Food webs
A food web describes feeding relationships among many species in an ecological community. It extends a single predator–prey pair into a network in which each species may interact with several others.
From a food chain to a food web
A simple chain might be
\[\text{plant}\longrightarrow\text{herbivore}\longrightarrow\text{predator}.\]But a real herbivore may eat several plants, a predator may consume several prey species, and two predators may share the same prey. The resulting network is a food web.
What does an arrow mean?
In this page, an arrow points from the organism being consumed toward the consumer:
\[\text{resource}\longrightarrow\text{consumer}.\]Thus an arrow from a plant to a herbivore means that energy or biomass is transferred from the plant to the herbivore through consumption.
A simple food-web diagram
Trophic levels
Species are often described approximately by their position in the flow of energy.
| Level | Typical role |
|---|---|
| basal species | primary producers or other resources that do not consume species within the web |
| primary consumers | consume basal resources |
| secondary consumers | consume primary consumers |
| top consumers | have few or no predators represented within the web |
Real food webs do not always fit into neat levels because omnivores can feed across several trophic positions.
Represent the web as a network
Suppose there are \(m\) species. Each species becomes a node. A directed link records a feeding relationship.
This separates two ideas:
Structure: who eats whom?
Dynamics: how do those interactions change population sizes through time?
An adjacency matrix
A food web can be stored in a matrix. Define
\[A_{ij}=\begin{cases}1,&\text{if species }j\text{ consumes species }i,\\0,&\text{otherwise}.\end{cases}\]With this convention, rows represent resources and columns represent consumers.
For a three-species chain \(1\to2\to3\),
\[A=\begin{pmatrix}0&1&0\\0&0&1\\0&0&0\end{pmatrix}.\]Binary versus weighted networks
A binary adjacency matrix records only whether an interaction exists. A weighted matrix can also record interaction strength.
For example, \(W_{ij}\) might represent attack rate, biomass flux, fraction of a consumer's diet, or another measured interaction strength.
Two webs can therefore have identical links but very different dynamics if their interaction strengths differ.
Population dynamics on the network
If \(N_i(t)\) is the abundance of species \(i\), a general food-web model can be written
\[\boxed{\frac{dN_i}{dt}=F_i(N_1,N_2,\ldots,N_m),\qquad i=1,\ldots,m.}\]The function \(F_i\) contains growth, natural mortality, gains from consuming resources, and losses to consumers or competitors.
A three-species chain model
Consider a resource \(R\), herbivore \(H\), and predator \(P\). A simple model is
\[\frac{dR}{dt}=rR\left(1-\frac{R}{K}\right)-aRH,\] \[\frac{dH}{dt}=e_1aRH-m_HH-bHP,\] \[\frac{dP}{dt}=e_2bHP-m_PP.\]The resource grows logistically. Herbivores gain from consuming the resource and lose individuals through mortality and predation. Predators gain from consuming herbivores and experience mortality.
Follow one interaction through the equations
The term \(aRH\) appears as a loss to the resource. A related term \(e_1aRH\) appears as a gain to the herbivore.
Likewise, \(bHP\) is a loss to the herbivore while \(e_2bHP\) contributes to predator growth.
Direct effects
If a predator consumes a herbivore, increasing predator abundance can directly increase herbivore mortality. This is a direct interaction because the two species share a link.
Indirect effects
Suppose a predator reduces herbivores, and herbivores consume plants. The predator can indirectly benefit plants even though it does not consume or directly interact with them.
\[\text{more predators}\Rightarrow\text{fewer herbivores}\Rightarrow\text{less grazing}\Rightarrow\text{more plants}.\]This is one of the central reasons food-web thinking is needed.
Trophic cascades
A trophic cascade occurs when a change at one trophic level propagates through other levels. For example, reducing a top predator can allow herbivores to increase, which can then reduce plant biomass.
Top-down and bottom-up control
Top-down control occurs when consumers strongly regulate populations below them. Bottom-up control occurs when resource availability strongly determines consumer abundance.
Real food webs can contain both simultaneously. Their relative importance can change with productivity, season or habitat.
