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A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence
Dynamics of a discrete-time stoichiometric optimal foraging model
1. | School of Science, Dalian Maritime University, 1 Linghai Road, Dalian, Liaoning, 116026, China |
2. | Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada |
In this paper, we discretize and analyze a stoichiometric optimal foraging model where the grazer's feeding effort depends on the producer's nutrient quality. We systematically make comparisons of the dynamical behaviors between the discrete-time model and the continuous-time model to study the robustness of model predictions to time discretization. When the maximum growth rate of producer is low, both model types admit similar dynamics including bistability and deterministic extinction of the grazer caused by low nutrient quality of the producer. Especially, the grazer is benefited from optimal foraging similarly in both discrete-time and continuous-time models. When the maximum growth rate of producer is high, dynamics of the discrete-time model are more complex including chaos. A phenomenal observation is that under extremely high light intensities, the grazer in the continuous-time model tends to perish due to poor food quality, however, the grazer in the discrete-time model persists in regular or irregular oscillatory ways. This significant difference indicates the necessity of studying discrete-time models which naturally include species' generations and are thus more popular in theoretical biology. Finally, we discuss how the shape of the quality-based feeding function regulates the beneficial or restraint effect of optimal foraging on the grazer population.
References:
[1] |
S. Busenberg and K. L. Cooke, Models of vertically transmitted diseases with sequential-continuous dynamics, Nonlinear Phenomena in Mathematical Sciences, (1982), 179-187. Google Scholar |
[2] |
M. Chen, M. Fan, C. B. Xie, A. Peace and H. Wang,
Stoichiometric food chain model on discrete time scale, Mathematical Biosciences and Engineering, 16 (2018), 101-118.
|
[3] |
M. Chen, L. Asik and A. Peace,
Stoichiometric knife-edge model on discrete time scale, Advances in Difference Equations, 2019 (2019), 1-16.
doi: 10.1186/s13662-019-2468-7. |
[4] |
K. L. Cooke and J. Wiener,
Retarded differential equations with piecewise constant delays, Journal of Mathematical Analysis and Applications, 99 (1984), 265-297.
doi: 10.1016/0022-247X(84)90248-8. |
[5] |
M. Fan, I. Loladze, Y. Kuang and J. J. Elser,
Dynamics of a stoichiometric discrete producer-grazer model, Journal of Difference Equations and Applications, 11 (2005), 347-364.
doi: 10.1080/10236190412331335427. |
[6] |
R. Frankham and B. W. Brook, The importance of time scale in conservation biology and ecology, Annales Zoologici Fennici, 41 (2004), 459-463. Google Scholar |
[7] |
W. Gurney and R. M. Nisbet, Ecological Dynamics, 1998. Google Scholar |
[8] |
J. J. Elser, M. Kyle, J. Learned, M. McCrackin, A. Peace and L. Steger,
Life on the stoichiometric knife-edge: Effects of high and low food C:P ratio on growth, feeding, and respiration in three Daphnia species, Inland Waters, 6 (2016), 136-146.
doi: 10.5268/IW-6.2.908. |
[9] |
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. |
[10] |
I. Loladze, Y. Kuang and J. J. Elser,
Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bulletin of Mathematical Biology, 62 (2000), 1137-1162.
doi: 10.1006/bulm.2000.0201. |
[11] |
I. Loladze, Y. Kuang, J. J. Elser and W. F. Fagan,
Competition and stoichiometry: Coexistence of two predators on one prey, Theoretical Population Biology, 65 (2004), 1-15.
doi: 10.1016/S0040-5809(03)00105-9. |
[12] |
A. Peace, Y. Zhao, I. Loladze, J. J. Elser and Y. Kuang,
A stoichiometric producer-grazer model incorporating the effects of excess food-nutrient content on consumer dynamics, Mathematical Biosciences, 244 (2013), 107-115.
doi: 10.1016/j.mbs.2013.04.011. |
[13] |
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, Bulletin of Mathematical Biology, 76 (2014), 2175-2197.
doi: 10.1007/s11538-014-0006-z. |
[14] |
A. Peace,
Effects of light, nutrients, and food chain length on trophic efficiencies in simple stoichiometric aquatic food chain models, Ecological Modelling, 312 (2015), 125-135.
doi: 10.1016/j.ecolmodel.2015.05.019. |
[15] |
A. Peace and H. Wang,
Compensatory foraging in stoichiometric producer-grazer models, Bulletin of Mathematical Biology, 81 (2019), 4932-4950.
