Stability and Hopf Bifurcation in a Three-Component Planktonic Model with Spatial Diffusion and Time Delay

Due to the different roles that nontoxic phytoplankton and toxin-producing phytoplankton play in the whole aquatic system, a delayed reaction-diffusion planktonic model under homogeneous Neumann boundary condition is investigated theoretically and numerically. This model describes the interactions between the zooplankton and two kinds of phytoplanktons. The long-time behavior of the model and existence of positive constant equilibrium solution are first discussed. Then, the stability of constant equilibrium solution and occurrence of Hopf bifurcation are detailed and analyzed by using the bifurcation theory. Moreover, the formulas for determining the bifurcation direction and stability of spatially bifurcating solutions are derived. Finally, some numerical simulations are performed to verify the appearance of the spatially homogeneous and nonhomogeneous periodic solutions.

[1]  J. Connolly,et al.  Model of Carbon Cycling in Planktonic Food Webs , 1995 .

[2]  Shovonlal Roy,et al.  Competing Effects of Toxin-Producing Phytoplankton on Overall Plankton Populations in the Bay of Bengal , 2006, Bulletin of mathematical biology.

[3]  J. Chattopadhyay,et al.  Role of competition in phytoplankton population for the occurrence and control of plankton bloom in the presence of environmental fluctuations , 2009 .

[4]  G. Hays,et al.  Climate change and marine plankton. , 2005, Trends in ecology & evolution.

[5]  Kamel Al-khaled,et al.  The effect of nanoparticles on plankton dynamics: A mathematical model , 2015, Biosyst..

[6]  Andrew M. Edwards,et al.  The role of higher predation in plankton population models , 2000 .

[7]  Junjie Wei,et al.  Dynamics in a diffusive plankton system with delay and toxic substances effect , 2015 .

[8]  Joydev Chattopadhyay,et al.  Mathematical modelling of harmful algal blooms supported by experimental findings , 2004 .

[9]  Junjie Wei,et al.  Persistence, Turing Instability and Hopf Bifurcation in a Diffusive Plankton System with Delay and Quadratic Closure , 2016, Int. J. Bifurc. Chaos.

[11]  Horst Malchow,et al.  Spatiotemporal Complexity of Plankton and Fish Dynamics , 2002, SIAM Rev..

[12]  B Mukhopadhyay,et al.  Role of gestation delay in a plankton-fish model under stochastic fluctuations. , 2008, Mathematical biosciences.

[13]  Dongwei Huang,et al.  Modelling algal densities in harmful algal blooms (HAB) with stochastic dynamics , 2008 .

[14]  R. Sarkar,et al.  Role of two toxin-producing plankton and their effect on phytoplankton-zooplankton system--a mathematical study supported by experimental findings. , 2005, Bio Systems.

[15]  T. K. Kar,et al.  Modeling and analysis of a marine plankton system with nutrient recycling and diffusion , 2015, Complex..

[16]  Tongqian Zhang,et al.  Dynamical analysis of a reaction-diffusion phytoplankton-zooplankton system with delay , 2017 .

[17]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[19]  Andrew M. Edwards,et al.  Generic dynamics of a simple plankton population model with a non-integer exponent of closure , 2001 .

[20]  Ming Liu,et al.  A diffusive toxin producing phytoplankton model with maturation delay and three-dimensional patch , 2017, Comput. Math. Appl..

[21]  Xia Liu,et al.  Spatio-temporal dynamics near the steady state of a planktonic system , 2018, Comput. Math. Appl..

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

[23]  Horst Malchow.  Nonlinear plankton dynamics and pattern formation in an ecohydrodynamic model system , 1996 .

[24]  K. Richards.  Viral infections of oceanic plankton blooms. , 2017, Journal of theoretical biology.

[25]  Junjie Wei,et al.  Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays , 2005 .