
- Previous Article
- NACO Home
- This Issue
-
Next Article
Passive control for a class of Nonlinear systems by using the technique of Adding a power integrator
Bifurcation analysis of a Singular Nutrient-plankton-fish model with taxation, protected zone and multiple delays
School of Science, Lanzhou University of Technology, Lanzhou, Gansu 730050, P. R. China |
A differential algebraic nutrient-plankton-fish model with taxation, free fishing zone, protected zone and multiple delays is investigated in this paper. First, the conditions of existence and control of singularity induced bifurcation are given by regarding economic interest as bifurcation parameter. Meanwhile, the existence of Hopf bifurcations are investigated when migration rates, taxation and the cost per unit harvest are taken as bifurcation parameters respectively. Next, the local stability of the interior equilibrium, existence and properties of Hopf bifurcation are discussed in the different cases of five delays. Furthermore, the optimal tax policy is obtained by using Pontryagin's maximum principle. Finally, some numerical simulations are presented to demonstrate analytical results.
References:
[1] |
K. Chakraborty, M. Chakraboty and T. K. Kar,
Bifurcation and control of a bioeconomic model of a prey-predator system with a time delay, Nonlinear Anal. Hybrid Syst., 5 (2011), 613-625.
doi: 10.1016/j.nahs.2011.05.004. |
[2] |
K. Chakraborty, S. Jana and T. K. Kar,
Effort dynamics of a delay-induced prey-predator system with reserve, Nonlinear Dyn., 70 (2012), 1805-1829.
doi: 10.1007/s11071-012-0575-z. |
[3] |
S. Chakraborty, S. Roy and J. Chattopadhyay, Nutrient-limited toxin production and the dynamics of two phytoplankton in culture media: A mathematical model, Ecol. Model., 213 (2008), 191-201. Google Scholar |
[4] |
C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, New York, 1976. |
[5] |
C. W. Clark, Bioeconomic Modelling and Fisheries Management, Wiley, New York, 1985. |
[6] |
L. Dai, Singular Control System, Springer, New York, 1989.
doi: 10.1007/BFb0002475. |
[7] |
K. Das and S. Ray, Effect of delay on nutrient cycling in phytoplankton-zooplankton interactions in estuarine system, Ecol. Model., 215 (2008), 69-76. Google Scholar |
[8] |
T. Das, R. N. Mukherjee and K. Chaudhuri, Harvesting of a prey-predator fishery in the presence of toxicity, Appl. Math. Model., 33 (2009), 2282-2292.
doi: 10.1016/j.apm.2008.06.008. |
[9] |
T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.
doi: 10.1006/jmaa.2000.7182. |
[10] |
H. S. Gordon,
The economic theory of a common-property resource: The fishery, J. P. Eco., 62 (1954), 124-142.
doi: 10.1007/s00199-010-0520-7. |
[11] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[12] |
R. P. Gupta, M. Banerjee and P. Chandra, Period doubling cascades of prey-predator model with nonlinear harvesting and control of over exploitation through taxation, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 2382-2405.
doi: 10.1016/j.cnsns.2013.10.033. |
[13] |
A. Hajihosseini, G. R. R. Lamooki, B. Beheshti and F. Maleki, The Hopf bifurcation analysis on a time-delayed recurrent neural network in the frequency domain, Neurocomputing, 73 (2010), 991-1005. Google Scholar |
[14] |
J. K. Hale, Theory of Functional Differential Equations, Springer, New York, 1977. |
[15] |
B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. |
[16] |
X. Z. He and S. G. Ruan, Global stability in chemostat-type plankton models with delayed nutrient recycling, J. Math. Biol., 37 (1998), 253-271.
doi: 10.1007/s002850050128. |
[17] |
S. V. Krishna, P. D. N. Srinivasu and B. Kaymakcalan, Conservation of an ecosystem through optimal taxation, Bull. Math. Biol., 60 (1998), 569-584. Google Scholar |
[18] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. |
[19] |
T. C. Liao, H. G. Yu and M. Zhao, Dynamics of a delayed phytoplankton-zooplankton system with Crowley-Martin functional response, Adv. Difference Equ., 2017 (2017), 5-35.
