# American Institute of Mathematical Sciences

January  2021, 26(1): 107-120. doi: 10.3934/dcdsb.2020264

## 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

* Corresponding Author (hao8@ualberta.ca)

Received  April 2020 Revised  June 2020 Published  January 2021 Early access  August 2020

Fund Project: Partially supported by NSFC-11801052, NSFLP-2019-ZD-1056, NSERC RGPIN-2020-03911 and NSERC RGPAS-2020-00090

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.

Citation: Ming Chen, Hao Wang. Dynamics of a discrete-time stoichiometric optimal foraging model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 107-120. doi: 10.3934/dcdsb.2020264
##### 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. [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. [7] W. Gurney and R. M. Nisbet, Ecological Dynamics, 1998. [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. [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.

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##### 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. [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. [7] W. Gurney and R. M. Nisbet, Ecological Dynamics, 1998. [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. [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.
Attractor of discrete-time optimal foraging model (4) in phase plane for different light intensities in two cases. Panels of ($a_i$) describe the case when the producer's growth rate is low $b = 1.2$, $i = 1,2,3$. Panels of ($b_i$) describe the case when the producer's growth rate is high $b = 3$, $i = 1,2,3$. The red dashed curves are defined by $F(x,y) = 0$, which denote the producer nullclines. The blue dotted curves are defined by $G(x,y) = 0$, which denote the grazer nullclines. Solid bullets denote stable equilibria while circles represent unstable equilibria
Bifurcation diagram of the population densities with respect to $K$ (light intensity) for the discrete-time model (4) ($a_i$), i = 1, 2, and the continuous-time model (1) ($b_i$), i = 1, 2. Shaded regions with $+$ represent the parameter regions of the optimal foraging behaviors benefiting the grazers. All parameters are provided in Table 1 with $b = 1.2$
Solution curves for system (4) and (1). ($a_i$) and ($b_i$) denote the dynamics of (4) and (1) with increasing $K$, respectively. Producer and grazer's densities ($\mathrm{mg\; C/L}$) are plotted by dashed and solid lines, respectively. All parameters are provided in Table 1 with $b = 1.2$
Solution curves for system (4) and (1). ($a_i$) and ($b_i$) denote the dynamics of (4) and (1) with increasing $K$, respectively. Producer and grazer densities ($\mathrm{mg\; C/L}$) are described by dashed and solid lines, respectively. All parameters are provided in Table 1 with $b = 3$
The bifurcation curves with respect to $K$ for the discrete-time model ($a_i$), i = 1, 2, and continuous-time model ($b_i$), i = 1, 2. All parameters are provided in Table 1 with $b = 3$
Spectrum of the maximum Lyapunov exponent (MLE) with respect to $K$ for the discrete-time model. All parameters are provided in Table 1 with $b = 3$
A two-parameter bifurcation diagram for varying light level $K$ and varying maximal growth rate of producer $b$ for the discrete-time model (a) and continuous-time model (b). All other parameter values are listed in Table 1 and the initial point is $x(0) = 0.2\; \mathrm{mg C L}^{-1}$ and $y(0) = 0.2\; \mathrm{mg C L}^{-1}$. Discrete-time model (4) exhibits periodic oscillations in blue region and chaotic behaviors in red region. Outside these regions, model (4) has stable equilibria
Bifurcation diagram of the grazer densities with respect to $K$ (light intensity) for the discrete-time model (4) ($a_i$), i = 1, 2, and the continuous-time model (1) ($b_i$), i = 1, 2. Specially, ($a_1$) and ($b_1$) denote the case with the low growth rate of producer ($b = 1.2$); ($a_2$) and ($b_2$) denote the case with the high growth rate of producer ($b = 3$). Light ($+$) and dark ($-$) shaded regions represent the parameter regions of the optimal foraging behaviors benefiting and restraining the grazers, respectively. All parameters are provided in Table 1 except the parameter $a_1 = 3.5$
Parameters of model (4) with default values and units
 Par. Description Value Unit $P_T$ Total phosphorus $0.02$ $\mathrm{mg P L}^{-1}$ $K$ Producer carrying capacity determined by light $0-3.5$ $\mathrm{mg C L}^{-1}$ $b$ Maximal growth rate of the producer $1.2$ or $3$ $\mathrm{day}^{-1}$ $\delta$ Grazer loss rate $0.12$ $\mathrm{day}^{-1}$ $\theta$ Grazer constant $\mathrm{P:C}$ 0.03 $\mathrm{mgP/mgC}$ $q$ Producer minimal $\mathrm{P:C}$ $0.0038$ $\mathrm{mgP/mgC}$ $e$ Maximal production efficiency in carbon terms for grazer $0.8$ $\alpha$ Phosphorus half saturation constant of the producer $0.008$ $\mathrm{mg C L}^{-1}$ $\mu$ Water cleared/mg C invested to generate filtering energy $700$ $\mathrm{L/mg C}$ $\tau$ Handling time (-inverse of max feeding rate) $1.23$ $\mathrm{day}$ $\xi(Q)$ Feeding cost, function for optimal foraging model $a_0=0.01$, $a_1=5.17$ $\xi(Q)=\min\{a_0,a_1Q^2+a_2Q+a_3\}$ $a_2=-0.31$, $a_3=0.007$
 Par. Description Value Unit $P_T$ Total phosphorus $0.02$ $\mathrm{mg P L}^{-1}$ $K$ Producer carrying capacity determined by light $0-3.5$ $\mathrm{mg C L}^{-1}$ $b$ Maximal growth rate of the producer $1.2$ or $3$ $\mathrm{day}^{-1}$ $\delta$ Grazer loss rate $0.12$ $\mathrm{day}^{-1}$ $\theta$ Grazer constant $\mathrm{P:C}$ 0.03 $\mathrm{mgP/mgC}$ $q$ Producer minimal $\mathrm{P:C}$ $0.0038$ $\mathrm{mgP/mgC}$ $e$ Maximal production efficiency in carbon terms for grazer $0.8$ $\alpha$ Phosphorus half saturation constant of the producer $0.008$ $\mathrm{mg C L}^{-1}$ $\mu$ Water cleared/mg C invested to generate filtering energy $700$ $\mathrm{L/mg C}$ $\tau$ Handling time (-inverse of max feeding rate) $1.23$ $\mathrm{day}$ $\xi(Q)$ Feeding cost, function for optimal foraging model $a_0=0.01$, $a_1=5.17$ $\xi(Q)=\min\{a_0,a_1Q^2+a_2Q+a_3\}$ $a_2=-0.31$, $a_3=0.007$
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