Food webs, interaction strength, and trophic cascades

Intuition Beginner

A food web is a map of who eats whom. Plants trap sunlight; herbivores eat plants; predators eat herbivores; a few top predators sit at the apex. Each feeding link is a channel through which energy and matter flow upward, step by step, through trophic levels. The shape of this web — how many links it has, how strong each one is, and how the levels are stacked — decides whether a community stays stable or unravels when something is disturbed.

Not every link matters equally. A sea star that eats only a few mussels can hold an entire rocky shore together, if those mussels would otherwise blanket the rock and crowd out everything else. A species like this, small in numbers but huge in effect, is a keystone. When a predator shapes the levels beneath it, and those effects ripple further down, ecologists call the chain of consequences a trophic cascade.

Why is the world green? Plants cover the land despite countless hungry herbivores. One answer, offered in 1960, is that predators hold herbivores in check, sparing the plants. This top-down control is one half of a long debate. The other half asks whether nutrient supply at the base of the web sets the limit instead. In real ecosystems both forces act at once, and their balance is what food-web ecology tries to measure.

Visual Beginner

Top predators      Pisaster (sea star)  ----+
              \                            | eats
               \                           v
Consumers      Mussels  --eats-->  (zoom: mussels crowd rock)
               Barnacles                    ^
               Snails                       | eaten by
Producers      Algae, plankton  -------------+

Removing Pisaster:  mussels UP  -->  barnacles, algae, snails DOWN
                   15 species  -->  ~8 species   (a trophic cascade)

The cascade is the indirect effect. Pisaster never touches the algae directly, yet removing the sea star ultimately wipes the algae off the rock, because the released mussels overgrow it.

Worked example Beginner

In 1966 Robert Paine carried out a now-classic experiment on the coast of Washington State. He chose two similar patches of rocky shore. On one patch he pried off every predatory sea star Pisaster ochraceus he could find, week after week, and threw them into the sea. On the other patch he left the shore alone. The untouched patch was the control.

Before the experiment each patch held about 15 species of invertebrates and seaweeds. Pisaster hunts mussels, and mussels are aggressive competitors for open rock space.

Without the sea star eating them, the mussels spread. They grew over the barnacles, limpets, snails, and algae, crowding each one off the rock. After roughly three years, the cleared patch had fallen from about 15 species to roughly 8 species, with mussels dominant.

One predator — a thin scattering of sea stars — had been holding 15 species' worth of diversity in place. A single feeding link reorganised the whole web. This is the signature of a keystone predator, and the chain predator-down, prey-up, competitor-down is a trophic cascade.

Check your understanding Beginner

Formal definition Intermediate+

Food web as a directed graph

A food web is a directed graph $G=(V,E)$ in which the vertices $V$ are taxa (usually species or lumped feeding groups) and the edges $E$ are trophic links, each pointing from a prey (resource) to its consumer. Write $S=|V|$ for the species richness and $L=|E|$ for the number of trophic links. The community matrix $A$ of per-capita interaction coefficients linearises the dynamics about an equilibrium ${\mathbf{N}}^{\ast }$:

$$\frac{\mathrm{d}{x}_i}{\mathrm{d}t}=\sum _j{A}_{ij}{x}_j,A_{ij}={\frac{\partial }{\partial N_j}\left(\frac{1}{N_i}\frac{\mathrm{d}{N}_i}{\mathrm{d}t}\right)|}_{{\mathbf{N}}^{\ast }},$$

where $x_i=N_i-N_i^{\ast }$ is a small perturbation. The diagonal $A_{ii}$ encodes density-dependent self-limitation (typically negative), and the off-diagonal $A_{ij}$ is the per-capita interaction strength of species $j$ upon the per-capita growth rate of species $i$ ^{[Begon et al. 2006]}.

Descriptors of web architecture

Connectance is the fraction of possible links that are realised:

$$C=\frac{L}{S^2}.$$

Some authors use $C=L/[S(S-1)]$ or $2L/[S(S-1)]$; the convention must be stated when comparing studies. Linkage density is $L/S$, the mean number of links per species. A web is compartmented if its taxa partition into blocks that link densely within a block and sparsely across blocks; modularity algorithms from network theory quantify this. Omnivory is present whenever a consumer feeds at more than one trophic level.

