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January  2021, 20(1): 427-448. doi: 10.3934/cpaa.2020275

Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food

1. 

College of Mathematics, Sichuan University, Sichuan 610065, China

2. 

College of Applied Mathematics, Chengdu University of Information Technology, Sichuan 610225, China

3. 

School of Mathematical Sciences, Sichuan Normal University, Sichuan 610068, China

* Corresponding author

Received  March 2020 Revised  September 2020 Published  January 2021 Early access  November 2020

Fund Project: The first author is supported by NSF grant 11901408 and 11711306

The article aims to investigate the dynamic transitions of a toxin-producing phytoplankton zooplankton model with additional food in a 2D-rectangular domain. The investigation is based on the dynamic transition theory for dissipative dynamical systems. Firstly, we verify the principle of exchange of stability by analysing the corresponding linear eigenvalue problem. Secondly, by using the technique of center manifold reduction, we determine the types of transitions. Our results imply that the model may bifurcate two new steady state solutions, which are either attractors or saddle points. In addition, the model may also bifurcate a new periodic solution as the control parameter passes critical value. Finally, some numerical results are given to illustrate our conclusions.

Citation: Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure and Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275
References:
[1]

S. ChakrabortyP. TiwariA. Misra and J. Chattopadhyay, Spatial dynamics of a nutrient-phytoplankton system with toxic effect on phytoplankton, Math. Bio., 264 (2015), 94-100.  doi: 10.1016/j.mbs.2015.03.010.

[2]

H. DijkstraT. SengulJ. Shen and S. Wang., Dynamic transition of quasi-geostrophic channel flow, SIAM J. Appl. Math., 75 (2015), 2361-2378.  doi: 10.1137/15M1008166.

[3]

M. Garvie, Finite-difference schemes for reaction-diffusion equations modelling predator-prey interactions in matlab, B. Math. Biol., 69 (2007), 931-956.  doi: 10.1007/s11538-006-9062-3.

[4]

R. Han and B. Dai, Spatiotemporal pattern formation and selection induced by nonlinear cross-diffusion in a toxic-phytoplankton-zooplankton model with allee effect, Nonlinear Anal. Real World Appl., 45 (2019), 822-853.  doi: 10.1016/j.nonrwa.2018.05.018.

[5]

D. HanM. Hernandez and Q. Wang, On the instabilities and transitions of the Western boundary current, Commun. Comput. Phys., 26 (2019), 35-56.  doi: 10.4208/cicp.oa-2018-0066.

[6]

D. HanM. Hernandez and Q. Wang, Dynamical transitions of a low-dimensional model for rayleigh-b$\acute{e}$nard convection under a vertical magnetic field, Chaos Solitons Fractals, 114 (2018), 370-380.  doi: 10.1016/j.chaos.2018.06.027.

[7]

C. Hsia, C. Lin, T. Ma and S. Wang, Tropical atmospheric circulations with humidity effects, Proc. A., 471 (2015), 20140353. doi: 10.1098/rspa.2014.0353.

[8]

C. HsiaT. Ma and S. Wang, Rotating boussinesq equations: dynamic stability and transition, Discrete Contin Dyn. Syst. Ser. A, 28 (2010), 99-130.  doi: 10.3934/dcds.2010.28.99.

[9]

S. JangJ. Baglama and W. Li, Dynamics of phytoplankton-zooplankton systems with toxin producing phytoplankton, Appl. Math. Comput., 227 (2014), 717-740.  doi: 10.1016/j.amc.2013.11.051.

[10]

Z. Jiang and T. Zhang, Dynamical analysis of a reaction-diffusion phytoplankton-zooplankton system with delay, Chaos Solitons Fractals, 104 (2017), 693-704.  doi: 10.1016/j.chaos.2017.09.030.

[11]

C. KieuT. SengulQ. Wang and D. Yan, On the Hopf (double Hopf) bifurcations and transitions of two layer western boundary currents, Commun. Nonlinear Sci. Numer. Simul., 65 (2018), 196-215.  doi: 10.1016/j.cnsns.2018.05.010.

