Food web

A food web records resource-consumer interactions among species. A linear food chain is a special case. Positive interaction coefficients alone do not imply persistence: the per-capita growth balances must admit feasible positive long-term mean abundances.

Two consumers with different break-even requirements cannot both have bounded persistent abundance on a single well-mixed limiting resource when their per-capita growth is linear in that resource. For $\dot{Y}/Y=cX-d$ and $\dot{Z}/Z=eX-f$, differentiation of $\log Y/c-\log Z/e$ gives the nonzero constant $-d/c+f/e$ when their break-even levels differ. This excludes simultaneous bounded persistence without assuming convergence to an equilibrium. Additional resources, environmental variation or different interactions can invalidate the simple one-resource argument.

A food chain orders species so that each consumer feeds on the level below and may be eaten by the level above. Basal resource growth, consumer mortality, conversion efficiency and density regulation determine whether all levels can persist.

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, $X_1^{b_2}{X}_3^{a_1}=C\exp [(b_{2}r-a_1{d}_3)t]$. A positive even-chain equilibrium gives a conserved coercive relative-entropy-type function, proving every species remains bounded above and away from zero.