Stoichiometric food chain model on discrete time scale

Ming Chen; Meng Fan; Congbo Xie; Angela Peace; Hao Wang; Ming Chen; Meng Fan; Congbo Xie; Angela Peace; Hao Wang

doi:10.3934/mbe.2019005

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 1: 101-118. doi: 10.3934/mbe.2019005

Previous Article Next Article

Research article Special Issues

Stoichiometric food chain model on discrete time scale

1.
School of Science, Dalian Maritime University, 1 Linghai Road, Dalian, Liaoning, 116026, P. R. China
2.
Center for Mathematical Biosciences, School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin, 130024, P. R. China
3.
College of Science, Dalian Minzu University, 18 Liaohe West Road, Dalian, Liaoning, 116600, P R. China
4.
Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409, United States
5.
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Received: 28 June 2018 Accepted: 09 September 2018 Published: 11 December 2018

Stoichiometry-based models can yield many new insights into producer - grazer systems. Many interesting results can be obtained from models continuous in time. There raises the question of how robust the model predictions are to time discretization. A discrete stoichiometric food-chain model is analyzed and compared with a corresponding continuous model. Theoretical and numerical results show that the discrete and continuous models have many properties in common but differences also exist. Stoichiometric impacts of producer nutritional quality also persist in the discrete system. Both types of models can exhibit qualitatively different behaviors with the same parameter sets. Discretization enlarges the parameter ranges for the existence of chaotic dynamics. Our results suggest that the stoichiometric mechanisms are robust to time discretization and the nutritional quality of the producer can have dramatic and counterintuitive impacts on population dynamics, which agrees with the existing analysis of pelagic systems.
- stoichiometry,
- predator-prey system,
- food quality,
- C:P ratio,
- discrete model
Citation: Ming Chen, Meng Fan, Congbo Xie, Angela Peace, Hao Wang. Stoichiometric food chain model on discrete time scale[J]. Mathematical Biosciences and Engineering, 2019, 16(1): 101-118. doi: 10.3934/mbe.2019005

Related Papers:

Abstract

Stoichiometry-based models can yield many new insights into producer - grazer systems. Many interesting results can be obtained from models continuous in time. There raises the question of how robust the model predictions are to time discretization. A discrete stoichiometric food-chain model is analyzed and compared with a corresponding continuous model. Theoretical and numerical results show that the discrete and continuous models have many properties in common but differences also exist. Stoichiometric impacts of producer nutritional quality also persist in the discrete system. Both types of models can exhibit qualitatively different behaviors with the same parameter sets. Discretization enlarges the parameter ranges for the existence of chaotic dynamics. Our results suggest that the stoichiometric mechanisms are robust to time discretization and the nutritional quality of the producer can have dramatic and counterintuitive impacts on population dynamics, which agrees with the existing analysis of pelagic systems.

References

[1]	G. Ahlgren, Temperature functions in biology and their application to algal growth constants, Oikos, 49 (1987), 177–190.
[2]	T. Andersen, Pelagic nutrient cycles: herbivores as sources and sinks, Berlin: Springer, 1997.
[3]	M. Chen, M. Fan and Y. Kuang, Global dynamics in a stoichiometric food chain model with two limiting nutrients, Math. Biosci., 289 (2017), 9–19.
[4]	K. L. Edelstein, Mathematical models in biology, Siam, 1988.
[5]	M. Fan, I. Loladze, Y. Kuang and J.E.James, Dynamics of a stoichiometric discrete producergrazer model, J. Differ. Equ. Appl., 11 (2005), 347–364.
[6]	R. Frankham and B.W. Brook, The importance of time scale in conservation biology and ecology, Ann. Zool. Fenn., 41 (2004), 459–463.
[7]	Y. Kuang, J. Huisman and J. J. Elser, Stoichiometric plant-herbivore models and their interpretation, Math. Biosci. Eng., 1 (2004), 215–222.
[8]	I. Loladze, Y. Kuang and J. J. Elser, Stoichiometry in producer-grazer systems: linking energy flow with element cycling, B. Math. Biol., 62 (2000), 1137–1162.
[9]	I. Loladze, Y. Kuang, J. J. Elser and W. F. Fagan, Competition and stoichiometry: coexistence of two predators on one prey, Theor. Popul. Biol., 65 (2004), 1–15.
[10]	A. Peace, Stoichiometric Producer-Grazer Models Incorporating the Effects of Excess Food- Nutrient Content on Grazer Dynamics, Arizona State University, 2014.
[11]	A. Peace, Effects of light, nutrients, and food chain length on trophic efficiencies in simple stoichiometric aquatic food chain models, Ecol. Model., 312 (2015), 125–135.
[12]	A. Peace, H.Wang and Y. Kuang, Dynamics of a Producer-Grazer Model Incorporating the Effects of Excess Food Nutrient Content on Grazer's Growth, B. Math. Biol., 76 (2014), 2175–2197.
[13]	R. W. Sterner and J. J. Elser, Ecological stoichiometry: the biology of elements from molecules to the biosphere, Princeton University Press, 2002.
[14]	G. Sui, M. Fan, Loladze I, and Y. Kuang, The dynamics of a stoichiometric plant-herbivore model and its discrete analog, Math. Biosci. Eng., 4 (2007), 1-18.
[15]	J. Urabe and R. W. Sterner, Regulation of herbivore growth by the balance of light and nutrients, Pro. Nat. Acad. Sci., 93 (1996), 8465–8469.
[16]	H. Wang, Y. Kuang and I. Loladze, Dynamics of a mechanistically derived stoichiometric producer-grazer model, J. Biol. Dynam., 2 (2008), 286–296.
[17]	C. Xie, M. Fan andW. Zhao, Dynamics of a discrete stoichiometric two predators one prey model, J. Biol. Syst., 18 (2010), 649–667.

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)