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May  2022, 27(5): 2791-2815. doi: 10.3934/dcdsb.2021160

The qualitative behavior of a plankton-fish interaction model with food limited growth rate and non-constant fish harvesting

Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221 005, India

*Corresponding author: R. P. Gupta; e-mail: ravipguptaiitk@gmail.com, ravigupta@bhu.ac.in

Received  October 2020 Revised  March 2021 Published  May 2022 Early access  June 2021

Fund Project: The work of the third author (Shivam Saxena) is supported by Council of Scientific and Industrial Research (No.09/013(0721)/2017)

The manuscript aims to investigate the qualitative analysis of a plankton-fish interaction with food limited growth rate of plankton population and non-constant harvesting of fish population. The ecological feasibility of population densities of both plankton and fish in terms of positivity and boundedness of solutions is shown. The conditions for the existence of various equilibrium points and their stability are derived thoroughly. This study mainly focuses on how the harvesting affects equilibrium points, their stability, periodic solutions and bifurcations in the proposed system. It is shown that the system exhibits saddle-node bifurcation in the form of a collision of two interior equilibrium points. Existence conditions for the occurrence of Hopf-bifurcation around interior equilibrium points are discussed. Lyapunov coefficients are examined to check the stability properties of these periodic solutions. We have also plotted the bifurcation diagrams for saddle-node, transcritical and Hopf bifurcations. A detailed algorithm for the occurrence of Bogdanov-Takens bifurcation is derived and finally some numerical simulations are also carried out to validate the theoretical results. This work suggests that the harvesting of fish population can change the dynamics of the system, which may be useful for the ecological management.

Citation: R. P. Gupta, Shristi Tiwari, Shivam Saxena. The qualitative behavior of a plankton-fish interaction model with food limited growth rate and non-constant fish harvesting. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2791-2815. doi: 10.3934/dcdsb.2021160
References:
[1]

N. D. Barlow, Harvesting models for resource-limited populations, N. Z. J. Ecol., 10 (1987), 129-133. 

[2]

G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, Ginn, Boston, 1982.

[3]

R. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plan, Selecta. Math. Soviet., 1 (1981), 373-388. 

[4]

R. Bogdanov, Versal deformations of a singular point on the plan in the case of zero eigenvalues, Selecta. Math. Soviet., 1 (1981), 389-421. 

[5]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, Berlin, 251, 1982.

[6]

S. N. Chow and H. D. Zhang, The qualitative analysis of two species predator-prey model with Holling's type III functional response, Appl. Math. Mech., 7 (1986), 73-80.  doi: 10.1007/BF01896254.

[7]

X. Dou and Y. Li, Almost periodic solution for a food-limited population model with delay and feedback control, Int. J. Comput. Math. Sci., 5 (2011), 174-179. 

[8]

K. GopalsamyM. R. S. Kulenovic and G. Ladas, Time lags in a food-limited population model, Appl. Anal., 31 (1988), 225-237.  doi: 10.1080/00036818808839826.

[9]

K. GopalsamyM. R. S. Kulenovic and G. Ladas, Environmental periodicity and time delays in a food-limited population model, J. Math. Anal. Appl., 147 (1990), 545-555.  doi: 10.1016/0022-247X(90)90369-Q.

[10]

R. P. Gupta and P. Chandra, Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295.  doi: 10.1016/j.jmaa.2012.08.057.

[11]

T. G. Hallam and J. T. De Luna, Effects of toxicants on populations: A qualitative approach III. Environmental and food chain pathways, J. Theor. Biol., 109 (1984), 411-429.  doi: 10.1016/S0022-5193(84)80090-9.

[12]

J. HuangY. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Disc. Cont. Dyn. Syst. B, 18 (2013), 2101-2121.  doi: 10.3934/dcdsb.2013.18.2101.

[13]

D. JiangN. Shi and Y. Zhao, Existence, uniqueness, and global stability of positive solutions to the food-limited population model with random perturbation, Math. and Comp. Modelling, 42 (2005), 651-658.  doi: 10.1016/j.mcm.2004.03.011.