Omnivory
An omnivore consumes resources from more than one trophic level. For example, a predator may consume both an herbivore and the plant resource used by that herbivore.
Omnivory adds alternative pathways and means trophic level is no longer a simple integer classification.
Apparent competition
Two prey species can affect one another even when they do not compete for resources. If both support the same predator, increasing prey 1 may increase predator abundance, which then increases predation on prey 2.
\[\text{prey 1 increases}\Rightarrow\text{predator increases}\Rightarrow\text{prey 2 decreases}.\]This indirect negative interaction is called apparent competition.
Intraguild predation
Species can simultaneously compete and consume one another. For example, two predators may share a prey species while one predator also consumes the other.
Such mixed interaction types can produce dynamics that cannot be understood from a simple trophic hierarchy.
Connectance
If \(L\) is the number of realised feeding links and \(L_{\max}\) is the number of possible links under the chosen convention, connectance is
\[\boxed{C=\frac{L}{L_{\max}}}.\]Connectance measures how densely linked the network is. It does not by itself describe interaction strength or stability.
Degree
The number of links associated with a species is its degree. In a directed food web, we can distinguish links to resources from links to consumers.
Species with many trophic links may occupy structurally important positions, but degree alone does not determine ecological importance.
Generalists and specialists
A specialist consumer uses a narrow set of resources, while a generalist uses many. Generalists may be less dependent on any single prey species, but they can also connect otherwise separate parts of the network.
Interaction strength matters
A diagram in which every arrow looks identical can hide large differences. One feeding link may account for most of a predator's diet while another is rarely used.
Quantitative food-web models therefore often require both network topology and interaction strengths.
Functional responses in food webs
Predation terms need not be linear. Consumers can saturate, switch prey or interfere with one another. A multi-prey functional response can make the consumption of one prey depend on the abundance of several alternative prey species.
This creates further indirect interactions through consumer behaviour.
Equilibria and stability
An equilibrium satisfies
\[F_i(N_1^*,\ldots,N_m^*)=0\qquad\text{for every species }i.\]To study local stability, the system can be linearised around an equilibrium using the Jacobian matrix
\[J_{ij}=\frac{\partial F_i}{\partial N_j}.\]The Jacobian records how a small change in species \(j\) changes the instantaneous growth rate of species \(i\) near the equilibrium.
Why the Jacobian resembles an interaction network
If species \(j\) has no direct effect on the equation for species \(i\), the corresponding derivative may be zero. Non-zero off-diagonal entries therefore encode local direct effects.
Unlike a binary adjacency matrix, the Jacobian also contains signs and local strengths and depends on the state at which it is evaluated.
Complexity and stability
More species and more links do not automatically make an ecosystem more stable or less stable. Stability depends on network structure, interaction signs, strengths, self-regulation and the operating equilibrium.
This is why mathematical food-web theory studies not only how many interactions exist, but how they are organised.
Species removal
Removing one species can produce direct losses for consumers that depend on it and indirect changes elsewhere in the web. Secondary extinctions can occur if other species lose essential resources or if interaction balances change strongly.
Keystone species
A species can have an ecological effect disproportionately large relative to its abundance. Such a species may strongly control a consumer, competitor or resource and thereby influence many indirect pathways.
Network position can help identify candidates, but whether a species is functionally keystone depends on dynamics as well as topology.
Stochastic food webs
Births, deaths, encounters and environmental conditions are random. In small populations, stochastic extinction of one species can alter the entire interaction network.
Stochastic models can estimate probabilities of persistence, extinction and cascade-like responses rather than producing only one deterministic trajectory.
Food webs and changing environments
Temperature, habitat loss, nutrient availability and seasonality can change population growth and interaction strengths. A food web is therefore not necessarily a fixed network through time.
Dynamic-network models can allow links or their strengths to change as species distributions and behaviours change.
What a food-web diagram cannot tell you alone
A network diagram tells us which interactions are represented. It does not automatically tell us population sizes, interaction rates, energy fluxes, stability, extinction probabilities or the size of indirect effects.