doi: 10.1186/s13662-016-1055-4. |
[20] |
C. Liu, Q. L. Zhang and X. D. Duan, Dynamical behavior in a harvested differential-algebraic prey-predator model with discrete time delay and stage structure, J. Franklin Inst., 346 (2009), 1038-1059.
doi: 10.1016/j.jfranklin.2009.06.004. |
[21] |
C. Liu, Q. L. Zhang, J. Huang and W. S. Tang, Dynamical analysis and control in a delayed differential-algebraic bioeconomic model with stage structure and diffusion, Int. J. Biomath., 5 (2012), 1-31.
doi: 10.1142/S1793524511001519. |
[22] |
W. Liu, C. J. Fu and B. S. Chen, Hopf bifurcation for a predator-prey biological economic system with Holling type Ⅱ functional response, J. Franklin Inst., 348 (2011), 1114-1127.
doi: 10.1016/j.jfranklin.2011.04.019. |
[23] |
W. M. Liu,
Criterion of Hopf bifurcations without using eigenvalues, J. Math. Anal. Appl., 182 (1994), 250-256.
doi: 10.1006/jmaa.1994.1079. |
[24] |
A. J. Lotka, Elements of Mathematical Biology, Econometrica, New York, 1956. Google Scholar |
[25] |
Y. F. Lv, R. Yuan and Y. Z. Pei, Stable coexistence mediated by specialist harvesting in a two zooplankton-phytoplankton system, Appl. Math. Model., 37 (2013), 9012-9030.
doi: 10.1016/j.apm.2013.03.076. |
[26] |
X. Y. Meng, H. F. Huo and X. B. Zhang, Stability and global Hopf bifurcation in a LeslieGower predator-prey model with stage structure for prey, J. Appl. Math. Comput., 60 (2019), 1-25.
doi: 10.1007/s12190-010-0383-x. |
[27] |
X. Y. Meng and J. G. Wang, Analysis of a delayed diffusive model with Beddington- DeAngelis functional response, Int. J. Biomathematics, 12 (2019), 1950047 (24 pages).
doi: 10.1142/S1793524519500475. |
[28] |
X. Y. Meng and Y. Q. Wu, Bifurcation and control in a singular phytoplankton- zooplanktonfish model with nonlinear fish harvesting and taxation, Int. J. Bifurc. Chaos, 28 (2018), 1850042.
doi: 10.1142/S0218127418500426. |
[29] |
X. Y. Meng and J. Li, Stability and Hopf bifurcation analysis of a delayed phytoplankton-zooplankton model with Allee effect and linear harvesting, Math. Biosci. Eng., 17 (2020), 1973-2002. Google Scholar |
[30] |
O. Pardo, Global stability for a phytoplankton-nutrient system, J. Biol. Systems, 8 (2000), 195-209. Google Scholar |
[31] |
S. G. Ruan and J. J. Wei,
On the zero of some transcendential functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Dis. Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874.
|
[32] |
T. Saha and M. Bandyopadhyay, Dynamical analysis of toxin producing phytoplankton- zooplankton interactions, Nonlinear Anal. Real. World Appl., 10 (2009), 314-332.
doi: 10.1016/j.nonrwa.2007.09.001. |
[33] |
P. Santra, G. S. Mahapatra and D. Pal, Analysis of differential-algebraic prey-predator dynamical model with super predator harvesting on economic perspective, Int. J. Dyn. and Control, 4 (2016), 266-274.
doi: 10.1007/s40435-015-0190-1. |
[34] |
V. Venkatasubramani, H. Schattler and J. Zaborszky, Local bifurcations and feasibility regions in differential-algebraic systems, IEEE Trans. Automat. Control., 40 (1995), 1992-2013.
doi: 10.1109/9.478226. |
[35] |
P. F. Wang, M. Zhao, H. G. Yu, C. J. Dai, N. Wang and B. B. Wang, Nonlinear dynamics of a marine phytoplankton-zooplankton system, Adv. Difference Equ., 2016 (2016), 212-227. Google Scholar |
[36] |
Y. Wang, H. B. Wang and W. H. Jiang, Stability switches and global Hopf bifurcation in a nutrient-plankton model, Nonlinear Dyn., 78 (2014), 981-994.
