Central and Corner Fixed Points of the Hypercycle

Under constant organization the total concentration of the hypercycle is held at the fixed value c₀, so every admissible state lies on the concentration simplex + … + . A state on that simplex is a fixed point precisely when the steady-state condition dxᵢ/dt = 0 holds simultaneously for every member , …, n. The simplex constraint alone only says which states are admissible — it is the vanishing of all growth rates that selects equilibria from among them.

Imposing this condition on an n-membered hypercycle produces a remarkably simple fixed-point set. The central fixed point is x̄₀ = (c₀/n)(1, …, 1)ᵀ, so every one of the n species carries the identical concentration c₀/n and the n shares sum exactly to c₀. Because all coordinates are equal, this point lies in the interior of the simplex and is permutation-symmetric: relabelling the species of the cycle leaves it unchanged. It is the equilibrium of full cooperation — no member is favoured over any other.

One Central Point and n Corner Points

Central fixed point: x̄₀ = (c₀/n)(1, …, 1)ᵀ — an equal-concentration, permutation-symmetric equilibrium in the interior of the simplex, where every member coexists.
n corner fixed points: one for each member, with that species at concentration c₀ and all remaining species at zero. Each species can, in principle, be the sole survivor.
Corners are pure states: they sit at the vertices of the simplex, so they describe a single dominating species rather than an integrated, cooperating community.

Worked Example:

For a three-membered hypercycle the concentration simplex is a triangle. The central fixed point is (c₀/3, c₀/3, c₀/3), and the three corner fixed points are (c₀, 0, 0), (0, c₀, 0) and (0, 0, c₀). The fixed-point set relevant to this step therefore contains one central equal-concentration equilibrium plus n pure-state corner equilibria — 1 + n points in total, four in this example. Note that the existence of the corners is immediate from the symmetry of the system; whether the interior point actually attracts trajectories depends on n and on the normal modes of the system, which is analysed as a separate step.

Reading the geometry

The central fixed point is the barycentre (arithmetic average) of the n vertices, which is exactly why its coordinates are all c₀/n. A frequent slip is to treat the corners as 'no equilibrium'; in fact they are genuine steady states on the boundary. The structural contrast is interior versus boundary and symmetry versus symmetry-breaking — not existence versus non-existence.

Central and Corner Fixed Points of the Hypercycle

A triangular concentration simplex for a three-membered hypercycle, with one permutation-symmetric central fixed point at (c 0 /3, c 0 /3, c 0 /3) in the interior and three pure-state corner fixed points at (c 0, 0, 0), (0, c 0, 0) and (0, 0, c 0) on the vertices. Schematic inward trajectories indicate that the centre is the equal-concentration equilibrium reached cooperatively, while each corner represents collapse onto a single species.

QUESTION 1

At the central fixed point of an n-membered hypercycle under constant organization, what concentration does each species carry?

c₀, because the total concentration is shared only at the corners

c₀/n, so that the n equal shares add up to c₀

n·c₀, because the cycle multiplies the total concentration

Zero, because no species survives at the centre

QUESTION 2

How many corner fixed points does an n-membered hypercycle possess?

Exactly one, the central equilibrium

n², one for every ordered pair of species

Exactly n — one pure state per member of the cycle

2n, counting each vertex twice

QUESTION 3

Where do the corner fixed points lie within the concentration simplex?

In the interior, where every concentration is equal

At the midpoints of the edges of the simplex

At the vertices, where one species equals c₀ and all others equal zero

Exactly at the barycentre of the simplex

QUESTION 4

Which condition must a point on the concentration simplex satisfy in order to be a fixed point of the hypercycle?

Σᵢxᵢ = c₀ is sufficient on its own

dxᵢ/dt > 0 for every species i

= … =

, which selects the interior only

dxᵢ/dt = 0 for every species i simultaneously

QUESTION 5

For

with

(arbitrary units), which triple is a corner fixed point of the hypercycle?

(0.5, 1, 1.5)

(1, 1, 1)

(1.5, 1.5, 0)

(3, 0, 0)

Case Study: Mapping the Fixed Points of a Three-Membered Hypercycle

Read the scenario below and answer the two questions.

A laboratory system implements a three-membered hypercycle (

, exponent

) under constant organization, with the total concentration clamped at

in arbitrary units. The experimenter wants to know every steady state the system can in principle occupy before asking whether the cooperative state is actually attracting. No species is externally fed, and the catalytic couplings are cyclically symmetric.

1. Enumerate all fixed points of this hypercycle on the concentration simplex and classify each as central or corner. Verify that every listed point satisfies both the steady-state condition and the total-concentration constraint.

Answer:
The fixed-point set is one central point plus

corner points, i.e. 4 steady states in total. (i) Central fixed point: x̄₀ = (c₀/n)(1, …, 1)ᵀ = (0.3, 0.3, 0.3). Its coordinates are equal, it lies in the interior of the simplex, and

, with dxᵢ/dt = 0 for all three species by symmetry. (ii) Corner fixed points (pure states): (0.9, 0, 0), (0, 0.9, 0) and (0, 0, 0.9). Each places a single species at the full concentration c₀ while the other two are zero, and each lies at a vertex of the simplex. Taking the vertex (0.9, 0, 0) as an example, the two absent species trivially have zero growth and the surviving species is also at steady state, so all three rates vanish and the sum equals

. Thus the relevant fixed-point structure is 1 central equal-concentration equilibrium + 3 pure-state corner equilibria.

2. A colleague objects: 'The corner points sit on the boundary of the simplex, so they are not genuine equilibria — only the central point counts.' Evaluate this claim and explain precisely what distinguishes the central fixed point from a corner fixed point structurally.

Answer:
The claim is incorrect. The corners are genuine fixed points: for a pure state such as (0.9, 0, 0), the growth rate of the two absent species is zero because their concentrations are zero, and the surviving species is also stationary, so dxᵢ/dt = 0 for every i. The simplex boundary is part of the state space, not outside it, so steady states there are legitimate equilibria rather than artefacts. What distinguishes the two kinds of fixed point is not existence but structure. The central point (0.3, 0.3, 0.3) is an interior equilibrium: all n species coexist at the equal concentration c₀/n, and the point is permutation-symmetric — relabelling the members of the cycle leaves it unchanged, so it represents integrated cooperative growth of the whole cycle. A corner point, by contrast, lies at a vertex of the simplex and corresponds to a pure state in which one species carries all of c₀ while the others are extinct; it breaks the permutation symmetry, singling out one member, and represents competitive exclusion / dominance by a single species rather than a joint function. In short, the difference is interior versus boundary geometry and symmetry versus symmetry-breaking, not whether an equilibrium exists at all.