For a two-membered hypercycle under constant organization, the state space is the concentration simplex . The two variables and are the concentrations of the hypercycle members, and the total concentration c is held constant by a dilution flux Ξ¦.
Rate Equations and Interior Fixed Point
Under the standard rate convention, the dynamics are written as and . Here Ξ¦ is the dilution or flux term that maintains the constant total concentration c. An interior fixed point requires with , . This gives , hence .
Combining this balance with the constraint yields the unique coexistence equilibrium:
and .
The equilibrium ratio is . Under this rate convention, the larger effective catalytic constant receives the larger equilibrium share.
Boundary Fixed Points and Assumptions
The boundary fixed points are , and , . These are not coexistence states. The interior fixed point is the coexistence fixed point relevant to the two-membered hypercycle. The calculation assumes positive rate constants, mass-action kinetics, a well-mixed system, and no mutation, parasites, or time-dependent forcing.
Algebraic Reduction and Source Insight
The structure of the differential equations suggests an affine transformation of the x-coordinate system, familiar from the theory of linear differential equations. One obtains a new set of equations for transformed population variables that separates the constant-total constraint from the relative-abundance dynamics. This reduction makes the balance and the partition , transparent.
Worked Example
Normalize . Then and . If , then and . Equivalently, set and solve for its nonzero root.
Key takeaway
At the interior fixed point the two catalytic production terms balance the common dilution flux, so ; together with this gives and , with ratio .
Two-Membered Hypercycle Simplex
The concentration simplex x 1 + x is drawn as a line segment from (c,0) to (0,c). The interior fixed point x 1 * = c k 1 /(k 1 + k 2), x 2 * = c k 2 /(k 1 + k 2) lies on this segment and partitions the total concentration according to the catalytic constants; the corners are boundary states with one member absent. A one-dimensional phase line for x 1 shows flow toward the coexistence point, with ratio x 1 */x 2 * = k 1 /k 2.
QUESTION 1
For a two-membered hypercycle under constant organization, which constraint defines the state space?
β Correct!
Correct. Constant organization keeps the total concentration fixed: .
β Incorrect
Reconsider the meaning of constant total concentration: the two concentrations sum to c.
QUESTION 2
Given and , what condition must hold at an interior fixed point , ?
β Correct!
At an interior fixed point both brackets vanish, so .
β Incorrect
Set with , . The two production terms must both balance the same dilution flux Ξ¦.
QUESTION 3
Combining with gives which interior fixed point?
,
, always
,
,
β Correct!
Correct. Solving the two linear relations gives and .
β Incorrect
Use in , then solve for . The larger k gets the larger share under this convention.
QUESTION 4
What does the ratio imply under the stated rate convention?
The ratio is independent of and .
The larger effective catalytic constant corresponds to the smaller equilibrium share.
Both members always have equal equilibrium concentrations.
The larger effective catalytic constant corresponds to the larger equilibrium share.
β Correct!
Yes. The ratio scales directly with , so a larger k gives a larger equilibrium share in this convention.
β Incorrect
Read the ratio: . If increases relative to , becomes larger relative to .
QUESTION 5
The source passage suggests carrying out the subdivision by an affine transformation of the x-coordinate system into transformed population variables . What is the main purpose of this transformation in the fixed-point analysis?
It separates the constant-total constraint from the relative-abundance dynamics.
It converts the deterministic hypercycle into a stochastic model.
It forces in the transformed system.
It eliminates the dilution flux Ξ¦ completely.
β Correct!
Correct. The affine transformation reorganizes the variables so that the constant-total constraint and the relative dynamics can be analyzed separately.
β Incorrect
The transformation is an algebraic reorganization; it helps isolate the constant-total constraint from the relative-abundance dynamics, not to remove Ξ¦ or change the kinetic constants.
Case Study: Two-Membered Hypercycle with Unequal Catalytic Strengths
Read the scenario and answer the questions.
Consider a two-membered hypercycle under constant organization with total concentration . The rate equations are and . Suppose the catalytic constants are and .
Q
1. Compute the interior coexistence fixed point (, ) and the ratio .
Answer: Using and with : and . The ratio is , which equals .
Q
2. If is increased to 4.0 while remains 4.0, what happens to the interior fixed point and the coexistence ratio?
Answer: The interior fixed point moves to the midpoint: and . The ratio becomes , so the two members coexist with equal equilibrium concentrations. The boundary fixed points (c,0) and (0,c) are unchanged; only the interior coexistence point shifts.
How to carry out this new subdivision is suggested by the structure of the differential equations (6). It corresponds actually to an affine transformation of the x-coordinate system, well known from the theory of linear differential equations. One obtains a new set of equations for the transformed population variables which reads:
