contains, who can eat whom, and their growth and feeding rates), not by its history.

In the special case where predators convert all the food they eat into growth (e = 1), we can prove this. We found a Lyapunov function: a quantity that only ever decreases as the community changes, so the system behaves like a ball rolling into a single valley. The construction extends one that Boukal and Křivan (1999) used for small predator–prey models. For realistic conversion efficiencies we have no proof yet, but every web we tested with Kondoh's parameters behaved the same way, and an independent check of the mathematics and code confirmed the results.

Flexible diets keep more species

In Kondoh's setup, adaptive webs usually kept every species, while fixed-diet webs lost about 5 to 20 percent. The few species that adaptive webs do lose are prey of predators with no alternative food, which cannot switch away.

Speed changes the route, not the destination

If the end state does not depend on speed, what does speed change? The route. Slow adapters keep feeding on a declining prey for too long, so it crashes to very low numbers before predators move on, and only then recovers. The slower the adaptation, the deeper the crash. Fixed-diet webs show no such crashes: species that survive never fall far, and those that are lost decline to extinction.

The Lyapunov function V also limits how deep a crash can be. Since V never increases, the density of a surviving species has a lower bound set by the initial value of V. The diet term of V is divided by G, so slow adaptation gives a large initial V and a weak bound. The bound is far from tight, but it has the observed form: crash depth, in powers of ten, grows roughly in proportion to 1/G.

Kondoh's speed effect comes from the extinction line

A real population that fell to one part in 10²⁷⁵ would be extinct. Models represent this with an extinction line, and that is where adaptation speed matters. We reran Kondoh's setup exactly. With his extinction line, the share of webs that kept every species rose steeply with adaptation speed, as he reported. Without it, every adaptive web kept every species at every speed. The end state is the same either way; the extinction line decides which species survive the journey to it.

Adapted diets get narrow

Each predator could eat many prey, but at the end state it eats only one or two. The links still in use form a tree-like network (a forest, in graph terms) with fewer links than twice the number of species. Matsuda and Namba (1991) proved this for coevolutionarily stable communities. It matters for theory, because calculations that are hard on dense networks are often exact on trees.

The argument is short. If a predator divides its effort between two prey, they must give it the same return:

Each such tie fixes the ratio of two prey densities. A loop of ties would constrain the same ratio along two different routes, which requires an exact coincidence among the feeding rates. So the links in use form no loops, and a network without loops has fewer links than nodes.

What about rewiring?

Our wider project also studies rewiring, in which a predator that loses its prey adds a new species to its diet (Gilljam et al. 2015). Does rewiring break the one-valley picture?

Gaining and losing links is not the problem in itself. Adaptive predators already drop links, by putting no effort into them, and can take them up again later, and the web still has a single end state. The difference lies in how choices are made. Continuous adaptation keeps every option open and keeps revising. Rewiring makes a one-off choice when a trigger fires, and keeps it. The web then retains a memory of the order in which events happened, so different runs can end differently. An extinction line has the same effect: a species removed during a crash never returns.

Why it matters

• It splits Kondoh's result into two separate effects. Adaptation changes where a food web ends up: more species survive at the end, at any speed. Adaptation speed only changes how the web gets there: slow switching drives species through deep crashes on the way, and in a real population (or a model with an extinction line) those crashes are permanent losses. So whether adaptation actually saves species depends on how fast predators switch compared with how fast populations change.
• Because the end state is set by the web, it should be possible to calculate it from a few summary statistics of the web, without simulation. Such calculations exist for food webs with fixed interactions (Bunin 2017; Barbier et al. 2018). Extending them to adaptive diets is our next step.

What we haven't shown yet

• In the setup used here, every species can grow without prey, which is why nearly all survive. We are now testing webs in which some species need prey to persist.
• Some results rest on a few dozen webs. Larger runs, with hundreds of webs of up to 500 species, are in progress.
• We have not finished checking the literature, so we make no claim of novelty yet.