Yield and dynamics of tritrophic food chains

Strong relationships between yield and dynamic behavior of tritrophic food chains are pointed out by analyzing the classical Rosenzweig-MacArthur model. On the one hand, food chains are subdivided into undersupplied and oversupplied categories, the first being those in which a marginal increase of nutrient supply to the bottom produces a marginal increase of mean yield al the top. On the other hand, a detailed bifurcation analysis proves that dynamic complexity first increases with nutrient supply (from stationary to a low-frequency cyclic regime and, finally, to chaos) and then decreases (from chaos to a high-frequency cyclic regime). A careful comparison of the two analyses supports the conclusion that food chains cycling at high frequency are oversupplied, while all others are undersupplied. A straightforward consequence of this result is that maximization of food yield requires a chaotic regime. This regime turns out to be very often on the edge of a potential catastrophic collapse of the top component of the food chain. In other words, optimality implies very complex and dangerous dynamics, as intuitively understood long ago for ditrophic food chains by Rosenzweig in his famous article on the paradox of enrichment.