# American Institute of Mathematical Sciences

December  2014, 7(6): 1305-1320. doi: 10.3934/dcdss.2014.7.1305

## 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, United States, United States 2 School of Mathematics and Statistical Sciences, Arizona State University, Tempe, AZ 85281

Received  February 2013 Revised  July 2013 Published  June 2014

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.
Citation: Hao Liu, Aaron Packer, Yang Kuang. Stoichiometric producer-grazer models with varying nitrogen pools and ammonia toxicity. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1305-1320. doi: 10.3934/dcdss.2014.7.1305
##### References:
 [1] T. Anderson, Pelagic Nutrient Cycles: Herbivores as Sources and Skinks, Springer, Berlin, 1997. [2] M. Boersma and J. Elser, Too much of a good thing: On stoichiometrically balanced diets and maximal growth, Ecology, 87 (2006), 1325-1330. [3] B. Buonomo, M. Falcucci, V. Hul and S. Rionero, A mathematical model for an integrated experimental aquaculture plant, Ecological Modelling, 183 (2005), 11-28. doi: 10.1016/j.ecolmodel.2004.07.019. [4] M. A. Burford and K. Lorenzen, Modeling nitrogen dynamics in intensive shrimp ponds: the role of sediment remineralization, Aquaculture, 229 (2004), 129-145. doi: 10.1016/S0044-8486(03)00358-2. [5] J. Elser, J. Watts, J. Schampell and J. Farmer, Early Cambrian food webs on a trophic knife-edge? A hypothesis and preliminary data from a modern stromatolite-based ecosystem, Ecology Letters, 9 (2006), 295-303. doi: 10.1111/j.1461-0248.2005.00873.x. [6] J. Elser, I. Loladze, A. Peace and Y. Kuang, Lotka re-loaded: Modeling trophic interactions under stoichiometric constraints, Ecological Modelling, 245 (2012), 3-11. doi: 10.1016/j.ecolmodel.2012.02.006. [7] K. J. Flynn, Ecological modelling in a sea of variable stoichiometry: Dysfunctionality and the legacy of Redfield and Monod, Progress in Oceanography, 84 (2010), 52-65. doi: 10.1016/j.pocean.2009.09.006. [8] D. M. Jamu and R. H. Piedrahita, Nitrogen biogeochemistry of aquaculture ponds, Aquaculture, 166 (1998), 181-212. [9] D. M. Jamu and R. H. Piedrahita, An organic matter and nitrogen dynamics model for the ecological analysis of integrated aquaculture/agriculture systems: I. model development and calibration, Environmental Modelling and Software, 17 (2002), 571-582. doi: 10.1016/S1364-8152(02)00016-6. [10] D. M. Jamu and R. H. Piedrahita, An organic matter and nitrogen dynamics model for the ecological analysis of integrated aquaculture/agriculture systems: II. Model evaluation and application, Environmental Modelling and Software, 17 (2002), 583-592. doi: 10.1016/S1364-8152(02)00017-8. [11] Y. Kuang, J. Huisman and J. J. Elser, Stoichiometric plant-herbivore models and their interpretation, Mathematical Biosciences and