[14]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Appl. Math. Sciences, Springer-Verlag, New York, 112, 2004. doi: 10.1007/978-1-4757-3978-7.

[15]

B. LeardC. Lewis and J. Rebaza, Dynamics of ratio-dependent predator prey models with non constant harvesting, Disc. Cont. Dyn. Syst. S, 1 (2008), 303-315.  doi: 10.3934/dcdss.2008.1.303.

[16]

P. Lenzini and J. Rebaza, Nonconstant predator harvesting on ratio-dependent predator-prey models, Appl. Math. Sciences, 4 (2010), 791-803. 

[17]

D. Li and M. Liu, Invariant measure of a stochastic food–limited population model with regime switching, Math. Comput. Simul., 178 (2020), 16-26.  doi: 10.1016/j.matcom.2020.06.003.

[18]

Z. Li and M. He, Hopf bifurcation in a delayed food-limited model with feedback control, Nonlinear Dyn., 76 (2014), 1215-1224.  doi: 10.1007/s11071-013-1205-0.

[19]

W. LiuC. Fu and B. Chen, Hopf bifurcation and center stability for a predator–prey biological economic model with prey harvesting, Commun. Nonlinear. Sci. Numer. Simulat., 17 (2012), 3989-3998.  doi: 10.1016/j.cnsns.2012.02.025.

[20]

P. LiuJ. Shi and Y. Wang, Periodic solutions of a logistic type population model with harvesting, J. Math. Anal. Appl., 369 (2010), 730-735.  doi: 10.1016/j.jmaa.2010.04.027.

[21]

O. P. Misra and R. Babu, A model for the effect of toxicant on a three species food chain system with Food–Limited growth of prey population, Glob. J. Math. Anal., 2 (2014), 120-145.  doi: 10.14419/gjma.v2i3.2990.

[22]

P. Panja and S. K. Mondal, Stability analysis of coexistence of three species prey-predator model, Nonlinear Dyn., 81 (2015), 373-382.  doi: 10.1007/s11071-015-1997-1.

[23]

P. PanjaS. K. Mondal and D. K. Jana, Effect of toxicants of phytoplankton-zooplankton-fish dynamics and harvesting, Chaos Soliton Fract., 104 (2017), 389-399.  doi: 10.1016/j.chaos.2017.08.036.

[24]

P. Panja, Plankton population and cholera disease transmission: A mathematical modeling study, Int. J. Bifurcat. Chaos, 30 (2020), 2050054(16). doi: 10.1142/S0218127420500546.

[25]

L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1996. doi: 10.1007/978-1-4684-0249-0.

[26]

E. C. Pielou, An Introduction to Mathematical Ecology, Wiley, New York, 1969.

[27]

F. E. Smith, Population dynamics in Daphnia magna and a new model for population growth, Ecology, 44 (1963), 651-663.  doi: 10.2307/1933011.

[28]

J. W.-H. So and J. S. Yu, On the uniform stability for a food-limited population model with time delay, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 991-1002.  doi: 10.1017/S0308210500022605.

[29]

S. Tang and L. Chen, Global attractivity in a food-limited population model with impulsive effects, J. Math. Anal. Appl., 292 (2004), 211-221.  doi: 10.1016/j.jmaa.2003.11.061.

[30]

Y. TaoX. Wang and X. Song, Effect of prey refuge on a harvested predator-prey model with generalized functional response, Commun. Nonlinear. Sci. Numer. Simulat., 16 (2011), 1052-1059.  doi: 10.1016/j.cnsns.2010.05.026.

[31]

A. Wan and J. Wei, Hopf bifurcation analysis of a food-limited population model with delay, Nonlinear Anal. Real World Appl., 11 (2010), 1087-1095.  doi: 10.1016/j.nonrwa.2009.01.052.

[32]

J. WangL. Zhou and Y. Tang, Asymptotic periodicity of a food-limited diffusive population model with time-delay, J. Math. Anal. Appl., 313 (2006), 381-399.  doi: 10.1016/j.jmaa.2005.03.036.

