The Lotka-Volterra Competition Equations

For two species, A and B, the equations are:

• Species A:

  dNAdt=rANA(KA−NA−αNBKA)dtdNA=rANA(KAKA−NA−αNB)

• Species B:

  dNBdt=rBNB(KB−NB−βNAKB)dtdNB=rBNB(KBKB−NB−βNA)

Where:

• NA,NBNA,NB: Population sizes of species A and B

• rA,rBrA,rB: Intrinsic growth rates

• KA,KBKA,KB: Carrying capacities

• $\alpha$: Effect of species B on species A (competition coefficient)

• $\beta$: Effect of species A on species B (competition coefficient)

Competitive Exclusion: When Does One Species Always Win?

The outcome of competition depends on the relative values of the carrying capacities and the competition coefficients. The zero-growth isoclines for each species define the boundaries within which populations can grow. The position of these isoclines determines whether both species coexist, or one excludes the other.

Species A always wins (and B is excluded) when:

• The isocline for species A lies entirely outside that of species B.

• This means that, for all possible population combinations, species A’s population can increase while species B’s cannot.

Mathematically, this is true when:

• K2<K1αK2<αK1

• K1>K2βK1>βK2

This set of inequalities ensures that species A has a competitive advantage, allowing it to persist while driving species B to extinction.

Matching the Correct Answer

Given the options:

1. K2>K1/βK2>K1/β and K1<K2/αK1<K2/α

2. K2>K1/αK2>K1/α and K1<K2/βK1<K2/β

3. K2<K1/βK2<K1/β and K1>K2/αK1>K2/α

4. K2<K1/αK2<K1/α and K1>K2/βK1>K2/β

The correct answer is:

(4) K2<K1/αK2<K1/α and K1>K2/βK1>K2/β

This condition directly reflects the scenario where species A always outcompetes and excludes species B, as derived from the Lotka-Volterra equations and supported by ecological theory13.

Conclusion

In the Lotka-Volterra competition model, the mathematical condition for one species (A) to always win and exclude the other species (B) is that the carrying capacity of B is less than the carrying capacity of A divided by the competition coefficient ($\alpha$), and the carrying capacity of A is greater than that of B divided by the competition coefficient ($\beta$). This ensures that species A can always increase its population even in the presence of B, leading to the exclusion of species B from the ecosystem.