×

Stability and bifurcation in a stoichiometric producer-grazer model with knife edge. (English) Zbl 1356.34057

Summary: All organisms are composed of multiple chemical elements such as nitrogen (N), phosphorus (P), and carbon (C). P is essential to build nucleic acids (DNA and RNA) and N is needed for protein production. To keep track of the mismatch between the P requirement in the consumer (grazer) and the P content in the provider (producer), stoichiometric models have been constructed to explicitly incorporate food quality and quantity. In addition to their fundamental applications in ecology and biology, stoichiometric models are especially suitable for medical applications where stoichiometrically distinct pathogens or cancer cells are competing with normal cells and suffer a higher death rate due to excessive chemotherapy agent or radiation uptake. Most stoichiometric models have suggested that the consumer dynamics heavily depends on the P content in the provider when the provider has low nutrient content (low P:C ratio). Motivated by recent lab experiments, researchers explored the effect of excess producer nutrient content (extremely high P:C ratio) on the consumer dynamics. This phenomenon is called the stoichiometric knife edge and its rich dynamics is yet to be appreciated due to the fact that a global analysis of a knife-edge model is challenging. The main challenge stems from the phase plane fragmentation and parameter space partitioning in order to carry out a detailed and complete case by case analysis of the model dynamics. The aim of this paper is to present a sample of a complete mathematical analysis of the dynamics of this model and to perform a bifurcation analysis for the model with Holling type-II functional response.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
92D40 Ecology
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] T. Andersen, J. J. Elser, and D. O. Hessen, {\it Stoichiometry and population dynamics}, Ecol. Lett., 7 (2004), pp. 884-900.
[2] M. R. Droop, {\it Vitamin B12 and marine ecology, IV: The kinetics of uptake, growth and inhibition in {\it Monochrysis lutheri}}, J. Mar. Biol. Assoc., 48 (1968), pp. 689-733.
[3] J. Elser, I. Loladze, A. Peace, and Y. Kuang, {\it Lotka re-loaded: Modeling trophic interactions under stoichiometric constraints}, Ecol. Model., 245 (2012), pp. 3-11.
[4] J. J. Elser, J. D. Nagy, and Y. Kuang, {\it Biological stoichiometry: An ecological perspective on tumor dynamics}, BioScience, 53 (2003), pp. 1112-1120.
[5] J. J. Elser, J. Watts, J. Schampell, and J. Farmer, {\it Early Cambrian food webs on a trophic knife-edge? A hypothesis and preliminary data from a modern stromatolite-based ecosystem}, Ecol. Lett., 9 (2006), pp. 295-303.
[6] R. A. Everett, J. D. Nagy, and Y. Kuang, {\it Dynamics of a data based ovarian cancer growth and treatment model with time delay}, J. Dynam. Differential Equations, 28 (2016), pp. 1393-1414, doi:10.1007/s10884-015-9498-y. · Zbl 1350.92025
[7] R. A. Everett, A. Packer, and Y. Kuang, {\it Can Mathematical models predict the outcomes of prostate cancer patients undergoing intermittent androgen deprivation therapy?}, Biophys. Rev. Lett., 9 (2014), pp. 173-191.
[8] R. A. Everett, Y. Zhao, K. B. Flores, and Y. Kuang, {\it Data and implication based comparison of two chronic myeloid leukemia models}, Math. Biosci. Eng., 10 (2013), pp. 1501-1518. · Zbl 1273.92026
[9] J. Grover, {\it Stoichiometry, herbivory and competition for nutrients: Simple models based on planktonic ecosystems}, J. Theoret. Biol., 214 (2002), pp. 599-618.
[10] J. Grover, {\it Predation, competition, and nutrient recycling: A stoichiometric approach with multiple nutrients}, J. Theoret. Biol., 229 (2004), pp. 31-43. · Zbl 1440.92071
[11] S. Hall, {\it Stoichiometrically explicit food webs: Feedbacks between resource supply, elemental constraints, and species diversity}, Annu. Rev. Ecol. Evol. Syst., 40 (2009), pp. 503-528.
[12] E. J. Kostelich, Y. Kuang, J. M. McDaniel, N. Z. Moore, N. L. Martirosyan, and M. C. Preul, {\it Accurate state estimation from uncertain data and models: An application of data assimilation to mathematical models of human brain tumors}, Biol. Direct, 6 (2011), 64, doi:10.1186/1745-6150-6-64.