Trophic levels

The integer trophic level of a producer is $1$; a herbivore is $2$; a carnivore eating herbivores is $3$. For a consumer $j$ with $n_j$ prey taxa, the fractional trophic level (Briand, Cohen) is

$${\tau }_j=1+\frac{1}{n_j}\sum _{i=1}^{n_j}{\tau }_i,$$

averaging over prey. A predator eating equal numbers of herbivores ($\tau =2$) and other carnivores ($\tau =3$) sits at $\tau =1+(2+3)/2=3.5$, and is therefore omnivorous.

Interaction strength

Empirical per-capita interaction strengths span several orders of magnitude. Following Paine's and Wootton's field-removal protocol, the effect of removing species $j$ on species $i$ is estimated as $s_{ij}=(N_i^{-j}-N_i^{\text{ctrl}})/N_i^{\text{ctrl}}$, scaled by the per-capita consumption rate where measurable. The distribution of $|s_{ij}|$ across a web is strongly right-skewed: many weak links, a handful of strong ones. This skew is the empirical hinge on which the resolution of the complexity-stability debate turns ^{[Paine 1966; Wootton 1997]}.

Counterexamples to common slips

• High connectance is not high stability. Intuition from engineering (redundant links buffer failure) fails for random model webs: May's criterion below shows that, all else equal, raising $C$ destabilises. Real persistence comes from structured, skewed links, not from sheer link count.

• A keystone is not a dominant. Dominance is measured in biomass or abundance; keystone status is measured in community-level effect per unit biomass. The sea otter is rare yet keystone; the dominant kelp is abundant but not, in the trophic sense, keystone.

• A trophic cascade is not a single predator-prey link. The term requires an indirect effect crossing at least two trophic links. A lion eating a zebra is predation; the lion lowering grass by lowering zebras is a cascade.

• Bottom-up and top-down are not mutually exclusive. Productivity sets the ceiling and predation sets the realised standing crop within it; most systems show both forces, with relative strength that varies along productivity gradients.

Key theorem with proof Intermediate+

Theorem (May's stability criterion, 1972). Let $A$ be the $S\times S$ community matrix of an ecosystem linearised about equilibrium. Set the diagonal $A_{ii}=-d$ for some self-limitation strength $d>0$. Each off-diagonal entry $A_{ij}$ ($i\ne j$) is zero with probability $1-C$ and, with probability $C$, is drawn independently from a distribution of mean $0$ and variance ${\sigma }^2$. Then, in the limit of large $S$ with $SC$ bounded away from zero, the equilibrium is locally asymptotically stable if and only if

$$\sigma \sqrt{SC}<d.$$

Proof. Write $A=-dI+M$, where $M$ carries the random off-diagonal entries. The diagonal shift $-dI$ translates every eigenvalue of $M$ leftward by $d$ in the complex plane, so it suffices to locate the eigenvalues of $M$.

Each row of $M$ has on average $SC$ nonzero entries, each of variance ${\sigma }^2$, and the entries are independent with mean zero. The variance of any single row-sum $r_i={\sum }_j{M}_{ij}{x}_j$ for a unit vector $\mathbf{x}$ is therefore $\mathbb{E}[r_i^2]\approx SC\cdot {\sigma }^2$, so the typical magnitude of $M\mathbf{x}$ is $\sigma \sqrt{SC}$. By Girko's circular law, for large $S$ the eigenvalues of $M$ are uniformly distributed in a disk of radius $\sigma \sqrt{SC}$ centred at the origin.

Adding the diagonal shifts this disk to be centred at $-d$. The rightmost eigenvalue of $A$ accordingly lies at approximately $-d+\sigma \sqrt{SC}$. Local asymptotic stability of the linear system $\mathrm{d}\mathbf{x}/\mathrm{d}t=A\mathbf{x}$ requires every eigenvalue to have negative real part, which holds precisely when the rightmost point of the disk is left of the imaginary axis:

$$-d+\sigma \sqrt{SC}<0⟺\sigma \sqrt{SC}<d.$$

The criterion is sharp in the large-$S$ limit: at equality the spectrum touches the imaginary axis and the equilibrium is neutrally stable. $\square$

Three readings follow immediately. First, holding $\sigma$ and $d$ fixed, increasing species richness $S$ or connectance $C$ eventually destabilises — the model prediction that complex webs should be unstable, contrary to their persistence in nature. Second, reducing the typical interaction strength $\sigma$ widens the stable region, foreshadowing the resolution via skewed weak links. Third, stronger self-limitation $d$ permits greater complexity, linking stability to life-history detail that the random model deliberately discards ^{[May 1972]}.