doi: 10.1007/s11538-019-00665-2. |
[16] |
G. H. Pyke, H. R. Pulliam and E. L. Charnov, Optimal Foraging: A selective review of theory and tests, Quarterly Review of Biology, 52 (1977), 137-154. Google Scholar |
[17] |
S. J. Simpson, R. M. Sibly, K. P. Lee, S. T. Behmer and D. Raubenheimer,
Optimal foraging when regulating intake of multiple nutrients, Animal Behaviour, 68 (2004), 1299-1311.
doi: 10.1016/j.anbehav.2004.03.003. |
[18] |
R. W. Sterner and J. J. Elser, Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere, Princeton University Press, 2002.
doi: 10.1515/9781400885695.![]() |
[19] |
G. Sui, M. Fan, I. Loladze and Y. Kuang,
The dynamics of a stoichiometric plant-herbivore model and its discrete analog, Mathematical Biosciences and Engineering, 4 (2007), 29-46.
doi: 10.3934/mbe.2007.4.29. |
[20] |
H. Wang, Y. Kuang and I. Loladze,
Dynamics of a mechanistically derived stoichiometric producer-grazer model, Journal of Biological Dynamics, 2 (2008), 286-296.
doi: 10.1080/17513750701769881. |
[21] |
H. Wang, K. Dunning, J. J. Elser and Y. Kuang,
Daphnia species invasion, competitive exclusion, and chaotic coexistence, Discrete & Continuous Dynamical Systems-B, 12 (2009), 481-493.
doi: 10.3934/dcdsb.2009.12.481. |
[22] |
H. Wang, R. W. Sterner and J. J. Elser,
On the "strict homeostasis" assumption in ecological stoichiometry, Ecological Modelling, 243 (2012), 81-88.
doi: 10.1016/j.ecolmodel.2012.06.003. |
[23] |
H. Wang, Z. Lu and A. Raghavan,
Weak dynamical threshold for the "strict homeostasis" assumption in ecological stoichiometry, Ecological Modelling, 384 (2018), 233-240.
doi: 10.1016/j.ecolmodel.2018.06.027. |
[24] |
C. Xie, M. Fan and W. Zhao,
Dynamics of a discrete stoichiometric two predators one prey model, Journal of Biological Systems, 18 (2010), 649-667.
doi: 10.1142/S0218339010003457. |
show all references
References:
[1] |
S. Busenberg and K. L. Cooke, Models of vertically transmitted diseases with sequential-continuous dynamics, Nonlinear Phenomena in Mathematical Sciences, (1982), 179-187. Google Scholar |
[2] |
M. Chen, M. Fan, C. B. Xie, A. Peace and H. Wang,
Stoichiometric food chain model on discrete time scale, Mathematical Biosciences and Engineering, 16 (2018), 101-118.
|
[3] |
M. Chen, L. Asik and A. Peace,
Stoichiometric knife-edge model on discrete time scale, Advances in Difference Equations, 2019 (2019), 1-16.
doi: 10.1186/s13662-019-2468-7. |
[4] |
K. L. Cooke and J. Wiener,
Retarded differential equations with piecewise constant delays, Journal of Mathematical Analysis and Applications, 99 (1984), 265-297.
doi: 10.1016/0022-247X(84)90248-8. |
[5] |
M. Fan, I. Loladze, Y. Kuang and J. J. Elser,
Dynamics of a stoichiometric discrete producer-grazer model, Journal of Difference Equations and Applications, 11 (2005), 347-364.
doi: 10.1080/10236190412331335427. |
[6] |
R. Frankham and B. W. Brook, The importance of time scale in conservation biology and ecology, Annales Zoologici Fennici, 41 (2004), 459-463. Google Scholar |
[7] |
W. Gurney and R. M. Nisbet, Ecological Dynamics, 1998. Google Scholar |
[8] |
J. J. Elser, M. Kyle, J. Learned, M. McCrackin, A. Peace and L. Steger,
Life on the stoichiometric knife-edge: Effects of high and low food C:P ratio on growth, feeding, and respiration in three Daphnia species, Inland Waters, 6 (2016), 136-146.
doi: 10.5268/IW-6.2.908. |
[9] |
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. |
[10] |
I. Loladze, Y. Kuang and J. J. Elser,
Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bulletin of Mathematical Biology, 62 (2000), 1137-1162.
doi: 10.1006/bulm.2000.0201. |
[11] |
I. Loladze, Y. Kuang, J. J. Elser and W. F. Fagan,
Competition and stoichiometry: Coexistence of two predators on one prey, Theoretical Population Biology, 65 (2004), 1-15.
