Survival and ergodicity of a stochastic phytoplankton-zooplankton model with toxin-producing phytoplankton in an impulsive polluted environment

Abstract To theoretically address the effects of random environmental change on the growth of plankton, we propose a stochastic phytoplankton–zooplankton model with toxin-producing phytoplankton and Markov switching in an impulsive polluted environment. We then analyse the extinction and persistence in mean, including weak persistence and non-persistence. By constructing Lyapunov function, we can show the model can be positive recurrent or ergodic. To conclude our study, we carry out some simulations, indicating that environmental fluctuations and exogenous toxicant input have a great influence on the survival fate of plankton.

[1]  Tonghua Zhang,et al.  Average break-even concentration in a simple chemostat model with telegraph noise , 2018, Nonlinear Analysis: Hybrid Systems.

[2]  M. Mimura,et al.  Avoiding toxic prey may promote harmful algal blooms , 2015 .

[3]  S. Petrovskii,et al.  Mathematical Modelling of Plankton–Oxygen Dynamics Under the Climate Change , 2015, Bulletin of mathematical biology.

[4]  Malay Bandyopadhyay,et al.  Dynamical analysis of toxin producing Phytoplankton-Zooplankton interactions , 2009 .

[5]  Qingling Zhang,et al.  Dynamic analysis of a hybrid bioeconomic plankton system with double time delays and stochastic fluctuations , 2018, Appl. Math. Comput..

[6]  Malay Banerjee,et al.  Modelling of phytoplankton allelopathy with Monod-Haldane-type functional response - A mathematical study , 2009, Biosyst..

[7]  Yun Kang,et al.  A stochastic SIRS epidemic model with nonlinear incidence rate , 2017, Appl. Math. Comput..

[8]  Chuanzhi Bai,et al.  Analysis of a stochastic tri-trophic food-chain model with harvesting , 2016, Journal of Mathematical Biology.

[9]  Bernd Blasius,et al.  A model for seasonal phytoplankton blooms. , 2005, Journal of theoretical biology.

[10]  G. Yin,et al.  Hybrid Switching Diffusions: Properties and Applications , 2009 .

[11]  S. Mondal,et al.  Stability analysis of coexistence of three species prey–predator model , 2015 .

[12]  Sanling Yuan,et al.  An analogue of break-even concentration in a simple stochastic chemostat model , 2015, Appl. Math. Lett..

[13]  R. Sarkar,et al.  Occurrence of planktonic blooms under environmental fluctuations and its possible control mechanism--mathematical models and experimental observations. , 2003, Journal of theoretical biology.

[14]  R. Sarkar,et al.  A delay differential equation model on harmful algal blooms in the presence of toxic substances. , 2002, IMA journal of mathematics applied in medicine and biology.

[15]  H. Hoppe,et al.  Bacterial growth and primary production along a north–south transect of the Atlantic Ocean , 2002, Nature.

[16]  Xiong Li,et al.  Stability and Bifurcation in a Stoichiometric Producer-Grazer Model with Knife Edge , 2016, SIAM J. Appl. Dyn. Syst..

[17]  Sanling Yuan,et al.  Persistence and ergodicity of a stochastic single species model with Allee effect under regime switching , 2018, Commun. Nonlinear Sci. Numer. Simul..

[18]  Jianhai Bao,et al.  Permanence and Extinction of Regime-Switching Predator-Prey Models , 2015, SIAM J. Math. Anal..

[19]  Tonghua Zhang,et al.  The effects of toxin-producing phytoplankton and environmental fluctuations on the planktonic blooms , 2018 .

[20]  Xuerong Mao,et al.  Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching , 2011 .

[21]  Huaiping Zhu,et al.  Analysis of a stochastic model for algal bloom with nutrient recycling , 2016 .

[22]  Qun Liu,et al.  Periodic Solution and Stationary Distribution of Stochastic Predator–Prey Models with Higher-Order Perturbation , 2018, J. Nonlinear Sci..

[23]  Lei Zhang,et al.  Dynamics of a two-species Lotka-Volterra competition system in a polluted environment with pulse toxicant input , 2009, Appl. Math. Comput..

[24]  A. M. Edwards,et al.  Zooplankton mortality and the dynamical behaviour of plankton population models , 1999, Bulletin of mathematical biology.

[25]  Li Wu,et al.  Dynamics of phytoplankton-zooplankton systems with toxin producing phytoplankton , 2014, Appl. Math. Comput..

[26]  S. Mondal,et al.  Effects of toxicants on Phytoplankton-Zooplankton-Fish dynamics and harvesting , 2017 .

[27]  Paul G. Falkowski,et al.  A consumer's guide to phytoplankton primary productivity models , 1997 .

[28]  Qimin Zhang,et al.  The effect of Lévy noise on the survival of a stochastic competitive model in an impulsive polluted environment , 2016 .

[29]  John Brindley,et al.  Ocean plankton populations as excitable media , 1994 .

[30]  Z. Teng,et al.  Threshold Behavior in a Class of Stochastic SIRS Epidemic Models With Nonlinear Incidence , 2015 .

[31]  G. Yin,et al.  Switching diffusion logistic models involving singularly perturbed Markov chains: Weak convergence and stochastic permanence , 2017 .

[32]  Gang George Yin,et al.  Asymptotic Properties of Hybrid Diffusion Systems , 2007, SIAM J. Control. Optim..

[33]  Shengqiang Liu,et al.  Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching , 2017, 1707.06380.

[34]  M. Fan,et al.  The dynamics of temperature and light on the growth of phytoplankton. , 2015, Journal of theoretical biology.

[35]  Tonghua Zhang,et al.  Dynamics analysis and numerical simulations of a stochastic non-autonomous predator–prey system with impulsive effects , 2017 .

[36]  Dianli Zhao,et al.  Study on the threshold of a stochastic SIR epidemic model and its extensions , 2016, Commun. Nonlinear Sci. Numer. Simul..

[37]  Adel Settati,et al.  Stationary distribution of stochastic population systems under regime switching , 2014, Appl. Math. Comput..

[38]  Zhen Jin,et al.  Weak average persistence and extinction of a predator–prey system in a polluted environment with impulsive toxicant input☆ , 2007 .

[39]  B. Moss,et al.  Ecology of Fresh Waters: Man and Medium , 1989 .

[40]  Xin-You Meng,et al.  Bifurcation and Control in a Singular Phytoplankton-Zooplankton-Fish Model with Nonlinear Fish Harvesting and Taxation , 2018, Int. J. Bifurc. Chaos.

[41]  M. Lehtiniemi,et al.  Feeding, reproduction and toxin accumulation by the copepods Acartia bifilosa and Eurytemora affinis in the presence of the toxic cyanobacterium Nodularia spumigena , 2003 .

[42]  S Mandal,et al.  Toxin-producing plankton may act as a biological control for planktonic blooms--field study and mathematical modelling. , 2002, Journal of theoretical biology.

[43]  Shengqiang Liu,et al.  The Evolutionary Dynamics of Stochastic Epidemic Model with Nonlinear Incidence Rate , 2015, Bulletin of Mathematical Biology.

[44]  Huaiping Zhu,et al.  Complex dynamics of a nutrient-plankton system with nonlinear phytoplankton mortality and allelopathy , 2016 .

[45]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[46]  Sanling Yuan,et al.  Stochastic periodic solution of a non-autonomous toxic-producing phytoplankton allelopathy model with environmental fluctuation , 2017, Commun. Nonlinear Sci. Numer. Simul..