= Lotka-Volterra food-chain parity
{c}
{title2=$N\text{ even: invertible interaction matrix; }N\text{ odd: compatibility condition}$}
For an unregulated exponential basal resource and nearest-neighbor <Lotka-Volterra equations>, the weighted interaction matrix is <skew-symmetric matrix>. An even chain has an invertible tridiagonal interaction matrix; its positive <equilibrium>, when feasible, fixes all long-term <means>. An odd chain has a null direction and requires a special balance between resource growth and mortality to permit bounded full coexistence. For a three-level chain, $X_1^{b_2}X_3^{a_1}=C\exp[(b_2r-a_1d_3)t]$. A positive even-chain <equilibrium> gives a conserved coercive relative-entropy-type function, proving every species remains bounded above and away from zero.
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