Check: Growth Degree Classification

Growth degree classification identifies the leading power p of the population-dependent growth term in the selection equations. Under constant organization, p determines whether selection is neutral, linear, or quadratic, and thus the qualitative outcome for species or hypercycle stability.

Reading the leading power

: no selective advantage between species; growth is independent of population, so the growth term does not discriminate variants.
: linear selection; the species with the highest rate constant is favored.
: quadratic selection; hypercyclic coupling can stabilize a hypercycle against parasites because growth depends on products of concentrations.

Interpretive check

The check requires reading the leading power of the growth function before interpreting selection behavior. The degree classifies the regime, while rate constants set the quantitative outcome. For example, a growth term constant in x gives ; a term proportional to gives ; a term proportional to x_ix_i-1 gives . These map to neutral, linear, and hypercyclic/quadratic selection outcomes.

Quick check

Before predicting winner or coexistence, identify p:

means no selection pressure,

means selection of the fittest, and

means hypercyclic stabilization can protect against parasitic replicates.

Growth-Degree Decision Tree

The leading power p of the population-dependent growth term classifies selection regimes: gives population-independent neutral growth, gives linear selection favoring the largest rate constant, and gives quadratic hypercyclic coupling that can stabilize a cycle against parasites.

QUESTION 1

What does the growth degree p classify in the selection equations?

The initial concentration of each species

The leading power of the population-dependent growth term

The numerical value of the rate constants

The dilution flux under constant organization

QUESTION 2

Under constant organization, what selection behavior corresponds to

No selective advantage between species; growth is population-independent.

Linear selection favors the largest rate constant.

Quadratic selection stabilizes hypercycles.

The fastest decomposing species always wins.

QUESTION 3

For

, which species is favored by linear selection?

A parasite species that couples to the hypercycle

The species with the lowest concentration

The species with the highest rate constant for growth

The species with the highest decomposition rate

QUESTION 4

Why can

quadratic selection stabilize a hypercycle against parasites?

Because growth depends on products of concentrations, coupling members in a cycle

Because growth is independent of population

Because the dilution flux vanishes for all species

Because growth depends linearly on each concentration

QUESTION 5

A growth term is proportional to

x_{i-1}. What p and regime does this indicate?

, linear selection

, neutral selection

, quadratic or hypercyclic selection

, cubic selection

Case Study: Classifying a Coupled Replicator System

Read the scenario and answer the questions.

A flow reactor maintains constant organization for a set of replicators. One mutant's growth contribution is proportional to its own concentration

. A coupled pair of replicators grows proportionally to

x_{i-1}. You must classify the selection regime and predict the qualitative outcome.

1. Identify the growth degree p for the term proportional to

and explain the expected selection behavior.

Answer:
The term proportional to

has

, so selection is linear. The species with the highest rate constant has the greatest relative growth rate and is favored. This is selection of the fittest among independently growing replicators.

2. Identify p for the term

x_{i-1} and explain how closing the coupling into a loop changes stability against parasites.

Answer:
The term

x_{i-1} has

, so selection is quadratic. Because growth depends on products of concentrations, closing the chain into a loop creates a hypercycle in which each member depends on the previous one. This coupling can stabilize the cycle against parasites: a parasite that does not participate in the full loop cannot exploit the same product-dependent growth advantage.