Lotka–Volterra communities
The abundances of species obey
where is the per-capita growth rate of species , its intrinsic growth rate, and the per-capita effect of species on . Although the dynamics are nonlinear, an equilibrium at which all species coexist solves the linear equations , so that with .
Press
A press on species is a sustained change of its intrinsic growth rate. It moves the equilibrium exactly, for any , to
where , the th column of , is the press response of : it sums all direct and indirect effects of on the other species.
Effective press
Now let the equation of species change in any way, : its growth rate, its self-regulation, its responses to other species, in any combination and by any amount. The other equations are untouched, so at equilibrium they still read , whereas species obeys with
These are the equilibrium conditions of a press of size , hence
The other species respond only to the number , the effective press. Whatever is done to species , the community moves along its press line.
Mutual linearity
Eliminating from two components gives, for any species and ,
The coefficient is a ratio of press responses, the same for every change of and every size of it. For a change of one interaction, , the effective press is . Every abundance is then a ratio of two affine functions of , with the same denominator for all species.
Why a line
Each unmodified species confines the equilibrium to its zero-growth hyperplane . These hyperplanes do not depend on the change, so their intersection is fixed. It is the press line, and species selects a point on it.
Extinctions and invasions
A large change drives species extinct or lets absent ones invade. An absent species has and cannot invade, . Written as with , this is a press of its own that holds at zero, like the reaction force that holds a particle at a wall. Like that force, it acts only at zero abundance, so .
Mutual linearity therefore holds piecewise: the path is straight while the composition stays the same, and kinks where it changes. When species is lost, the direction changes to
If is a P-matrix (all principal minors positive), every change of moves the community along one piecewise-straight press path.
Cycles and chaos
On limit cycles and chaotic attractors the per-capita growth rates of the unmodified species still vanish on average, because is affine and equals :
Time averages therefore obey the same relations, with replaced by the time-averaged effective press , whether the attractor is periodic or chaotic.
Beyond one species
Changing the equations of a set of species confines the response to the affine subspace spanned by the press responses of the species in , with one dimension per changed equation. Nonlinear functional responses of the other species bend the line into a curve, which all changes of still share. The same straight lines appear in MacArthur's consumer–resource model, whose per-capita growth rates are linear in the state.
Two measurements
As long as the composition stays the same, the response of one species to any manipulation of species predicts the responses of all others. Because two points fix a line, two equilibrium measurements determine the mutual-linearity relations without inferring the interaction matrix.