Engineering, 1 (2004), 215-222. doi: 10.3934/mbe.2004.1.215. [12] G. Lemarie, A. Dosdat, D. Coves, G. Dutto, E. Gasset and J. Person-Le Ruyet, Effect of chronic ammonia exposure on growth of European seabass (Dicentrarchus labrax) juveniles, Aquaculture, 229 (2004), 479-491. doi: 10.1016/S0044-8486(03)00392-2. [13] X. Li, H. Wang and Y. Kuang, Global analysis of a stoichiometric producer-grazer model with Holling type functional responses, Journal of Mathematical Biology, 63 (2011), 901-932. doi: 10.1007/s00285-010-0392-2. [14] I. Loladze, Y. Kuang and J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bull. Math Bio., 62 (2000), 1137-1162. doi: 10.1006/bulm.2000.0201. [15] K. Lorenzen, J. Struve and V. J. Cowan, Impact of farming intensity and water management on nitrogen dynamics in intensive pond culture: A mathematical model applied to Thai commercial shrimp farms, Aquaculture Research, 28 (1997), 493-507. doi: 10.1046/j.1365-2109.1997.00875.x. [16] L. Markus, Asymptotically autonomous differential systems, Contributions to the Theory of Nonlinear Oscillations, Annals of Mathematics Studies, 3 (1956), 17-29. [17] A. Packer, Y. Li, T. Andersen, Q. Hu, Y. Kuang and M. Sommerfeld, Growth and neutral lipid synthesis in green microalgae: A mathematical model, Bioresource technology, 102 (2011), 111-117. doi: 10.1016/j.biortech.2010.06.029. [18] A. Peace, Y. Zhao, I. Loladze, J. Elser and Y. Kuang, A stoichiometric producer-grazer model incorporating the effects of excess food-nutrient content on cunsumer dynamics, Mathematical Biosciences, 244 (2013), 107-115. doi: 10.1016/j.mbs.2013.04.011. [19] R. V. Rodrigues, M. H. Schwarz, B. C. Delbos and L. A. Sampaio, Acute toxicity and sublethal effects of ammonia and nitrite for juvenile cobia Rachycentron canadum, Aquaculture, 271 (2007), 553-557. doi: 10.1016/j.aquaculture.2007.06.009. [20] R. W. Sterner and J. J. Elser, Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere, Princeton University Press, 2002. [21] H. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992), 755-763. doi: 10.1007/BF00173267. [22] J. R. Tomasso, Toxicity of nitrogenous wastes to aquaculture animals. Reviews in Fisheries Science, Reviews in Fisheries Science, 2 (1994), 291-314. [23] S.-J. Tsai and J.-C. Chen, Acute toxicity of nitrate on Penaeus monodon juveniles at different salinity levels, Aquaculture, 213 (2002), 163-170. doi: 10.1016/S0044-8486(02)00023-6. [24] J. Urabe and R. Sterner, Regulation of herbivore growth by the balance of light and nutrients, Proc. Natl. Acad. Sci. USA, 93 (1996), 8465-8469. doi: 10.1073/pnas.93.16.8465. [25] H. Wang, Y. Kuang and I. Loladze, Dynamics of a mechanistically derived stoichiometric producer-grazer model, Jounal of Biological Dynamics, 2 (2008), 286-296. doi: 10.1080/17513750701769881.