Bridge. May's criterion builds toward the resolution of the complexity-stability paradox developed in the Advanced results, where skewed interaction strengths and compartmentalised structure rescue stability. The circular-law bound appears again in 19.10.01 as the random-matrix shadow cast over the competitive-exclusion discussion; the foundational reason connectance destabilises is exactly that each extra link broadens the eigenvalue disk, and this is dual to the resource-competition picture in 19.10.01 in which more shared resources shrink the coexistence window. Putting these together, the criterion generalises to structured (non-random) interaction matrices, and the bridge is between the topology of who-eats-whom and the dynamical question of whether the resulting web persists.

Exercises Intermediate+

Advanced results Master

Interaction-strength skew and the weak-link principle

The empirical resolution of May's paradox came from measuring interaction strengths in the field rather than assuming them. Paine's removal protocol, extended by Wootton on the rocky shore of Tatoosh Island and by Fagan and Berlow in mesocosms, showed that the distribution of per-capita effects is consistently right-skewed: the median link is weak, and a small minority of strong links account for most of the biomass flux. McCann, Hastings and Huxel (1998) formalised this in model webs by drawing interaction strengths from a skewed distribution rather than May's equal-variance Gaussian. Their central result is that weak links damp the oscillatory coupling between strong consumers and their resources, preventing the overcompensatory cycles that would otherwise drive extinctions. The weak-link principle reconciles May's local-stability bound with observed persistence: complexity is tolerable when most links are weak, and the few strong links are precisely the keystone interactions identified experimentally ^{[McCann et al. 1998]}.

Trophic cascades: strength, asymmetry, and context dependence

Cascades are not all-or-nothing. Strong cascades, in which a top predator controls two lower levels, are best documented in systems with simple linear chains and fast turnover at the base: lake pelagics (Carpenter, Kitchell and Hodgson's whole-lake manipulations), kelp forests (sea otter-urchin-kelp, Estes and Palmisano), and rocky intertidals (Pisaster-mussels-algae). Terrestrial cascades tend to be weaker and more diffuse, because plant defences, omnivory, and detrital subsidises dilute the signal. Schmitz, Hambäck and Beckerman's meta-analysis estimated that the average cascade effect on plant biomass is roughly four times larger in aquatic than in terrestrial systems. The asymmetry follows from food-chain architecture: short, linear chains transmit predator effects cleanly; reticulate webs with omnivory leak the signal across alternative pathways. This context dependence is the content, not the failure, of cascade theory.

Bottom-up, top-down, and the productivity gradient

The HSS hypothesis that the world is green because predators limit herbivores was generalised by Fretwell and by Oksanen, Fretwell, Arruda and Niemelä into the exploitation ecosystem hypothesis. The prediction is that the number of trophic levels set by a system increases with primary productivity: unproductive systems support only plants (level 1), consumed by no-one; moderately productive systems add herbivores that suppress plant biomass (level 2, top-down on plants); productive systems add predators that suppress herbivores and release the plants (level 3, green world); very productive systems add a fourth level that suppresses predators and again releases herbivores. The green-world prediction thus alternates with productivity. Chase's and Borer's experimental tests along fertilisation gradients confirm that top-down and bottom-up forces interact: productivity sets how many levels can be sustained, and the top level then determines whether the cascade runs downward.

The energetic equivalence rule

Damuth (1981) compiled abundance and body-mass data across mammalian herbivores and found that population density scales as $N\propto M^{-3/4}$. Combined with Kleiber's $B\propto M^{3/4}$ for individual metabolic rate (and the West-Brown-Enquist derivation from fractal transport networks), population energy use $E=N\cdot B$ scales as $M^0$. Nee, Read, Greenwood and Harvey (1991) confirmed the prediction across local assemblages: coexisting populations of very different body sizes channel approximately equal energy. The rule connects food-web structure to metabolic scaling — the energy available to a trophic level is partitioned roughly equally among body-size classes — and explains why abundance distributions are so regularly right-skewed toward small, numerous organisms. Currie and Fritz extended the analysis and found the rule strongest within trophic groups and weaker across them.

