Ecological networks
Ecological networks represent interactions among species, populations or functional groups. They allow community ecology to be studied as a system of many connected interactions rather than as isolated species pairs.
Nodes and edges
A node usually represents a species or ecological group. An edge represents an interaction such as predation, competition, mutualism, parasitism or facilitation.
The biological meaning of an edge must always be specified because different ecological networks require different directions and signs.
Food webs
A food web represents trophic interactions. If an edge is defined from resource to consumer,
\[i\to j\]means that species \(j\) consumes species \(i\).
Other authors use the opposite convention. Therefore the direction convention should be stated explicitly before interpreting in-degree or out-degree.
Signed interactions
Ecological interactions can be classified by their effects on the two participating species.
| Interaction | Effect on species 1 | Effect on species 2 |
|---|---|---|
| competition | negative | negative |
| mutualism | positive | positive |
| consumer–resource | positive for consumer | negative for resource |
| commensalism | positive | approximately neutral |
A signed network records these qualitative effects.
Adjacency and interaction matrices
A binary adjacency matrix records whether interactions exist:
\[A_{ij}=\begin{cases}1,&\text{if an interaction from }j\text{ to }i\text{ is included},\\0,&\text{otherwise.}\end{cases}\]A weighted interaction matrix instead uses coefficients \(a_{ij}\) that quantify the effect of species \(j\) on species \(i\).
The two matrices should not be confused: adjacency describes topology, whereas interaction coefficients describe dynamical strength.
Generalised Lotka–Volterra dynamics
A common population model is
\[\boxed{\frac{dN_i}{dt}=N_i\left(r_i+\sum_{j=1}^{S}a_{ij}N_j\right)},\]where \(N_i\) is the abundance of species \(i\), \(r_i\) is its intrinsic growth contribution and \(a_{ij}\) is the effect of species \(j\) on species \(i\).
If no direct interaction is represented, one may set
\[a_{ij}=0.\]Interpretation of interaction coefficients
With this convention,
\[a_{ij}>0\]means species \(j\) has a positive direct effect on the per-capita growth of species \(i\), while
\[a_{ij}<0\]means the direct effect is negative.
The coefficients need not be symmetric:
\[a_{ij}\ne a_{ji}.\]Competition networks
In a competitive community, off-diagonal interaction coefficients are often negative:
\[a_{ij}<0,\qquad a_{ji}<0.\]Network structure can represent which species compete directly, while the coefficient magnitudes describe competition strength.
Mutualistic networks
Mutualistic interactions benefit both participants. Plant–pollinator systems are commonly represented as bipartite networks with plants in one node set and pollinators in another.
If the sets are \(P\) and \(Q\),
\[E\subseteq P\times Q.\]No plant–plant or pollinator–pollinator edge is required in the basic bipartite representation.
Host–parasite networks
Bipartite networks can also connect hosts to parasites. Degree then has a direct interpretation: a parasite's degree may count host species used, while a host's degree may count associated parasite species.
The same graph concept therefore acquires different biological meaning depending on which node set is examined.
Connectance
Connectance measures the fraction of possible links that are realised.
For a directed network of \(S\) species without self-links, there are \(S(S-1)\) possible directed edges. If \(L\) links are present,
\[\boxed{C=\frac{L}{S(S-1)}}.\]For an undirected simple network, the denominator is instead
\[\binom S2.\]Bipartite connectance
If a bipartite network contains \(S_1\) nodes in one set and \(S_2\) in the other, there are
\[S_1S_2\]possible cross-group links. Hence
\[\boxed{C=\frac{L}{S_1S_2}}.\]Degree and ecological specialisation
In some ecological networks, low degree may indicate specialisation and high degree may indicate generalism.
This interpretation must be made cautiously because observed degree depends on sampling effort, interaction definition and species abundance.
Nestedness
A bipartite network is described as nested when the interaction partners of specialists tend to form subsets of the partners of more generalist species.
Nestedness is frequently studied in mutualistic networks, but its ecological implications depend on the dynamical model and should not be inferred from topology alone.
