Lotka-Volterra Dynamics

The most obvious thing about equation (1) is its fixed point

(2)

where . For this point to be biologically meaningful, all components of must be positive, giving rise to the following inequalities:

(3)

The stability of this point is related to the negative definiteness of derivative of at . The components of the derivative are given by

(4)

Substituting eq (2) gives

(5)

Stability of the fixed point requires that this matrix should be negative definite. Since the are all negative by virtue of (3), each minor determinant of this matrix is equal to a minor determinant of multiplied by a positive number, stability of the equilibrium is equivalent to being negative definite.

A weaker condition is to require that the system remain bounded with time:

(6)

As becomes large in any direction, this functional is dominated by the quadratic term, so this implies that . Negative definiteness of is sufficient, but not necessary for this condition. For example, the predator-prey relations (heavily normalised) have the following matrix as : which has eigenvalues . If we let , then , which is clearly non-positive for all .

Consider adding a new row and column to . What is condition is the new row and column required to satisfy such that equation (6) is satisfied. Break up in the following way:

Condition (6) becomes:

(7)

Let

Then a sufficient but not necessary condition for condition (7) is

The maximum value with respect to is , so this requires that

(8)