May's paradox and the structure of real webs

May's criterion predicted instability for any sufficiently complex web; field webs are both complex and persistent. The resolution operates on several axes: interaction strengths are skewed (most weak), webs are compartmented into weakly interacting subwebs (Pimm and Lawton 1977 found compartmentalisation reduces effective connectance), body-size ratios constrain the feasible interaction strengths (Brose, Williams and Martinez showed allometric scaling narrows the parameter region of instability), and adaptive foraging rotates interaction strengths in real time toward stabilising configurations. None of these rescue an arbitrary random web; they collectively show that real webs are not random. The contemporary synthesis, due to Allesina, Tang, Grilli and others, replaces May's circular law with structured random-matrix ensembles in which predator-prey sign structure, body-size scaling, and weak-link skew are built in; the resulting stability boundary matches observed web complexity within a factor of two.

Synthesis. Food-web dynamics are governed by three facts working in concert: topology, who links to whom; strength, how hard each link pulls; and energy, how much biomass each level can carry. The foundational reason complex natural webs persist despite May's warning is that real interaction strengths are skewed — many weak, few strong — and weak links act as dampers; this is exactly the McCann mechanism, which generalises May's equal-variance model to heterogeneous interactions. The central insight of trophic-cascade theory, that a single predator can restructure two lower levels, is dual to the energetic-equivalence rule, which says population energy use is body-mass invariant; putting these together, the bridge is from the network architecture of feeding links to the dynamical persistence of whole ecosystems, and the whole picture appears again in 19.11.01 as the energy-flow backbone of ecosystem ecology and in 19.10.01 as the multi-species competition substrate on which every cascade is built.

Full proof set Master

Proposition (Trophic-transfer efficiency caps chain length). Let $B_0$ be the energy or biomass entering the producer level and let $\epsilon \in (0,1)$ be the ecological efficiency, the fraction of energy at one level transferred to the next. Then the biomass supported at trophic level $k$ satisfies $B_k\le {\epsilon }^k{B}_0$, and the maximum sustainable chain length $k_{\max }$ for a threshold biomass $B_{\min }$ obeys $k_{\max }\le (\ln B_0-\ln B_{\min })/\ln (1/\epsilon )$.

Proof. Energy balance at level $i$ gives $B_i=\epsilon B_{i-1}$ exactly when respiration and non-predatory losses take the remainder $1-\epsilon$; solving the recurrence from $B_0$ yields $B_k={\epsilon }^k{B}_0$ for the ideal chain, and any real losses only reduce $B_k$, giving the inequality. The chain ends when biomass drops below the viable threshold $B_{\min }$, i.e. ${\epsilon }^k{B}_0<B_{\min }$. Taking logarithms (both $\epsilon$ and $B_{\min }/B_0$ lie in $(0,1)$ so signs are consistent): $k\ln \epsilon <\ln (B_{\min }/B_0)$, hence $k>\ln (B_{\min }/B_0)/\ln \epsilon =(\ln B_0-\ln B_{\min })/\ln (1/\epsilon )$. With the empirical $\epsilon \approx 0.10$, four trophic levels already retain only $0.10^3=0.001$ of the producer energy, which is why natural chains almost never exceed five levels. $\square$

Proposition (Energetic equivalence). Suppose population density scales with body mass as $N(M)=a{M}^{-3/4}$ and individual metabolic rate scales as $B(M)=b{M}^{3/4}$ for positive constants $a,b$. Then the total energy flux through a population, $E(M)=N(M)\cdot B(M)$, is independent of $M$.

