Dynamics of a stoichiometric producer-grazer model with maturation delay

Ecological stoichiometry provides a multi-scale approach to study macroscopic phenomena via microscopic lens. A stoichiometric producer-grazer model with maturation delay is proposed and studied in this paper. The interaction between stoichiometry and delay is novel and leads to more interesting insights beyond classical delay-driven periodic solutions. For example, the period doubling route to chaos can occur as the minimal phosphorous:carbon ratio in producer decreases. Mathematically, we establish the conditions for the existence and stability of positive equilibria, and study the occurrence of Hopf bifurcation at positive equilibria. Analytic results show that delay can change the number and stability of positive equilibria through transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation, and it further determines the grazer's extinction. Our model with a small delay behaves like LKE model in terms of light intensity, and Rosenzweig's paradox of enrichment exists in a suitable light intensity. We plot bifurcation diagrams and show rich dynamics driven by delay, light intensity, phosphorous availability, and conversion efficiency, including that a large delay can drive the grazer to go extinct in an intermediate light intensity that is favorable for the survival of the grazer when there is no delay; a limit cycle can appear, then disappear as the delay increases; given the same initial condition, solutions with different delay values can tend to different attractors.