[33]

G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defence, SIAM J. Appl. Math., 48 (1988), 592-606.  doi: 10.1137/0148033.

[34]

D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator prey systems with constant rate harvesting, Fields Inst. Commun., 21 (1999), 493-506. 

[35]

J. Zhou and J. Shi, The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie-Gower predator-prey model with Holling-type II functional responses, J. Math. Anal. Appl., 405 (2013), 618-630.  doi: 10.1016/j.jmaa.2013.03.064.

show all references

References:
[1]

N. D. Barlow, Harvesting models for resource-limited populations, N. Z. J. Ecol., 10 (1987), 129-133. 

[2]

G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, Ginn, Boston, 1982.

[3]

R. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plan, Selecta. Math. Soviet., 1 (1981), 373-388. 

[4]

R. Bogdanov, Versal deformations of a singular point on the plan in the case of zero eigenvalues, Selecta. Math. Soviet., 1 (1981), 389-421. 

[5]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, Berlin, 251, 1982.

[6]

S. N. Chow and H. D. Zhang, The qualitative analysis of two species predator-prey model with Holling's type III functional response, Appl. Math. Mech., 7 (1986), 73-80.  doi: 10.1007/BF01896254.

[7]

X. Dou and Y. Li, Almost periodic solution for a food-limited population model with delay and feedback control, Int. J. Comput. Math. Sci., 5 (2011), 174-179. 

[8]

K. GopalsamyM. R. S. Kulenovic and G. Ladas, Time lags in a food-limited population model, Appl. Anal., 31 (1988), 225-237.  doi: 10.1080/00036818808839826.

[9]

K. GopalsamyM. R. S. Kulenovic and G. Ladas, Environmental periodicity and time delays in a food-limited population model, J. Math. Anal. Appl., 147 (1990), 545-555.  doi: 10.1016/0022-247X(90)90369-Q.

[10]

R. P. Gupta and P. Chandra, Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295.  doi: 10.1016/j.jmaa.2012.08.057.

[11]

T. G. Hallam and J. T. De Luna, Effects of toxicants on populations: A qualitative approach III. Environmental and food chain pathways, J. Theor. Biol., 109 (1984), 411-429.  doi: 10.1016/S0022-5193(84)80090-9.

[12]

J. HuangY. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Disc. Cont. Dyn. Syst. B, 18 (2013), 2101-2121.  doi: 10.3934/dcdsb.2013.18.2101.

[13]

D. JiangN. Shi and Y. Zhao, Existence, uniqueness, and global stability of positive solutions to the food-limited population model with random perturbation, Math. and Comp. Modelling, 42 (2005), 651-658.  doi: 10.1016/j.mcm.2004.03.011.

[14]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Appl. Math. Sciences, Springer-Verlag, New York, 112, 2004. doi: 10.1007/978-1-4757-3978-7.

[15]

B. LeardC. Lewis and J. Rebaza, Dynamics of ratio-dependent predator prey models with non constant harvesting, Disc. Cont. Dyn. Syst. S, 1 (2008), 303-315.  doi: 10.3934/dcdss.2008.1.303.

[16]

P. Lenzini and J. Rebaza, Nonconstant predator harvesting on ratio-dependent predator-prey models, Appl. Math. Sciences, 4 (2010), 791-803. 

[17]

D. Li and M. Liu, Invariant measure of a stochastic food–limited population model with regime switching, Math. Comput. Simul., 178 (2020), 16-26.  doi: 10.1016/j.matcom.2020.06.003.

[18]

Z. Li and M. He, Hopf bifurcation in a delayed food-limited model with feedback control, Nonlinear Dyn., 76 (2014), 1215-1224.  doi: 10.1007/s11071-013-1205-0.

[19]

W. LiuC. Fu and B. Chen, Hopf bifurcation and center stability for a predator–prey biological economic model with prey harvesting, Commun. Nonlinear. Sci. Numer. Simulat., 17 (2012), 3989-3998.  doi: 10.1016/j.cnsns.2012.02.025.