doi: 10.1016/S0040-5809(03)00105-9. |
[12] |
A. Peace, Y. Zhao, I. Loladze, J. J. Elser and Y. Kuang,
A stoichiometric producer-grazer model incorporating the effects of excess food-nutrient content on consumer dynamics, Mathematical Biosciences, 244 (2013), 107-115.
doi: 10.1016/j.mbs.2013.04.011. |
[13] |
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, Bulletin of Mathematical Biology, 76 (2014), 2175-2197.
doi: 10.1007/s11538-014-0006-z. |
[14] |
A. Peace,
Effects of light, nutrients, and food chain length on trophic efficiencies in simple stoichiometric aquatic food chain models, Ecological Modelling, 312 (2015), 125-135.
doi: 10.1016/j.ecolmodel.2015.05.019. |
[15] |
A. Peace and H. Wang,
Compensatory foraging in stoichiometric producer-grazer models, Bulletin of Mathematical Biology, 81 (2019), 4932-4950.
doi: 10.1007/s11538-019-00665-2. |
[16] |
G. H. Pyke, H. R. Pulliam and E. L. Charnov, Optimal Foraging: A selective review of theory and tests, Quarterly Review of Biology, 52 (1977), 137-154. Google Scholar |
[17] |
S. J. Simpson, R. M. Sibly, K. P. Lee, S. T. Behmer and D. Raubenheimer,
Optimal foraging when regulating intake of multiple nutrients, Animal Behaviour, 68 (2004), 1299-1311.
doi: 10.1016/j.anbehav.2004.03.003. |
[18] |
R. W. Sterner and J. J. Elser, Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere, Princeton University Press, 2002.
doi: 10.1515/9781400885695.![]() |
[19] |
G. Sui, M. Fan, I. Loladze and Y. Kuang,
The dynamics of a stoichiometric plant-herbivore model and its discrete analog, Mathematical Biosciences and Engineering, 4 (2007), 29-46.
doi: 10.3934/mbe.2007.4.29. |
[20] |
H. Wang, Y. Kuang and I. Loladze,
Dynamics of a mechanistically derived stoichiometric producer-grazer model, Journal of Biological Dynamics, 2 (2008), 286-296.
doi: 10.1080/17513750701769881. |
[21] |
H. Wang, K. Dunning, J. J. Elser and Y. Kuang,
Daphnia species invasion, competitive exclusion, and chaotic coexistence, Discrete & Continuous Dynamical Systems-B, 12 (2009), 481-493.
doi: 10.3934/dcdsb.2009.12.481. |
[22] |
H. Wang, R. W. Sterner and J. J. Elser,
On the "strict homeostasis" assumption in ecological stoichiometry, Ecological Modelling, 243 (2012), 81-88.
doi: 10.1016/j.ecolmodel.2012.06.003. |
[23] |
H. Wang, Z. Lu and A. Raghavan,
Weak dynamical threshold for the "strict homeostasis" assumption in ecological stoichiometry, Ecological Modelling, 384 (2018), 233-240.
doi: 10.1016/j.ecolmodel.2018.06.027. |
[24] |
C. Xie, M. Fan and W. Zhao,
Dynamics of a discrete stoichiometric two predators one prey model, Journal of Biological Systems, 18 (2010), 649-667.
doi: 10.1142/S0218339010003457. |






Par. | Description | Value | Unit |
Total phosphorus | |||
Producer carrying capacity determined by light | |||
Maximal growth rate of the producer | |||
Grazer loss rate | |||
Grazer constant |
0.03 | ||
Producer minimal |
|||
Maximal production efficiency in carbon terms for grazer | |||
Phosphorus half saturation constant of the producer | |||
Water cleared/mg C invested to generate filtering energy | |||
Handling time (-inverse of max feeding rate) | |||
Feeding cost, function for optimal foraging model | |||
Par. | Description | Value | Unit |
Total phosphorus | |||
Producer carrying capacity determined by light | |||
Maximal growth rate of the producer | |||
Grazer loss rate | |||
Grazer constant |
0.03 | ||
Producer minimal |
|||
Maximal production efficiency in carbon terms for grazer | |||
Phosphorus half saturation constant of the producer | |||
Water cleared/mg C invested to generate filtering energy | |||
Handling time (-inverse of max feeding rate) | |||
Feeding cost, function for optimal foraging model | |||
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