doi: 10.1007/s11071-014-1491-1. |
[37] |
H. Xiang, Y. Y. Wang and H. F. Huo,
Analysis of the binge drinking models with demographics and nonlinear infectivity on networks, J. Appl. Anal. Comput., 8 (2018), 1535-1554.
|
[38] |
G. D. Zhang, B. S. Chen, L. L. Zhu and Y. Shen, Hopf bifurcation for a differential-algebraic biological economic system with time delay, Appl. Math. Comput., 218 (2012), 7717-7726.
doi: 10.1016/j.amc.2011.12.096. |
[39] |
G. D. Zhang, Y. Shen and B. S. Chen, Hopf bifurcation of a predator-prey system with predator harvesting and two delays, Nonlinear Dyn., 73 (2013), 2119-2131.
doi: 10.1007/s11071-013-0928-2. |
[40] |
J. Z. Zhang, Z. Jin, J. R. Yan and G. Q. Sun, Stability and Hopf bifurcation in a delayed competition system, Nonlinear Anal.: Theo., Meth. Appl., 70 (2009), 658-670.
doi: 10.1016/j.na.2008.01.002. |
[41] |
Y. Zhang, J. Li, Y. Jie and X. G. Yan,
Optimal taxation policy for a prey-predator fishery model with reserves, Pac. J. Optim., 11 (2015), 137-155.
|
[42] |
Y. Zhang, Q. L. Zhang and X. G. Yan, Complex dynamics in a singular Leslie-Gower predatorprey bioeconomic model with time delay and stochastic fluctuations, Physica A., 404 (2014), 180-191.
doi: 10.1016/j.physa.2014.02.013. |
[43] |
Z. Z. Zhang and M. Rehim, Global qualitative analysis of a phytoplankton-zooplankton model in the presence of toxicity, Int. J. Dynam. Control, 5 (2017), 799-810.
doi: 10.1007/s40435-016-0230-5. |
show all references
References:
[1] |
K. Chakraborty, M. Chakraboty and T. K. Kar,
Bifurcation and control of a bioeconomic model of a prey-predator system with a time delay, Nonlinear Anal. Hybrid Syst., 5 (2011), 613-625.
doi: 10.1016/j.nahs.2011.05.004. |
[2] |
K. Chakraborty, S. Jana and T. K. Kar,
Effort dynamics of a delay-induced prey-predator system with reserve, Nonlinear Dyn., 70 (2012), 1805-1829.
doi: 10.1007/s11071-012-0575-z. |
[3] |
S. Chakraborty, S. Roy and J. Chattopadhyay, Nutrient-limited toxin production and the dynamics of two phytoplankton in culture media: A mathematical model, Ecol. Model., 213 (2008), 191-201. Google Scholar |
[4] |
C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, New York, 1976. |
[5] |
C. W. Clark, Bioeconomic Modelling and Fisheries Management, Wiley, New York, 1985. |
[6] |
L. Dai, Singular Control System, Springer, New York, 1989.
doi: 10.1007/BFb0002475. |
[7] |
K. Das and S. Ray, Effect of delay on nutrient cycling in phytoplankton-zooplankton interactions in estuarine system, Ecol. Model., 215 (2008), 69-76. Google Scholar |
[8] |
T. Das, R. N. Mukherjee and K. Chaudhuri, Harvesting of a prey-predator fishery in the presence of toxicity, Appl. Math. Model., 33 (2009), 2282-2292.
doi: 10.1016/j.apm.2008.06.008. |
[9] |
T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.
doi: 10.1006/jmaa.2000.7182. |
[10] |
H. S. Gordon,
The economic theory of a common-property resource: The fishery, J. P. Eco., 62 (1954), 124-142.
doi: 10.1007/s00199-010-0520-7. |
[11] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[12] |
R. P. Gupta, M. Banerjee and P. Chandra, Period doubling cascades of prey-predator model with nonlinear harvesting and control of over exploitation through taxation, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 2382-2405.