[20]

P. LiuJ. Shi and Y. Wang, Periodic solutions of a logistic type population model with harvesting, J. Math. Anal. Appl., 369 (2010), 730-735.  doi: 10.1016/j.jmaa.2010.04.027.

[21]

O. P. Misra and R. Babu, A model for the effect of toxicant on a three species food chain system with Food–Limited growth of prey population, Glob. J. Math. Anal., 2 (2014), 120-145.  doi: 10.14419/gjma.v2i3.2990.

[22]

P. Panja and S. K. Mondal, Stability analysis of coexistence of three species prey-predator model, Nonlinear Dyn., 81 (2015), 373-382.  doi: 10.1007/s11071-015-1997-1.

[23]

P. PanjaS. K. Mondal and D. K. Jana, Effect of toxicants of phytoplankton-zooplankton-fish dynamics and harvesting, Chaos Soliton Fract., 104 (2017), 389-399.  doi: 10.1016/j.chaos.2017.08.036.

[24]

P. Panja, Plankton population and cholera disease transmission: A mathematical modeling study, Int. J. Bifurcat. Chaos, 30 (2020), 2050054(16). doi: 10.1142/S0218127420500546.

[25]

L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1996. doi: 10.1007/978-1-4684-0249-0.

[26]

E. C. Pielou, An Introduction to Mathematical Ecology, Wiley, New York, 1969.

[27]

F. E. Smith, Population dynamics in Daphnia magna and a new model for population growth, Ecology, 44 (1963), 651-663.  doi: 10.2307/1933011.

[28]

J. W.-H. So and J. S. Yu, On the uniform stability for a food-limited population model with time delay, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 991-1002.  doi: 10.1017/S0308210500022605.

[29]

S. Tang and L. Chen, Global attractivity in a food-limited population model with impulsive effects, J. Math. Anal. Appl., 292 (2004), 211-221.  doi: 10.1016/j.jmaa.2003.11.061.

[30]

Y. TaoX. Wang and X. Song, Effect of prey refuge on a harvested predator-prey model with generalized functional response, Commun. Nonlinear. Sci. Numer. Simulat., 16 (2011), 1052-1059.  doi: 10.1016/j.cnsns.2010.05.026.

[31]

A. Wan and J. Wei, Hopf bifurcation analysis of a food-limited population model with delay, Nonlinear Anal. Real World Appl., 11 (2010), 1087-1095.  doi: 10.1016/j.nonrwa.2009.01.052.

[32]

J. WangL. Zhou and Y. Tang, Asymptotic periodicity of a food-limited diffusive population model with time-delay, J. Math. Anal. Appl., 313 (2006), 381-399.  doi: 10.1016/j.jmaa.2005.03.036.

[33]

G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defence, SIAM J. Appl. Math., 48 (1988), 592-606.  doi: 10.1137/0148033.

[34]

D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator prey systems with constant rate harvesting, Fields Inst. Commun., 21 (1999), 493-506. 

[35]

J. Zhou and J. Shi, The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie-Gower predator-prey model with Holling-type II functional responses, J. Math. Anal. Appl., 405 (2013), 618-630.  doi: 10.1016/j.jmaa.2013.03.064.

