Analysis of a toxin producing phytoplankton–zooplankton interaction with Holling IV type scheme and time delay

The present paper aims to study the interaction of toxin producing phytoplankton (TPP)–zooplankton (a prey–predator interaction) and its role in plankton ecology. The delay in the zooplankton predation is considered and its effect on the overall dynamic of phytoplankton–zooplankton interaction is studied. Moreover, Holling IV type response function is used for zooplankton grazing to account for the effect of toxication by the TPP population. It is shown that time delay can destabilise the given system and induce oscillation in population due to Hopf-bifurcation. Further stability of the bifurcating periodic solution is determined by using normal form theory and centre manifold arguments. Some numerical simulations are executed to validate the analytical findings.

[1]  Edward J. Buskey,et al.  Effects of the Texas (USA) \'brown tide\' alga on planktonic grazers , 1995 .

[2]  H. I. Freedman,et al.  Models of competition in the chemostat with instantaneous and delayed nutrient recycling , 1993 .

[3]  S. Ray,et al.  Effect of delay on nutrient cycling in phytoplankton-zooplankton interactions in estuarine system , 2008 .

[4]  J. Wanzenböck,et al.  How plankton copepods avoid fish predation: From individual responses to variations of the life cycle , 2006, Journal of Ichthyology.

[5]  G. Hallegraeff.  A review of harmful algal blooms and their apparent global increase , 1993 .

[6]  Nurul Huda Gazi,et al.  Effect of time delay on a harvested predator-prey model , 2008 .

[7]  Xue-Zhong He,et al.  Global stability in chemostat-type plankton models with delayed nutrient recycling , 1998 .

[8]  Sue Ann Campbell,et al.  A closed NPZ model with delayed nutrient recycling , 2014, Journal of mathematical biology.

[9]  F. C. Hansen.  Trophic interactions between zooplankton andPhaeocystis cf.globosa , 1995, Helgoländer Meeresuntersuchungen.

[10]  M. Imran,et al.  Dynamical analysis of a delay model of phytoplankton–zooplankton interaction , 2012 .

[11]  J. Brindley,et al.  Excitability of an age-structured plankton ecosystem , 1999, Journal of mathematical biology.

[12]  Shigui Ruan,et al.  Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling , 1993 .

[13]  Haihong Liu,et al.  Hopf bifurcations in a predator-prey system of population allelopathy with a discrete delay and a distributed delay , 2012 .

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

[15]  Teresa Faria,et al.  Stability and Bifurcation for a Delayed Predator–Prey Model and the Effect of Diffusion☆ , 2001 .

[16]  Yong Wang,et al.  Stability and global Hopf bifurcation in toxic phytoplankton–zooplankton model with delay and selective harvesting , 2013, Nonlinear Dynamics.

[17]  J. Ives,et al.  Possible mechanisms underlying copepod grazing responses to levels of toxicity in red tide dinoflagellates , 1987 .

[18]  P. Hartman.  Ordinary Differential Equations , 1965 .

[19]  N. Macdonald.  Biological Delay Systems: Linear Stability Theory , 1989 .

[20]  Amit Sharma,et al.  The dynamic of plankton-nutrient interaction with delay , 2014, Appl. Math. Comput..

[21]  Joydev Chattopadhyay,et al.  Nutrient-limited toxin production and the dynamics of two phytoplankton in culture media: A mathematical model , 2008 .

[22]  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.

[23]  Mimmo Iannelli,et al.  Stability and Bifurcation for a Delayed PredatorPrey Model and the Effect of Diffusion , 2001 .

[24]  Jens C. Nejstgaard,et al.  Predation by copepods upon natural populations of Phaeocystis pouchetii as a function of the physiological state of the prey , 1990 .

[25]  Yang Kuang,et al.  Geometric Stability Switch Criteria in Delay Differential Systems with Delay Dependent Parameters , 2002, SIAM J. Math. Anal..

[26]  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.

[27]  Samrat Chatterjee,et al.  Role of toxin and nutrient for the occurrence and termination of plankton bloom - Results drawn from field observations and a mathematical model , 2007, Biosyst..

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

[29]  Torkel Gissel Nielsen,et al.  Effects of a Chrysochromulina polylepis subsurface bloom on the planktonic community , 1990 .

[30]  Peng Han,et al.  Global dynamics of a nutrient-plankton system in the water ecosystem , 2013, Appl. Math. Comput..

[31]  M. Grasso.  Ecological–economic model for optimal mangrove trade off between forestry and fishery production: comparing a dynamic optimization and a simulation model , 1998 .

[32]  T. Smayda,et al.  What is a bloom? A commentary , 1997 .

[33]  Y. Kuang.  Delay Differential Equations: With Applications in Population Dynamics , 2012 .

[34]  Junjie Wei,et al.  Stability and bifurcation in a two harmful phytoplankton–zooplankton system ☆ , 2009 .

[35]  J. Baglama,et al.  Nutrient-plankton models with nutrient recycling , 2005 .

[36]  R. Bhattacharyya,et al.  Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity , 2006 .

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

[38]  Jim M Cushing,et al.  Integrodifferential Equations and Delay Models in Population Dynamics , 1977 .

[39]  K. Gopalsamy.  Stability and Oscillations in Delay Differential Equations of Population Dynamics , 1992 .