doi: 10.1016/j.cnsns.2013.10.033. |
[13] |
A. Hajihosseini, G. R. R. Lamooki, B. Beheshti and F. Maleki, The Hopf bifurcation analysis on a time-delayed recurrent neural network in the frequency domain, Neurocomputing, 73 (2010), 991-1005. Google Scholar |
[14] |
J. K. Hale, Theory of Functional Differential Equations, Springer, New York, 1977. |
[15] |
B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. |
[16] |
X. Z. He and S. G. Ruan, Global stability in chemostat-type plankton models with delayed nutrient recycling, J. Math. Biol., 37 (1998), 253-271.
doi: 10.1007/s002850050128. |
[17] |
S. V. Krishna, P. D. N. Srinivasu and B. Kaymakcalan, Conservation of an ecosystem through optimal taxation, Bull. Math. Biol., 60 (1998), 569-584. Google Scholar |
[18] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. |
[19] |
T. C. Liao, H. G. Yu and M. Zhao, Dynamics of a delayed phytoplankton-zooplankton system with Crowley-Martin functional response, Adv. Difference Equ., 2017 (2017), 5-35.
doi: 10.1186/s13662-016-1055-4. |
[20] |
C. Liu, Q. L. Zhang and X. D. Duan, Dynamical behavior in a harvested differential-algebraic prey-predator model with discrete time delay and stage structure, J. Franklin Inst., 346 (2009), 1038-1059.
doi: 10.1016/j.jfranklin.2009.06.004. |
[21] |
C. Liu, Q. L. Zhang, J. Huang and W. S. Tang, Dynamical analysis and control in a delayed differential-algebraic bioeconomic model with stage structure and diffusion, Int. J. Biomath., 5 (2012), 1-31.
doi: 10.1142/S1793524511001519. |
[22] |
W. Liu, C. J. Fu and B. S. Chen, Hopf bifurcation for a predator-prey biological economic system with Holling type Ⅱ functional response, J. Franklin Inst., 348 (2011), 1114-1127.
doi: 10.1016/j.jfranklin.2011.04.019. |
[23] |
W. M. Liu,
Criterion of Hopf bifurcations without using eigenvalues, J. Math. Anal. Appl., 182 (1994), 250-256.
doi: 10.1006/jmaa.1994.1079. |
[24] |
A. J. Lotka, Elements of Mathematical Biology, Econometrica, New York, 1956. Google Scholar |
[25] |
Y. F. Lv, R. Yuan and Y. Z. Pei, Stable coexistence mediated by specialist harvesting in a two zooplankton-phytoplankton system, Appl. Math. Model., 37 (2013), 9012-9030.
doi: 10.1016/j.apm.2013.03.076. |
[26] |
X. Y. Meng, H. F. Huo and X. B. Zhang, Stability and global Hopf bifurcation in a LeslieGower predator-prey model with stage structure for prey, J. Appl. Math. Comput., 60 (2019), 1-25.
doi: 10.1007/s12190-010-0383-x. |
[27] |
X. Y. Meng and J. G. Wang, Analysis of a delayed diffusive model with Beddington- DeAngelis functional response, Int. J. Biomathematics, 12 (2019), 1950047 (24 pages).
doi: 10.1142/S1793524519500475. |
[28] |
X. Y. Meng and Y. Q. Wu, Bifurcation and control in a singular phytoplankton- zooplanktonfish model with nonlinear fish harvesting and taxation, Int. J. Bifurc. Chaos, 28 (2018), 1850042.
doi: 10.1142/S0218127418500426. |
[29] |
X. Y. Meng and J. Li, Stability and Hopf bifurcation analysis of a delayed phytoplankton-zooplankton model with Allee effect and linear harvesting, Math. Biosci. Eng., 17 (2020), 1973-2002. Google Scholar |
[30] |
O. Pardo, Global stability for a phytoplankton-nutrient system, J. Biol. Systems, 8 (2000), 195-209. Google Scholar |
[31] |
S. G. Ruan and J. J. Wei,
On the zero of some transcendential functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Dis. Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874.
|
[32] |
T. Saha and M. Bandyopadhyay, Dynamical analysis of toxin producing phytoplankton- zooplankton interactions, Nonlinear Anal. Real. World Appl., 10 (2009), 314-332.
doi: 10.1016/j.nonrwa.2007.09.001. |
[33] |
P. Santra, G. S. Mahapatra and D. Pal, Analysis of differential-algebraic prey-predator dynamical model with super predator harvesting on economic perspective, Int. J. Dyn. and Control, 4 (2016), 266-274.