show all references

##### References:
 [1] T. Anderson, Pelagic Nutrient Cycles: Herbivores as Sources and Skinks, Springer, Berlin, 1997. [2] M. Boersma and J. Elser, Too much of a good thing: On stoichiometrically balanced diets and maximal growth, Ecology, 87 (2006), 1325-1330. [3] B. Buonomo, M. Falcucci, V. Hul and S. Rionero, A mathematical model for an integrated experimental aquaculture plant, Ecological Modelling, 183 (2005), 11-28. doi: 10.1016/j.ecolmodel.2004.07.019. [4] M. A. Burford and K. Lorenzen, Modeling nitrogen dynamics in intensive shrimp ponds: the role of sediment remineralization, Aquaculture, 229 (2004), 129-145. doi: 10.1016/S0044-8486(03)00358-2. [5] J. Elser, J. Watts, J. Schampell and J. Farmer, Early Cambrian food webs on a trophic knife-edge? A hypothesis and preliminary data from a modern stromatolite-based ecosystem, Ecology Letters, 9 (2006), 295-303. doi: 10.1111/j.1461-0248.2005.00873.x. [6] J. Elser, I. Loladze, A. Peace and Y. Kuang, Lotka re-loaded: Modeling trophic interactions under stoichiometric constraints, Ecological Modelling, 245 (2012), 3-11. doi: 10.1016/j.ecolmodel.2012.02.006. [7] K. J. Flynn, Ecological modelling in a sea of variable stoichiometry: Dysfunctionality and the legacy of Redfield and Monod, Progress in Oceanography, 84 (2010), 52-65. doi: 10.1016/j.pocean.2009.09.006. [8] D. M. Jamu and R. H. Piedrahita, Nitrogen biogeochemistry of aquaculture ponds, Aquaculture, 166 (1998), 181-212. [9] D. M. Jamu and R. H. Piedrahita, An organic matter and nitrogen dynamics model for the ecological analysis of integrated aquaculture/agriculture systems: I. model development and calibration, Environmental Modelling and Software, 17 (2002), 571-582. doi: 10.1016/S1364-8152(02)00016-6. [10] D. M. Jamu and R. H. Piedrahita, An organic matter and nitrogen dynamics model for the ecological analysis of integrated aquaculture/agriculture systems: II. Model evaluation and application, Environmental Modelling and Software, 17 (2002), 583-592. doi: 10.1016/S1364-8152(02)00017-8. [11] Y. Kuang, J. Huisman and J. J. Elser, Stoichiometric plant-herbivore models and their interpretation, Mathematical Biosciences and Engineering, 1 (2004), 215-222. doi: 10.3934/mbe.2004.1.215. [12] G. Lemarie, A. Dosdat, D. Coves, G. Dutto, E. Gasset and J. Person-Le Ruyet, Effect of chronic ammonia exposure on growth of European seabass (Dicentrarchus labrax) juveniles, Aquaculture, 229 (2004), 479-491. doi: 10.1016/S0044-8486(03)00392-2. [13] X. Li, H. Wang and Y. Kuang, Global analysis of a stoichiometric producer-grazer model with Holling type functional responses, Journal of Mathematical Biology, 63 (2011), 901-932. doi: 10.1007/s00285-010-0392-2. [14] I. Loladze, Y. Kuang and J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bull. Math Bio., 62 (2000), 1137-1162. doi: 10.1006/bulm.2000.0201. [15] K. Lorenzen, J. Struve and V. J. Cowan, Impact of farming intensity and water management on nitrogen dynamics in intensive pond culture: A mathematical model applied to Thai commercial shrimp farms, Aquaculture Research, 28 (1997), 493-507. doi: 10.1046/j.1365-2109.1997.00875.x. [16] L. Markus, Asymptotically autonomous differential systems, Contributions to the Theory of Nonlinear Oscillations, Annals of Mathematics Studies, 3 (1956), 17-29. [17] A. Packer, Y. Li, T. Andersen, Q. Hu, Y. Kuang and M. Sommerfeld, Growth and neutral lipid synthesis in green microalgae: A mathematical model, Bioresource technology, 102 (2011), 111-117. doi: 10.1016/j.biortech.2010.06.029. [18] A. Peace, Y. Zhao, I. Loladze, J. Elser and Y. Kuang, A stoichiometric producer-grazer model incorporating the effects of excess food-nutrient content on cunsumer dynamics, Mathematical Biosciences, 244 (2013), 107-115. doi: 10.1016/j.mbs.2013.04.011. [19] R. V. Rodrigues, M. H. Schwarz, B. C. Delbos and L. A. Sampaio, Acute toxicity and sublethal effects of ammonia and nitrite for juvenile cobia Rachycentron canadum, Aquaculture, 271 (2007), 553-557. doi: 10.1016/j.aquaculture.2007.06.009. [20] R. W. Sterner and J. J. Elser, Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere, Princeton University Press, 2002. [21] H. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992), 755-763. doi: 10.1007/BF00173267. [22] J. R. Tomasso, Toxicity of nitrogenous wastes to aquaculture animals. Reviews in Fisheries Science, Reviews in Fisheries Science, 2 (1994), 291-314. [23] S.-J. Tsai and J.-C. Chen, Acute toxicity of nitrate on Penaeus monodon juveniles at different salinity levels, Aquaculture, 213 (2002), 163-170. doi: 10.1016/S0044-8486(02)00023-6. [24] J. Urabe and R. Sterner, Regulation of herbivore growth by the balance of light and nutrients, Proc. Natl. Acad. Sci. USA, 93 (1996), 8465-8469. doi: 10.1073/pnas.93.16.8465. [25] H. Wang, Y. Kuang and I. Loladze, Dynamics of a mechanistically derived stoichiometric producer-grazer model, Jounal of Biological Dynamics, 2 (2008), 286-296. doi: 10.1080/17513750701769881.
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