Stoichiometric producer-grazer models with varying nitrogen pools and ammonia toxicity

Hao Liu; Aaron Packer; Yang Kuang

doi:10.3934/dcdss.2014.7.1305

Article Contents

2014, Volume 7, Issue 6: 1305-1320. Doi: 10.3934/dcdss.2014.7.1305

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Stoichiometric producer-grazer models with varying nitrogen pools and ammonia toxicity

1.
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804
2.
School of Mathematics and Statistical Sciences, Arizona State University, Tempe, AZ 85281

Received: January 31, 2013

Revised: June 30, 2013

Published: November 2014

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Abstract

We formulate and analyze a stoichiometric model of producer-grazer systems with excess nutrient recycling (waste) that may inhibit grazer survival and growth. Specifically, we model the intoxication dynamics caused by accumulation of grazer waste and dead biomass decay. This system has a range of applications, but we focus on those in which the producers are microalgae and the limiting nutrient is nitrogen. High levels of ammonia (and to a lesser extent nitrite) have been observed to increase grazer death, especially in aquaculture systems. We assume that all nitrification is due to nitrogen uptake and assimilation by the producer; therefore, the model explores systems in which the producer serves the dual role of grazer food and water treatment. The model exhibits three equilibria corresponding to total extinction, grazer-only extinction, and coexistence. While a sufficient condition is found under which grazer extinction equilibrium is globally stable, we propose a conjecture for neccessary and sufficient conditions, which remains an open mathematical problem. Local stability of grazer extinction equilibrium is ensured under a sharp necessary and sufficient condition. Local stability for the coexistence equilibrium is studied algebraically and numerically. Bifurcation diagrams with respect to total nitrogen and its implications are also presented.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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