Figure 1.  Zero-growth isoclines along with the interior equilibrium states of the system (6) are depicted in this figure. Here, red and green color curves represent the plankton and fish nullclines respectively. For the values of the parameters $ r = 0.1 $, $ k = 1 $, $ a = 0.3 $, $ m = 0.03 $, $ d = 0.07 $, $ n = 0.23 $, $ c = 0.4 $, we get two equilibrium (A), then the two unique equilibrium collide to reach unique equilibrium (B) and then no equilibrium (C), for $ h = 0.09, 0.1116729526 $ and $ 0.13 $ respectively. If we take $ r = 0.25 $, $ k = 10 $, $ a = 0.2 $, $ m = 0.5 $, $ d = 0.9 $, $ h = 0.2 $, $ c = 0.2 $, $ n = 0.2 $, then the system has unique interior equilibrium point (D)
Figure 2.  In (A) $ L_{2*} $ is stable for $ r = 0.1 $, $ k = 1 $, $ a = 0.3 $, $ m = 0.03 $, $ d = 0.07 $, $ h = 0.11 $, $ c = 0.4 $ and $ n = 0.25 $, if we increase the bifurcation parameter $ n $ to $ n = n^{[H]} = 0.3068476027 $, the system possesses a periodic solution about $ L_{2*} $ which is represented in (B) and finally for $ n = 0.307 $ and keeping rest of the parameters same, the periodic solution collide with the saddle point $ L_{1*} $ to give a homoclinic orbit about $ L_{2*} $ which is shown in (C) where as $ L_{1*} $ remains saddle in each case. Figure (D) represents the bifurcation diagram for $ r = 0.1 $, $ k = 1 $, $ a = 0.3 $, $ m = 0.03 $, $ d = 0.07 $, $ h = 0.11 $ and $ c = 0.4 $ with respect to the bifurcation parameter $ n $
Figure 3.  (A) Bifurcation diagram w.r.t. parameter $ a $ in case of two interior equilibrium. (B) Bifurcation diagram w.r.t. parameter $ a $ in case of one interior equilibrium
Figure 4.  (A) For $ r = 0.1 $, $ k = 1 $, $ a = 0.3 $, $ m = 0.03 $, $ n = 0.23 $, $ d = 0.07 $, $ c = 0.4 $ and $ h = 0.09 $ there are two interior equilibrium points of the system (6). (B) The model (6) attains a unique instantaneous equilibrium point for $ h = h^{[SN]} = 0.1116729526 $ and keeping rest of the values same. (C) For $ h = 0.13 $ and keeping other parameter fixed the system (6) has no equilibrium point. (D) Represents bifurcation curves in $ \lambda_1 $$ \lambda_2 $-plane
Figure 6.  (A) The components of interior equilibrium are plotted to show their stability. The red curves stand for stable branch and green curves stand for unstable branch. (B) This figure depicts the P-components of $ L_1 $ and $ L_{2*} $ for $ r = 2.0 $, $ k = 22 $, $ a = 0.01 $, $ m = 0.02 $, $ n = 0.1 $, $ d = 0.08 $, $ h = 0.05 $, $ c = 0.2 $ as d varies. When $ d<1.51 $ the equilibrium $ L_1 $ is stable while the interior equilibrium $ L_{2*} $ is unstable and when $ d>1.51 $, the interior equilibrium $ L_{2*} $ is stable while the equilibrium $ L_1 $ is unstable
Figure 5.  (A) For $ r = 0.1 $, $ k = 1 $, $ a = 0.3 $, $ m = 0.03 $, $ d = 0.07 $, $ c = 0.4 $, $ h = 0.1116729526 $, $ n = 0.2313139014 $ and $ h = 0.13 $ the system attains an unique instantaneous equilibrium point which is a cusp of co-dimension 2. (B) Here, the trace and determinant of the variational matrix at $ (N_{2*},P_{2*}) $ are plotted in green and red color where their intersection is $ (n_0, h_0) = (0.2313139014, 0.1116729526) $
Figure 7.  In (A) there is a stable point for $ r = 0.25 $, $ k = 10 $, $ a = 0.2 $, $ m = 0.5 $, $ d = 0.9 $, $ h = 0.2 $, $ c = 0.2 $ and $ n = 0.194 $, if we increase the bifurcation parameter $ n $ to $ n = n^{[H]} = 0.1952313043 $, the system possesses a periodic solution which is shown in figure (B) and finally for $ n = 0.23 $ and keeping rest of the parameters same, the periodic solution collide with the saddle point to give a homoclinic orbit which is given in figure (C) and $ L_{1*} $ remains saddle in each case. (D) represents the bifurcation diagram for $ r = 0.25 $, $ k = 10 $, $ a = 0.2 $, $ m = 0.5 $, $ d = 0.9 $, $ h = 0.2 $ and $ c = 0.2 $ with respect to the bifurcation parameter $ n $
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