[12]

H. Liu, T. Sengul and S. Wang, Dynamic transition for quasilinear system and Cahn-Hilliard equation with onsager mobility, J. Math. Phys., 53 (2012), 023518, 31. doi: 10.1063/1.3687414.

[13]

H. LiuT. SengulS. Wang and P. Zhang, Dynamic transition and pattern formation for a Cahn-Hilliard model with long-range repulsive interactions, Commun. Math. Sci., 13 (2015), 1289-1315.  doi: 10.4310/CMS.2015.v13.n5.a10.

[14]

C. Lu, Y. Mao, T. Sengul and Q. Wang, On the spectral instability and bifurcation of the 2D-quasi-geostrophic potential vorticity equation with a generalized Kolmogorov forcing, Phys. D, 403 (2020), 132296. doi: 10.1016/j.physd.2019.132296.

[15]

Y. Mao, Dynamic transitions of the fitzhugh-nagumo equations on a finite domain, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3935-3947.  doi: 10.3934/dcdsb.2018118.

[16]

T. Ma and S. Wang, Dynamic transition for thermohaline circulation, Phys. D, 239 (2010), 167-189.  doi: 10.1016/j.physd.2009.10.014.

[17]

T. Ma and S. Wang, Phase Transition Dynamics, Springer, New York, 2014. doi: 10.1007/978-1-4614-8963-4.

[18]

T. Ma and S. Wang, Dynamic transition and pattern formation for chemotactic systems, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2809-2835.  doi: 10.3934/dcdsb.2014.19.2809.

[19]

Y. MaoD. Yan and C. Lu, Dynamic transitions and stability for the acetabularia whorl formation, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5989-6004.  doi: 10.3934/dcdsb.2019117.

[20]

A. MedvinskyS. PetrovskiiI. TikhonovaH. Malchow and B. Liu, Spatiotemporal complexity of plankton and fish dynamics, SIAM Rev., 3 (2002), 311-370.  doi: 10.1137/S0036144502404442.

[21]

Z. Pan, T. Sengul and Q. Wang, On the viscous instabilities and transitions of two-layer model with a layered topography, Commun. Nonlinear Sci. Numer. Simul., 80 (2020), 104978. doi: 10.1016/j.cnsns.2019.104978.

[22]

F. Rao, Spatiotemporal dynamics in a reaction-diffusion toxic-phytoplankton zooplankton model, J. Stat. Mech. Theory Exp., (2013), 08014. doi: 10.1088/1742-5468/2013/08/p08014.

[23]

T. Saha and M. Bandyopahyay, 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.

[24]

Q. Song, R. Yang, C. Zhang and L. Tang, Bifurcation Analysis of a Diffusive Predator-Prey Model with Monod-Haldane Functional Response, Int. J. Bifurcat. Chaos, 29 (2019), 1950152. doi: 10.1142/S0218127419501529.

[25]

W. WangS. LiuD. Tian and D. Wang, Pattern dynamics in a toxin-producing phytoplankton-zooplankton model with additional food, Nonlinear Dyn., 94 (2018), 211-228. 

[26]

R. Yang and C. Zhang, Dynamics in a diffusive predator-prey system with a constant prey refuge and delay, Nonlinear Anal. Real World Appl., 31 (2016), 1-22. doi: 10.1016/j.nonrwa.2016.01.005.

[27]

X. YuS. Yuan and T. Zhang, Survival and ergodicity of a stochastic phytoplankton-zooplankton model with toxin-producing phytoplankton in an impulsive polluted environment, Appl. Math. Comput., 347 (2019), 249-264.  doi: 10.1016/j.amc.2018.11.005.

[28]

D. Zhang and R. Liu, Dynamical transition for S-K-T biological competing model with cross-diffusion, Math. Methods Appl. Sci., 41 (2018), 4641-4658.  doi: 10.1002/mma.4919.