Proof. Substitute the scaling relations: $E(M)=a{M}^{-3/4}\cdot b{M}^{3/4}=ab\cdot M^{(-3/4+3/4)}=ab\cdot M^0=ab$. Since $M^0=1$ for every $M>0$, $E(M)$ is the constant $ab$, independent of body mass. Two populations whose body masses differ by a factor $f$ therefore channel equal energy: the larger is rarer by $f^{-3/4}$ and each individual burns more by $f^{3/4}$, and these factors cancel exactly. $\square$

Proposition (Connectance scaling under allometric constraints). If the number of prey a consumer can exploit is bounded above by a constant $k_{\max }$ set by foraging and handling-time limits, and each web has roughly equal numbers of predators and prey, then total links scale as $L\le k_{\max }S/2$, and connectance scales at most as $C=L/S^2\le k_{\max }/(2S)$, i.e. $C\to 0$ as $S\to \infty$.

Proof. Order the $S$ taxa and let $k_i$ denote the number of prey of consumer $i$. The allometric handling-time bound gives $k_i\le k_{\max }$ for every $i$. Summing links over all consumers, $L={\sum }_i{k}_i\le {\sum }_i{k}_{\max }=S{k}_{\max }$; halving to avoid double-counting directed links in an undirected approximation gives $L\le k_{\max }S/2$. Dividing by $S^2$, $C\le k_{\max }/(2S)$, which vanishes as $S$ grows. This reproduces the empirical constant-links conjecture (Martinez 1992, Brose et al. 2004) that linkage density $L/S$ is roughly constant across webs and that connectance consequently declines as $1/S$, keeping $\sigma \sqrt{SC}$ within the stable region as richness rises. $\square$

Connections Master

• 19.10.01 provides the species-interaction framework — competition, mutualism, predation, the competitive-exclusion principle, and the Lotka-Volterra coefficients ${\alpha }_{ij}$ — that the present unit specialises into per-capita trophic interaction strengths $A_{ij}$. The May random-matrix criterion restated here is the spectral refinement of the stability discussion sketched in the community-ecology unit, and every cascade in this unit is built on a Lotka-Volterra consumer-resource pairing inherited from there.

• 19.10.02 pending supplies the temporal axis that food-web structure lacks: succession determines which species are present to form links, so the connectance and interaction-strength distributions measured here are themselves successional snapshots. A late-successional community typically has higher connectance and more omnivory than a pioneer assemblage, linking the static topology of this unit to the dynamic replacement of the succession unit.

• 19.09.01 population ecology underwrites every link in a food web as a pair of coupled population-growth equations. The logistic self-limitation $d$ in May's diagonal is the same density dependence that bounds a single population, and the consumer-resource cycles dampened by weak links are Lotka-Volterra orbits from the population unit.

• 19.11.01 ecosystem ecology receives the trophic-transfer efficiency $\epsilon \approx 0.10$ derived in the Full proof set as the backbone of energy-flow pyramids and productivity budgets. The energetic equivalence rule connects directly to ecosystem-level metabolism and to the carbon and nutrient fluxes that the ecosystem unit formalises.

• 19.14.01 conservation biology applies cascade theory to management: protecting or restoring top predators (sea otters, wolves, sharks) is a lever for whole-ecosystem recovery, and the keystone concept justifies single-species interventions with community-wide goals. The productivity-dependent cascade strength derived here predicts where such interventions will and will not succeed.

Historical & philosophical context Master

Robert Paine's 1966 paper "Food Web Complexity and Species Diversity" (American Naturalist 100, 65-75) introduced the keystone-predator concept through the Pisaster removal experiment on Makah Bay, Washington. Paine's central claim — that a species low in biomass could nonetheless govern community diversity — overturned the intuition that importance tracks abundance, and it reframed predation from a pairwise interaction into a community-structuring force ^{[Paine 1966]}. His later work with Wootton on Tatoosh Island extended the programme to quantitative measurement of per-capita interaction strengths, launching the empirical interaction-strength tradition that ultimately resolved the complexity-stability debate.

The trophic-cascade idea originates with Nelson Hairston, Frederick Smith and Lawrence Slobodkin's 1960 paper "Community Structure, Population Control, and Competition" (American Naturalist 94, 421-425), which posed the "green-world" question: why is the terrestrial world visibly green, given the abundance of herbivores ready to consume it? Their answer — predators limit herbivores, releasing plants — was contentious from the start. Murdoch (1966) and Ehrlich and Birch (1967) objected that the argument ignored plant defences and climatic limitation, a debate that drove four decades of field experimentation ^{[Hairston, Smith & Slobodkin 1960]}. The cascade was generalised into the exploitation ecosystem hypothesis by Fretwell (1977, Oecologia 22) and by Oksanen, Fretwell, Arruda and Niemelä (1981, American Naturalist 118), who predicted alternating top-down and bottom-up control along productivity gradients.