Modularity
Ecological networks may contain modules: groups of species interacting more strongly or more frequently with one another than with species outside the group.
Modules can influence the spread of disturbances and may partially compartmentalise ecological effects.
Trophic levels
Food-web direction can be used to organise species according to trophic position. Basal species obtain resources without consuming other represented species, while consumers occupy higher trophic positions.
Real food webs can contain omnivory and cycles, so trophic level is not always a simple integer hierarchy.
Indirect effects
A network contains paths through which one species can influence another indirectly.
For example, if a predator suppresses a herbivore and the herbivore consumes a plant, the predator can indirectly benefit the plant. Such trophic cascades illustrate why pairwise interactions can produce community-level consequences.
Paths do not determine effect signs automatically
Multiplying qualitative signs along a simple pathway can suggest an indirect effect, but real dynamical responses can involve multiple competing paths, nonlinearities and feedback.
Therefore full ecological effects generally require analysis of the dynamical system, not only inspection of the graph.
Equilibria
For the generalised Lotka–Volterra model, an equilibrium with all \(N_i^*>0\) satisfies
\[r_i+\sum_j a_{ij}N_j^*=0\]for every species.
In vector form,
\[\mathbf r+A\mathbf N^*=0.\]If \(A\) is invertible, the formal solution is
\[\boxed{\mathbf N^*=-A^{-1}\mathbf r}.\]However, this is biologically feasible only when every required equilibrium abundance is positive.
Feasibility and stability are different
An equilibrium is feasible if the species abundances are biologically admissible, typically
\[N_i^*>0.\]Stability asks whether small perturbations return toward the equilibrium.
A feasible equilibrium need not be stable, and a mathematically stable equilibrium may fail to be biologically feasible.
Community matrix
Near an equilibrium, local dynamics are determined by the Jacobian or community matrix
\[J_{ij}=\left.\frac{\partial}{\partial N_j}\frac{dN_i}{dt}\right|_{\mathbf N=\mathbf N^*}.\]For an interior equilibrium of the generalised Lotka–Volterra system, the equilibrium condition simplifies this to
\[\boxed{J_{ij}=N_i^*a_{ij}}.\]Thus network interaction coefficients contribute directly to local stability.
Local stability
The equilibrium is locally asymptotically stable when all eigenvalues of \(J\) have negative real parts:
\[\boxed{\operatorname{Re}(\lambda_m)<0\quad\text{for all }m}.\]This connects ecological network structure to the linear-stability methods developed earlier in mathematical biology.
Complexity and stability
Increasing the number of species or interactions does not have a universally positive or negative effect on ecological stability.
Classical random-matrix results show that sufficiently strong random interactions can destabilise large complex systems, but real ecological networks contain structured signs, strengths and correlations that can substantially change this conclusion.
Interaction strength matters
A binary network treats all observed links equally, but a weak interaction and a strong interaction can have very different dynamical consequences.
Weighted ecological networks are therefore often necessary when the aim is to predict population responses rather than merely describe topology.
Dynamic ecological networks
Ecological interactions can change through seasons, environmental conditions, abundance changes or evolution.
A time-dependent interaction matrix can be written
\[A=A(t),\]while adaptive models may allow interactions themselves to respond to population states.
Species loss and secondary effects
Removing a species deletes a node and its incident interactions. Other species may then change abundance or become extinct because of lost resources, lost mutualists or altered competitive pressure.
Network robustness therefore depends on both topology and population dynamics.
Ecological network data
Observed food webs and mutualistic networks are affected by sampling effort. Rare species and weak interactions are particularly easy to miss.
Apparent connectance, degree and specialisation can therefore depend on how and when the ecological community was observed.
Relation to earlier ecological models
Earlier population and ecological modelling sections studied equations for interacting populations directly. The emphasis here is complementary: many-species interactions are organised explicitly as a network, allowing topology and dynamics to be analysed together.
Transition to network centrality
Ecological networks raise a natural question: which nodes occupy especially influential structural positions? The next lesson develops centrality measures that quantify different meanings of network importance.