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 & 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, (1997). Google Scholar

[2]

M. Boersma and J. Elser, Too much of a good thing: On stoichiometrically balanced diets and maximal growth,, Ecology, 87 (2006), 1325. Google Scholar

[3]

B. Buonomo, M. Falcucci, V. Hul and S. Rionero, A mathematical model for an integrated experimental aquaculture plant,, Ecological Modelling, 183 (2005), 11. doi: 10.1016/j.ecolmodel.2004.07.019. Google Scholar

[4]

M. A. Burford and K. Lorenzen, Modeling nitrogen dynamics in intensive shrimp ponds: the role of sediment remineralization,, Aquaculture, 229 (2004), 129. doi: 10.1016/S0044-8486(03)00358-2. Google Scholar

[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. doi: 10.1111/j.1461-0248.2005.00873.x. Google Scholar

[6]

J. Elser, I. Loladze, A. Peace and Y. Kuang, Lotka re-loaded: Modeling trophic interactions under stoichiometric constraints,, Ecological Modelling, 245 (2012), 3. doi: 10.1016/j.ecolmodel.2012.02.006. Google Scholar

[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. doi: 10.1016/j.pocean.2009.09.006. Google Scholar

[8]

D. M. Jamu and R. H. Piedrahita, Nitrogen biogeochemistry of aquaculture ponds,, Aquaculture, 166 (1998), 181. Google Scholar

[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. doi: 10.1016/S1364-8152(02)00016-6. Google Scholar

[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. doi: 10.1016/S1364-8152(02)00017-8. Google Scholar

[11]

Y. Kuang, J. Huisman and J. J. Elser, Stoichiometric plant-herbivore models and their interpretation,, Mathematical Biosciences and Engineering, 1 (2004), 215. doi: 10.3934/mbe.2004.1.215. Google Scholar

[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. doi: 10.1016/S0044-8486(03)00392-2. Google Scholar

[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. doi: 10.1007/s00285-010-0392-2. Google Scholar

[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. doi: 10.1006/bulm.2000.0201. Google Scholar

[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. doi: 10.1046/j.1365-2109.1997.00875.x. Google Scholar

[16]

L. Markus, Asymptotically autonomous differential systems,, Contributions to the Theory of Nonlinear Oscillations, 3 (1956), 17. Google Scholar

[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. doi: 10.1016/j.biortech.2010.06.029. Google Scholar

[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. doi: 10.1016/j.mbs.2013.04.011. Google Scholar

[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. doi: 10.1016/j.aquaculture.2007.06.009. Google Scholar

[20]

R. W. Sterner and J. J. Elser, Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere,, Princeton University Press, (2002). Google Scholar

[21]

H. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations,, Journal of Mathematical Biology, 30 (1992), 755. doi: 10.1007/BF00173267. Google Scholar

[22]

J. R. Tomasso, Toxicity of nitrogenous wastes to aquaculture animals. Reviews in Fisheries Science,, Reviews in Fisheries Science, 2 (1994), 291. Google Scholar

[23]

S.-J. Tsai and J.-C. Chen, Acute toxicity of nitrate on Penaeus monodon juveniles at different salinity levels,, Aquaculture, 213 (2002), 163. doi: 10.1016/S0044-8486(02)00023-6. Google Scholar

[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. doi: 10.1073/pnas.93.16.8465. Google Scholar

[25]

H. Wang, Y. Kuang and I. Loladze, Dynamics of a mechanistically derived stoichiometric producer-grazer model,, Jounal of Biological Dynamics, 2 (2008), 286. doi: 10.1080/17513750701769881. Google Scholar

show all references

References:
[1]

T. Anderson, Pelagic Nutrient Cycles: Herbivores as Sources and Skinks,, Springer, (1997). Google Scholar

[2]

M. Boersma and J. Elser, Too much of a good thing: On stoichiometrically balanced diets and maximal growth,, Ecology, 87 (2006), 1325. Google Scholar

[3]

B. Buonomo, M. Falcucci, V. Hul and S. Rionero, A mathematical model for an integrated experimental aquaculture plant,, Ecological Modelling, 183 (2005), 11. doi: 10.1016/j.ecolmodel.2004.07.019. Google Scholar

[4]

M. A. Burford and K. Lorenzen, Modeling nitrogen dynamics in intensive shrimp ponds: the role of sediment remineralization,, Aquaculture, 229 (2004), 129. doi: 10.1016/S0044-8486(03)00358-2. Google Scholar

[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. doi: 10.1111/j.1461-0248.2005.00873.x. Google Scholar

[6]

J. Elser, I. Loladze, A. Peace and Y. Kuang, Lotka re-loaded: Modeling trophic interactions under stoichiometric constraints,, Ecological Modelling, 245 (2012), 3. doi: 10.1016/j.ecolmodel.2012.02.006. Google Scholar

[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. doi: 10.1016/j.pocean.2009.09.006. Google Scholar

[8]

D. M. Jamu and R. H. Piedrahita, Nitrogen biogeochemistry of aquaculture ponds,, Aquaculture, 166 (1998), 181. Google Scholar

[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. doi: 10.1016/S1364-8152(02)00016-6. Google Scholar

[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. doi: 10.1016/S1364-8152(02)00017-8. Google Scholar

[11]

Y. Kuang, J. Huisman and J. J. Elser, Stoichiometric plant-herbivore models and their interpretation,, Mathematical Biosciences and Engineering, 1 (2004), 215. doi: 10.3934/mbe.2004.1.215. Google Scholar

[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. doi: 10.1016/S0044-8486(03)00392-2. Google Scholar

[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. doi: 10.1007/s00285-010-0392-2. Google Scholar

[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. doi: 10.1006/bulm.2000.0201. Google Scholar

[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. doi: 10.1046/j.1365-2109.1997.00875.x. Google Scholar

[16]

L. Markus, Asymptotically autonomous differential systems,, Contributions to the Theory of Nonlinear Oscillations, 3 (1956), 17. Google Scholar

[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. doi: 10.1016/j.biortech.2010.06.029. Google Scholar

[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. doi: 10.1016/j.mbs.2013.04.011. Google Scholar

[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. doi: 10.1016/j.aquaculture.2007.06.009. Google Scholar

[20]

R. W. Sterner and J. J. Elser, Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere,, Princeton University Press, (2002). Google Scholar

[21]

H. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations,, Journal of Mathematical Biology, 30 (1992), 755. doi: 10.1007/BF00173267. Google Scholar

[22]

J. R. Tomasso, Toxicity of nitrogenous wastes to aquaculture animals. Reviews in Fisheries Science,, Reviews in Fisheries Science, 2 (1994), 291. Google Scholar

[23]

S.-J. Tsai and J.-C. Chen, Acute toxicity of nitrate on Penaeus monodon juveniles at different salinity levels,, Aquaculture, 213 (2002), 163. doi: 10.1016/S0044-8486(02)00023-6. Google Scholar

[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. doi: 10.1073/pnas.93.16.8465. Google Scholar

[25]

H. Wang, Y. Kuang and I. Loladze, Dynamics of a mechanistically derived stoichiometric producer-grazer model,, Jounal of Biological Dynamics, 2 (2008), 286. doi: 10.1080/17513750701769881. Google Scholar

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