Robert May's 1972 one-page paper "Will a Large Complex System be Stable?" (Nature 238, 413-414) and his 1973 monograph Stability and Complexity in Model Ecosystems (Princeton University Press) imported random-matrix theory into ecology. May showed that, under random interactions, increasing species richness or connectance destabilises a community, in direct tension with the classical Eltonian intuition (Charles Elton 1958, The Ecology of Invasions by Animals and Plants, Methuen) that diverse communities are more stable. The resulting complexity-stability paradox became the central problem of theoretical community ecology ^{[May 1972; May 1973]}.

Stuart Pimm's 1982 monograph Food Webs (Chapman & Hall) consolidated the empirical and theoretical programme, cataloguing regularities — connectance scaling, chain length, omnivory — across published webs and sharpening the questions May had posed. Pimm and Lawton (1977, Nature 268) and (1980, Journal of Animal Ecology 49) tested compartmentalisation and predicted that shorter, less-connected model webs should be more stable, results later refined by McCann, Hastings and Huxel (1998, Nature 395) whose skewed-interaction-strength model showed how weak links stabilise complex webs ^{[Pimm 1982; McCann et al. 1998]}. The contemporary structured random-matrix synthesis, due to Allesina and Tang (2012, Nature 483) and Grilli, Rogers and Allesina (2016, Nature Ecology & Evolution), folds predator-prey sign structure and body-size constraints into May's original ensemble and recovers the observed boundary between stable and unstable webs.

Bibliography Master

1. Paine, R. T., "Food web complexity and species diversity", American Naturalist 100 (1966), 65-75.

2. Hairston, N. G., Smith, F. E. & Slobodkin, L. B., "Community structure, population control, and competition", American Naturalist 94 (1960), 421-425.

3. May, R. M., "Will a large complex system be stable?", Nature 238 (1972), 413-414.

4. May, R. M., Stability and Complexity in Model Ecosystems, Princeton University Press (1973).

5. Pimm, S. L., Food Webs, Chapman & Hall (1982).

6. Pimm, S. L. & Lawton, J. H., "Number of trophic levels in ecological communities", Nature 268 (1977), 329-331.

7. Pimm, S. L. & Lawton, J. H., "Are food webs divided into compartments?", Journal of Animal Ecology 49 (1980), 879-898.

8. McCann, K., Hastings, A. & Huxel, G. R., "Weak trophic interactions and the balance of nature", Nature 395 (1998), 794-798.

9. Estes, J. A. & Palmisano, J. F., "Sea otters: their role in structuring nearshore communities", Science 185 (1974), 1058-1060.

10. Oksanen, L., Fretwell, S. D., Arruda, J. & Niemelä, P., "Exploitation ecosystems in gradients of primary productivity", American Naturalist 118 (1981), 240-261.

11. Damuth, J., "Population density and body size in mammals", Nature 290 (1981), 699-700.

12. Nee, S., Read, A. F., Greenwood, J. J. D. & Harvey, P. H., "The relationship between abundance and body size in British birds", Nature 351 (1991), 312-313.

13. Wootton, J. T., "Estimates and tests of per-capita interaction strength: diet, abundance, and impact of intertidally foraging birds", Ecological Monographs 67 (1997), 45-64.

14. Carpenter, S. R., Kitchell, J. F. & Hodgson, J. R., "Cascading trophic interactions and lake productivity", BioScience 35 (1985), 634-639.

15. Schmitz, O. J., Hambäck, P. A. & Beckerman, A. P., "Trophic cascades in terrestrial systems: a review of the effects of carnivore removals on plants", American Naturalist 155 (2000), 141-153.

16. Allesina, S. & Tang, S., "Stability criteria for complex ecosystems", Nature 483 (2012), 205-208.

17. Elton, C. S., The Ecology of Invasions by Animals and Plants, Methuen (1958).

18. Begon, M., Harper, J. L. & Townsend, C. R., Ecology: Individuals, Populations and Communities, 4th ed., Blackwell (2006), Ch. 9-10.