[13] Y. Kuang, J. D. Nagy, and S. E. Eikenberry, {\it Introduction to Mathematical Oncology}, CRC Press, London, 2016. · Zbl 1341.92002
[14] Y. Kuang, J. D. Nagy, and J. J. Elser, {\it Biological stoichiometry of tumor dynamics: Mathematical models and analysis}, Discrete Contin. Dyn. Syst. B., 4 (2004), pp. 221-240. · Zbl 1056.34074
[15] X. Li and H. Wang, {\it A stoichiometerically derived algal growth model and its global analysis}, Math. Biosci. Eng., 7 (2010), pp. 825-836. · Zbl 1259.92081
[16] X. Li, H. Wang, and Y. Kuang, {\it Global analysis of a stoichiometric producer-grazer model with Holling type functional responses}, J. Math. Biol., 63 (2011), pp. 901-932. · Zbl 1234.92071
[17] J. Liebig, {\it The Natural Laws of Husbandry}, Walton Maberly, London, 1863.
[18] I. Loladze, {\it Rising atmospheric CO \(2\) and human nutrition: Toward globally imbalanced plant stoichiometry?}, Trends Ecol. Evol., 17 (2002), pp. 457-461.
[19] I. Loladze, Y. Kuang, and J. J. Elser, {\it Stoichiometry in producer-grazer systems: Linking energy flow with element cycling}, Bull. Math. Biol., 62 (2000), pp. 1137-1162. · Zbl 1323.92098
[20] I. Loladze, Y. Kuang, J. J. Elser, and W. F. Fagan, {\it Competition and stoichiometry: Coexistence of two predators on one prey}, Theoret. Popul. Biol., 65 (2004), pp. 1-15. · Zbl 1105.92044
[21] A. Lotka, {\it Elements of Physical Biology}, Williams and Wilkins, Baltimore, MD, 1925. · JFM 51.0416.06
[22] J. McDaniel, E. Kostelich, Y. Kuang, J. Nagy, M. Preul, N. Z. Moore, and N. Matirosyan, {\it Data assimilation in brain tumor models}, in Mathematical Methods and Models in Biomedicine, U. Ledzewicz, H. Schattler, A. Friedman, and E. Kashdan eds., New York, Springer, 2012, pp. 227-254. · Zbl 1345.92080
[23] C. Miller, Y. Kuang, W. Fagan, and J. J. Elser, {\it Modeling and analysis of stoichiometric two-patch consumer-resource systems}, Math. Biosci., 189 (2004), pp. 153-184. · Zbl 1047.92046
[24] J. D. Morken, A. Packer, R. A. Everett, J. D. Nagy, and Y. Kuang, {\it Mechanisms of resistance to intermittent androgen deprivation therapy identified in prostate cancer patients by novel computational method}, Cancer Res., 74 (2014), pp. 2673-2683.
[25] A. Packer, Y. Li, T. Andersen, Q. Hu, Y. Kuang, and M. Sommerfeld, {\it Growth and neutral lipid synthesis in green microalgae: A mathematical model}, Bioresour. Technol., 102 (2011), pp. 111-117.
[26] A. Peace, {\it Effects of light, nutrients, and food chain length on trophic efficiencies in simple stoichiometric aquatic food chain models}, Ecol. Model., 312 (2015), pp. 125-135.
[27] A. Peace, H. Wang, and Y. Kuang, {\it Dynamics of a producer-grazer model incorporating the effects of excess food nutrient content on grazer’s growth}, Bull. Math. Biol., 76 (2014), pp. 2175-2197. · Zbl 1300.92086
[28] A. Peace, Y. Zhao, I. Loladze, J. Elser, and Y. Kuang, {\it A stoichiometric producer-grazer model incorporating the effects of excess food-nutrient content on consumer dynamics}, Math. Biosci., 244 (2013), pp. 107-115. · Zbl 1280.92077
[29] T. Portz, Y. Kuang, and J. D. Nagy, {\it A clinical data validated mathematical model of prostate cancer growth under intermittent androgen suppression therapy}, AIP Adv., 2 (2012), 011002, doi:10.1063/1.3697848
[30] H. Stech, B. Peckham, and J. Pastor, {\it Enrichment in a general class of stoichiometric producer-consumer population growth models}, Theoret. Popul. Biol., 81 (2012), pp. 210-222. · Zbl 1322.92063
[31] R. Sterner and J. J. Elser, {\it Ecological Stoichiometry}, Princeton University Press, Princeton, NJ, 2002.
[32] V. Volterra, {\it Fluctuations in the abundance of a species considered mathematically}, Nature, 118 (1926), pp. 558-600. · JFM 52.0453.03
[33] H. Wang, Y. Kuang, and I. Loladze, {\it Dynamics of a mechanistically derived stoichiometric producer-grazer model}, J. Biol. Dyn., 2 (2008), pp. 286-296. · Zbl 1402.92372
[34] H. Wang, R. Sterner, and J. J. Elser, {\it On the “strict homeostasis” assumption in ecological stoichiometry}, Ecol. Model., 243 (2012), pp. 81-88.
[35] F. Touratier, J. Field, and C. Moloney, {\it A stoichiometric model relating growth substrate quality (C:N:P ratios) to N:P ratios in the products of heterotrophic release and excretion}, Ecol. Model., 139 (2001), pp. 265-291.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.