[29]

W. Zheng and J. Sugie, Global asymptotic stability and equiasymptotic stability for time-varying phytoplankton-zooplankton-fish system, Nonlinear Anal. Real World Appl., 46 (2019), 116-136.  doi: 10.1016/j.nonrwa.2018.09.015.

show all references

References:
[1]

S. ChakrabortyP. TiwariA. Misra and J. Chattopadhyay, Spatial dynamics of a nutrient-phytoplankton system with toxic effect on phytoplankton, Math. Bio., 264 (2015), 94-100.  doi: 10.1016/j.mbs.2015.03.010.

[2]

H. DijkstraT. SengulJ. Shen and S. Wang., Dynamic transition of quasi-geostrophic channel flow, SIAM J. Appl. Math., 75 (2015), 2361-2378.  doi: 10.1137/15M1008166.

[3]

M. Garvie, Finite-difference schemes for reaction-diffusion equations modelling predator-prey interactions in matlab, B. Math. Biol., 69 (2007), 931-956.  doi: 10.1007/s11538-006-9062-3.

[4]

R. Han and B. Dai, Spatiotemporal pattern formation and selection induced by nonlinear cross-diffusion in a toxic-phytoplankton-zooplankton model with allee effect, Nonlinear Anal. Real World Appl., 45 (2019), 822-853.  doi: 10.1016/j.nonrwa.2018.05.018.

[5]

D. HanM. Hernandez and Q. Wang, On the instabilities and transitions of the Western boundary current, Commun. Comput. Phys., 26 (2019), 35-56.  doi: 10.4208/cicp.oa-2018-0066.

[6]

D. HanM. Hernandez and Q. Wang, Dynamical transitions of a low-dimensional model for rayleigh-b$\acute{e}$nard convection under a vertical magnetic field, Chaos Solitons Fractals, 114 (2018), 370-380.  doi: 10.1016/j.chaos.2018.06.027.

[7]

C. Hsia, C. Lin, T. Ma and S. Wang, Tropical atmospheric circulations with humidity effects, Proc. A., 471 (2015), 20140353. doi: 10.1098/rspa.2014.0353.

[8]

C. HsiaT. Ma and S. Wang, Rotating boussinesq equations: dynamic stability and transition, Discrete Contin Dyn. Syst. Ser. A, 28 (2010), 99-130.  doi: 10.3934/dcds.2010.28.99.

[9]

S. JangJ. Baglama and W. Li, Dynamics of phytoplankton-zooplankton systems with toxin producing phytoplankton, Appl. Math. Comput., 227 (2014), 717-740.  doi: 10.1016/j.amc.2013.11.051.

[10]

Z. Jiang and T. Zhang, Dynamical analysis of a reaction-diffusion phytoplankton-zooplankton system with delay, Chaos Solitons Fractals, 104 (2017), 693-704.  doi: 10.1016/j.chaos.2017.09.030.

[11]

C. KieuT. SengulQ. Wang and D. Yan, On the Hopf (double Hopf) bifurcations and transitions of two layer western boundary currents, Commun. Nonlinear Sci. Numer. Simul., 65 (2018), 196-215.  doi: 10.1016/j.cnsns.2018.05.010.

[12]

H. Liu, T. Sengul and S. Wang, Dynamic transition for quasilinear system and Cahn-Hilliard equation with onsager mobility, J. Math. Phys., 53 (2012), 023518, 31. doi: 10.1063/1.3687414.

[13]

H. LiuT. SengulS. Wang and P. Zhang, Dynamic transition and pattern formation for a Cahn-Hilliard model with long-range repulsive interactions, Commun. Math. Sci., 13 (2015), 1289-1315.  doi: 10.4310/CMS.2015.v13.n5.a10.

[14]

C. Lu, Y. Mao, T. Sengul and Q. Wang, On the spectral instability and bifurcation of the 2D-quasi-geostrophic potential vorticity equation with a generalized Kolmogorov forcing, Phys. D, 403 (2020), 132296. doi: 10.1016/j.physd.2019.132296.

[15]

Y. Mao, Dynamic transitions of the fitzhugh-nagumo equations on a finite domain, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3935-3947.  doi: 10.3934/dcdsb.2018118.

[16]

T. Ma and S. Wang, Dynamic transition for thermohaline circulation, Phys. D, 239 (2010), 167-189.  doi: 10.1016/j.physd.2009.10.014.

