• Previous Article
    Flocking of non-identical Cucker-Smale models on general coupling network
  • DCDS-B Home
  • This Issue
  • Next Article
    Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media
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  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 & Continuous Dynamical Systems - B, 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. Google Scholar

[2]

M. ChenM. FanC. B. XieA. Peace and H. Wang, Stoichiometric food chain model on discrete time scale, Mathematical Biosciences and Engineering, 16 (2018), 101-118.   Google Scholar

[3]

M. ChenL. 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.  Google Scholar

[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.  Google Scholar

[5]

M. FanI. LoladzeY. 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.  Google Scholar

[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. ElserM. KyleJ. LearnedM. McCrackinA. 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.  Google Scholar

[9]

Y. KuangJ. 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.  Google Scholar

[10]

I. LoladzeY. 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.  Google Scholar

[11]

I. LoladzeY. KuangJ. 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.  Google Scholar

[12]

A. PeaceY. ZhaoI. LoladzeJ. 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.  Google Scholar

[13]

A. PeaceH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

G. H. PykeH. 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. SimpsonR. M. SiblyK. P. LeeS. 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.  Google Scholar

[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.  Google Scholar
[19]

G. SuiM. FanI. 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.  Google Scholar

[20]

H. WangY. 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.  Google Scholar

[21]

H. WangK. DunningJ. 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.  Google Scholar

[22]

H. WangR. 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.  Google Scholar

[23]

H. WangZ. 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.  Google Scholar

[24]

C. XieM. 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.  Google Scholar

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. ChenM. FanC. B. XieA. Peace and H. Wang, Stoichiometric food chain model on discrete time scale, Mathematical Biosciences and Engineering, 16 (2018), 101-118.   Google Scholar

[3]

M. ChenL. 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.  Google Scholar

[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.  Google Scholar

[5]

M. FanI. LoladzeY. 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.  Google Scholar

[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. ElserM. KyleJ. LearnedM. McCrackinA. 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.  Google Scholar

[9]

Y. KuangJ. 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.  Google Scholar

[10]

I. LoladzeY. 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.  Google Scholar

[11]

I. LoladzeY. KuangJ. 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.  Google Scholar

[12]

A. PeaceY. ZhaoI. LoladzeJ. 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.  Google Scholar

[13]

A. PeaceH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

G. H. PykeH. 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. SimpsonR. M. SiblyK. P. LeeS. 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.  Google Scholar

[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.  Google Scholar
[19]

G. SuiM. FanI. 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.  Google Scholar

[20]

H. WangY. 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.  Google Scholar

[21]

H. WangK. DunningJ. 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.  Google Scholar

[22]

H. WangR. 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.  Google Scholar

[23]

H. WangZ. 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.  Google Scholar

[24]

C. XieM. 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.  Google Scholar

Figure 1.  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
Figure 2.  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 $
Figure 3.  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 $
Figure 4.  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 $
Figure 5.  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 $
Figure 6.  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 $
Figure 7.  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
Figure 8.  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 $
Table 1.  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 $
[1]

Sie Long Kek, Mohd Ismail Abd Aziz, Kok Lay Teo, Rohanin Ahmad. An iterative algorithm based on model-reality differences for discrete-time nonlinear stochastic optimal control problems. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 109-125. doi: 10.3934/naco.2013.3.109

[2]

Ka Chun Cheung, Hailiang Yang. Optimal investment-consumption strategy in a discrete-time model with regime switching. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 315-332. doi: 10.3934/dcdsb.2007.8.315

[3]

Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023

[4]

Sie Long Kek, Kok Lay Teo, Mohd Ismail Abd Aziz. Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 207-222. doi: 10.3934/naco.2012.2.207

[5]

Sie Long Kek, Mohd Ismail Abd Aziz. Output regulation for discrete-time nonlinear stochastic optimal control problems with model-reality differences. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 275-288. doi: 10.3934/naco.2015.5.275

[6]

Veena Goswami, Gopinath Panda. Optimal information policy in discrete-time queues with strategic customers. Journal of Industrial & Management Optimization, 2019, 15 (2) : 689-703. doi: 10.3934/jimo.2018065

[7]

Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019112

[8]

Eduardo Liz. A new flexible discrete-time model for stable populations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2487-2498. doi: 10.3934/dcdsb.2018066

[9]

H. L. Smith, X. Q. Zhao. Competitive exclusion in a discrete-time, size-structured chemostat model. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 183-191. doi: 10.3934/dcdsb.2001.1.183

[10]

Yuefen Chen, Yuanguo Zhu. Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems. Journal of Industrial & Management Optimization, 2018, 14 (3) : 913-930. doi: 10.3934/jimo.2017082

[11]

Angelica Pachon, Federico Polito, Costantino Ricciuti. On discrete-time semi-Markov processes. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020170

[12]

Yun Kang. Permanence of a general discrete-time two-species-interaction model with nonlinear per-capita growth rates. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2123-2142. doi: 10.3934/dcdsb.2013.18.2123

[13]

Ferenc A. Bartha, Ábel Garab. Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model. Journal of Computational Dynamics, 2014, 1 (2) : 213-232. doi: 10.3934/jcd.2014.1.213

[14]

Deepak Kumar, Ahmad Jazlan, Victor Sreeram, Roberto Togneri. Partial fraction expansion based frequency weighted model reduction for discrete-time systems. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 329-337. doi: 10.3934/naco.2016015

[15]

Dan Zhang, Xiaochun Cai, Lin Wang. Complex dynamics in a discrete-time size-structured chemostat model with inhibitory kinetics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3439-3451. doi: 10.3934/dcdsb.2018327

[16]

John E. Franke, Abdul-Aziz Yakubu. Periodically forced discrete-time SIS epidemic model with disease induced mortality. Mathematical Biosciences & Engineering, 2011, 8 (2) : 385-408. doi: 10.3934/mbe.2011.8.385

[17]

S. R.-J. Jang. Allee effects in a discrete-time host-parasitoid model with stage structure in the host. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 145-159. doi: 10.3934/dcdsb.2007.8.145

[18]

Jingang Zhao, Chi Zhang. Finite-horizon optimal control of discrete-time linear systems with completely unknown dynamics using Q-learning. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020030

[19]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183

[20]

Lih-Ing W. Roeger, Razvan Gelca. Dynamically consistent discrete-time Lotka-Volterra competition models. Conference Publications, 2009, 2009 (Special) : 650-658. doi: 10.3934/proc.2009.2009.650

2019 Impact Factor: 1.27

Article outline

Figures and Tables

[Back to Top]