# American Institute of Mathematical Sciences

September  2020, 10(3): 391-423. doi: 10.3934/naco.2020010

## 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

* Corresponding author: Xin-You Meng

Received  May 2019 Revised  October 2019 Published  February 2020

Fund Project: The first author is supported by the National Natural Science Foundation of China (Grant No.11661050, 11861044), and the HongLiu First-class Disciplines Development Program of Lanzhou University of Technology

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.

Citation: Xin-You Meng, Yu-Qian Wu, Jie Li. Bifurcation analysis of a Singular Nutrient-plankton-fish model with taxation, protected zone and multiple delays. Numerical Algebra, Control and Optimization, 2020, 10 (3) : 391-423. doi: 10.3934/naco.2020010
##### 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. [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. 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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. [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. [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. [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. [30] O. Pardo, Global stability for a phytoplankton-nutrient system, J. Biol. Systems, 8 (2000), 195-209. [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. [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. [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. [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. [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. [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. [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. [30] O. Pardo, Global stability for a phytoplankton-nutrient system, J. Biol. Systems, 8 (2000), 195-209. [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. [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.
Dynamical responses of system (3) around $X_{0}$. (a) system (7) without control under $v = -0.1$; (b) system (7) without control under $v = 0$; (c) system (7) with control
Dynamical responses of system (7) with different $\delta_{1}$. (a) N(t); (b) P(t); (c) Z(t); (d) $F_{1}(t)$; (e)$F_{2}(t)$
Phase diagram of system (7) around $\bar{X}$: (a)$\delta_{1} = 1$; (b)$\delta_{1} = 2.5$; (c) $\delta_{1} = 7$
Dynamical responses of system (7) with different $\delta_{2}$: (a) N(t); (b) P(t); (c) Z(t); (d) $F_{1}(t)$; (e)$F_{2}(t)$; (f)-(h) $\delta_{2} = 0.06,0.3$
Dynamical responses of system (7) with different $c$: (a) N(t); (b) P(t); (c) Z(t); (d) $F_{1}(t)$; (e)$F_{2}(t)$; (f)$c = 0.2$; (g)$c = 1$; (h)$c = 2.6$
Dynamical responses of system (7) with different $T$: (a) N(t); (b) P(t); (c) Z(t); (d) $F_{1}(t)$; (e)$F_{2}(t)$; (f)-(h) $T = 0.85,2$
Dynamical response and phase plot of system at $\bar{X}$. (a)and(d): system (17) with $\tau_{2} = 0.8,\tau_{4} = 1<\tau^{*}_{4}$; (b)and(e): system (17) with $\tau_{2} = 0.8,\tau_{4} = 1.38>\tau^{*}_{4}$; (c)and(f): the control system with $\tau_{2} = 0.8,\tau_{4} = 1.38>\tau^{*}_{4}$
Dynamical response of system (3) with $\tau = 0.003<\tau_{0}$ and $\tau = 0.09>\tau_{0}$ at $\bar{X}$.(a): dynamical response of system; (b)and(c): phase plot of system
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