doi: 10.1007/s40435-015-0190-1. |
[34] |
V. Venkatasubramani, H. Schattler and J. Zaborszky, Local bifurcations and feasibility regions in differential-algebraic systems, IEEE Trans. Automat. Control., 40 (1995), 1992-2013.
doi: 10.1109/9.478226. |
[35] |
P. F. Wang, M. Zhao, H. G. Yu, C. J. Dai, N. Wang and B. B. Wang, Nonlinear dynamics of a marine phytoplankton-zooplankton system, Adv. Difference Equ., 2016 (2016), 212-227. Google Scholar |
[36] |
Y. Wang, H. B. Wang and W. H. Jiang, Stability switches and global Hopf bifurcation in a nutrient-plankton model, Nonlinear Dyn., 78 (2014), 981-994.
doi: 10.1007/s11071-014-1491-1. |
[37] |
H. Xiang, Y. Y. Wang and H. F. Huo,
Analysis of the binge drinking models with demographics and nonlinear infectivity on networks, J. Appl. Anal. Comput., 8 (2018), 1535-1554.
|
[38] |
G. D. Zhang, B. S. Chen, L. L. Zhu and Y. Shen, Hopf bifurcation for a differential-algebraic biological economic system with time delay, Appl. Math. Comput., 218 (2012), 7717-7726.
doi: 10.1016/j.amc.2011.12.096. |
[39] |
G. D. Zhang, Y. Shen and B. S. Chen, Hopf bifurcation of a predator-prey system with predator harvesting and two delays, Nonlinear Dyn., 73 (2013), 2119-2131.
doi: 10.1007/s11071-013-0928-2. |
[40] |
J. Z. Zhang, Z. Jin, J. R. Yan and G. Q. Sun, Stability and Hopf bifurcation in a delayed competition system, Nonlinear Anal.: Theo., Meth. Appl., 70 (2009), 658-670.
doi: 10.1016/j.na.2008.01.002. |
[41] |
Y. Zhang, J. Li, Y. Jie and X. G. Yan,
Optimal taxation policy for a prey-predator fishery model with reserves, Pac. J. Optim., 11 (2015), 137-155.
|
[42] |
Y. Zhang, Q. L. Zhang and X. G. Yan, Complex dynamics in a singular Leslie-Gower predatorprey bioeconomic model with time delay and stochastic fluctuations, Physica A., 404 (2014), 180-191.
doi: 10.1016/j.physa.2014.02.013. |
[43] |
Z. Z. Zhang and M. Rehim, Global qualitative analysis of a phytoplankton-zooplankton model in the presence of toxicity, Int. J. Dynam. Control, 5 (2017), 799-810.
doi: 10.1007/s40435-016-0230-5. |








[1] |
Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344 |
[2] |
Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020110 |
[3] |
Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020342 |
[4] |
Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 |
[5] |
Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021013 |
[6] |
Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173 |
[7] |
Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020048 |
[8] |
Giuseppe Capobianco, Tom Winandy, Simon R. Eugster. The principle of virtual work and Hamilton's principle on Galilean manifolds. Journal of Geometric Mechanics, 2021 doi: 10.3934/jgm.2021002 |
[9] |
Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032 |
[10] |
Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3031-3055. doi: 10.3934/dcds.2020053 |
[11] |
Kuo-Chih Hung, Shin-Hwa Wang. Classification and evolution of bifurcation curves for a porous-medium combustion problem with large activation energy. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020281 |
[12] |
Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020293 |
[13] |
Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020360 |
[14] |
Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387 |
[15] |
Yi-Ming Tai, Zhengyang Zhang. Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021027 |
[16] |
Linfeng Mei, Feng-Bin Wang. Dynamics of phytoplankton species competition for light and nutrient with recycling in a water column. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020359 |
[17] |
Chang-Yuan Cheng, Shyan-Shiou Chen, Rui-Hua Chen. Delay-induced spiking dynamics in integrate-and-fire neurons. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020363 |
[18] |
Stefan Siegmund, Petr Stehlík. Time scale-induced asynchronous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1011-1029. doi: 10.3934/dcdsb.2020151 |
[19] |
Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050 |
[20] |
Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]