[17]

T. Ma and S. Wang, Phase Transition Dynamics, Springer, New York, 2014. doi: 10.1007/978-1-4614-8963-4.

[18]

T. Ma and S. Wang, Dynamic transition and pattern formation for chemotactic systems, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2809-2835.  doi: 10.3934/dcdsb.2014.19.2809.

[19]

Y. MaoD. Yan and C. Lu, Dynamic transitions and stability for the acetabularia whorl formation, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5989-6004.  doi: 10.3934/dcdsb.2019117.

[20]

A. MedvinskyS. PetrovskiiI. TikhonovaH. Malchow and B. Liu, Spatiotemporal complexity of plankton and fish dynamics, SIAM Rev., 3 (2002), 311-370.  doi: 10.1137/S0036144502404442.

[21]

Z. Pan, T. Sengul and Q. Wang, On the viscous instabilities and transitions of two-layer model with a layered topography, Commun. Nonlinear Sci. Numer. Simul., 80 (2020), 104978. doi: 10.1016/j.cnsns.2019.104978.

[22]

F. Rao, Spatiotemporal dynamics in a reaction-diffusion toxic-phytoplankton zooplankton model, J. Stat. Mech. Theory Exp., (2013), 08014. doi: 10.1088/1742-5468/2013/08/p08014.

[23]

T. Saha and M. Bandyopahyay, 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.

[24]

Q. Song, R. Yang, C. Zhang and L. Tang, Bifurcation Analysis of a Diffusive Predator-Prey Model with Monod-Haldane Functional Response, Int. J. Bifurcat. Chaos, 29 (2019), 1950152. doi: 10.1142/S0218127419501529.

[25]

W. WangS. LiuD. Tian and D. Wang, Pattern dynamics in a toxin-producing phytoplankton-zooplankton model with additional food, Nonlinear Dyn., 94 (2018), 211-228. 

[26]

R. Yang and C. Zhang, Dynamics in a diffusive predator-prey system with a constant prey refuge and delay, Nonlinear Anal. Real World Appl., 31 (2016), 1-22. doi: 10.1016/j.nonrwa.2016.01.005.

[27]

X. YuS. Yuan and T. Zhang, Survival and ergodicity of a stochastic phytoplankton-zooplankton model with toxin-producing phytoplankton in an impulsive polluted environment, Appl. Math. Comput., 347 (2019), 249-264.  doi: 10.1016/j.amc.2018.11.005.

[28]

D. Zhang and R. Liu, Dynamical transition for S-K-T biological competing model with cross-diffusion, Math. Methods Appl. Sci., 41 (2018), 4641-4658.  doi: 10.1002/mma.4919.

[29]

W. Zheng and J. Sugie, Global asymptotic stability and equiasymptotic stability for time-varying phytoplankton-zooplankton-fish system, Nonlinear Anal. Real World Appl., 46 (2019), 116-136.  doi: 10.1016/j.nonrwa.2018.09.015.

Figure 1.  The topological structure of phase portrait of continuous transition as control parameter $ \Lambda>\Lambda_{c} $
Figure 2.  The topological structure of phase portrait of continuous transition as control parameter $ \Lambda<\Lambda_{c} $
Figure 3.  The topological structure of phase portrait of jump transition as control parameter $ \Lambda<\Lambda_{c} $
Figure 4.  The topological structure of phase portrait of jump transition as control parameter $ \Lambda>\Lambda_{c} $
Figure 5.  The graph of critical parameter $ \Lambda_{c} $ and $ \lambda_{c} $ as $ n\in [0.1,1.5] $ and $ d\in [6,13] $
Figure 6.  The regions separating two types of transitions. Region A, continuous transitions from a real simple eigenvalue; Region B, jump transitions from a pair of simple complex eigenvalues
Figure 7.  The numerical solutions $ u_{1} $ and $ u_{2} $ at time T = 600. The parameter $ \lambda = 0.52 $
Figure 8.  The numerical solutions $ u_{1} $ and $ u_{2} $ at time T = 600. The parameter $ \lambda = 0.58 $
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