Analysis of a Spatiotemporal Phytoplankton Dynamics: Higher Order Stability and Pattern Formation
In this paper, for the understanding of the phytoplankton dynamics in marine ecosystem, a susceptible and an infected class of phytoplankton population is considered in spatiotemporal domain.
Here, the susceptible phytoplankton is growing logistically and the
growth of infected phytoplankton is due to the instantaneous Holling
type-II infection response function. The dynamics are studied in terms of the local and global stabilities for the system and further
explore the possibility of Hopf -bifurcation, taking the half saturation period as (i.e., ) the bifurcation parameter in temporal domain.
It is also observe that the reaction diffusion system exhibits spatiotemporal
chaos and pattern formation in phytoplankton dynamics,
which is particularly important role play for the spatially extended phytoplankton system. Also the effect of the diffusion coefficient
on the spatial system for both one and two dimensional case is obtained. Furthermore, we explore the higher-order stability analysis
of the spatial phytoplankton system for both linear and no-linear system. Finally, few numerical simulations are carried out for pattern
formation.
[1] O. Bergh, K.Y. Borsheim and G. Bratbak, High abundance of viruses is
found in aquatic environment, Nature. 340 (1989), pp. 467 - 468.
[2] S.Brenner and L. Scott, The Mathematical Theory of Finite Element
Methods, Applied Mathematics, Springer, New York, 1994.
[3] J. Chattopadhayay, R. Sarkar and S. Mandal, Toxin producing phytoplankton
may act as biology control for planktonic bloom eld study and mathematical modelling. J.Theor.Biol. 215 (2002), pp. 333 - 344.
[4] P. Ciarlet, The Finite Element Method for Elliptic Problems., Studies in
Mathematics and its Applications, North-Holland, Amsterdam, 1979.
[5] J. Dhar and A. K. Sharma, The role of viral infection in phytoplankton
dynamics with the inclusion of incubation class, Nonlinear Analysis:
Hybrid Systems. 4 (2010), pp. 9 - 15 .
[6] J.A. Furman, Marine viruses and their biogeochemical and ecological
effects, Nature. 399 (1990), pp. 541 - 548.
[7] W. Gentleman, A. Leising, B. Frost and S. Strom, J. Murray, Functional
responses for zooplankton feeding on multiple resources: A review of
assumptions and biological dynamics, Deep Sea Res. 50 (2003), pp.
2847 - 2875.
[8] C. Holling, Some characteristics of simple types of predation and
parasitism, Can. Entomol. 91 (1959), pp. 385 - 398.
[9] E. Holmes, M. Lewis, J. Banks and R. Veit, Partial differential equations
in ecology: Spatial interactions and population dynamics, Ecology. 75
(1994), no. 1, pp. 17 - 29.
[10] M. Horst, F.M. Hilker, V. Sergei and S.V. Petrovskii, Oscilation and
waves in a virally infected plankton system, Ecological complexity1.
(3)(2004), pp. 211 - 223.
[11] J. Jeschke, M. Kopp and R. Tollrian, Predator functional responses:
Discriminating between handling and digesting prey, Ecol. Monogr. 72
(2002), no. 1, pp. 95 - 112.
[12] A. Medvinsky, S. Petrovskii, I. Tikhonova and H. Malchow, Spatiotemporal
complexity of plankton and fish dynamics,SIAM Rev. 44 (2002),
no. 3, pp. 311 - 370.
[13] J. Murray, Mathematical Biology, Biomathematics Texts. Springer,
Berlin, 1993.
[14] S. Petrovskii and H. Malchow, A minimal model of pattern formation
in a prey-predator system, Math. Comput. Model. 29 (1999), pp. 49 -
63.
[15] V. Rai and G. Jayaraman, Is diffusion-induced chaos robust?, Curr. Sci.
India. 84 (2003), no. 7, pp. 925 - 929.
[16] G. Skalski and J.F. Gilliam, Functional responses with predator interference:
Viable alternatives to the Holling type II model, Ecology. 82(2001), no. 11, pp. 3083 - 3092.
[17] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren
der mathematischen Wissenschaften, Springer-Verlag, New York, 1983.
[18] C.A.Suttle, A.M. Chan and M.T. Cottrell, Infection of phytoplankton by
viruses and reduction of primary productivity, Nature. 347 (1990), pp.467 - 469.
[19] C. Suttle and A.M. Chan, Marine Cyanophages infecting oceanic and
coastal strain of Synechococcus: Abundance , morphology, cross infectivity
and growth characteristic, Mar Eco Prog.Ser. 92 (1993), pp. 99 -109.
[20] J.L. Vanetten, L.C. Lane and R.H. Meints, Viruses and virus like particles of eukaryotic algae, Microbiol Rev. 55 (1991), pp. 586 - 620.
[21] S. S. Riaz, R. Sharma, S. P. Bhattacharya and D. S. Ray, Instability and
pattern formation in reaction-diffusion systems: a higher order analysis., J. Chem. Phys.,(2007), 126, 064503.
[22] Platt, T., Local phytoplankton abundance and turbulence., Deep-Sea Res. 19 (1972), 183-187.
[23] J. Thomas, Numerical Partial Differential Equations: Finite Difference
Methods., Texts in Applied Mathematics. Springer, New York (1995).
[24] K.S. Cheng, S.B. Hsu and S. S. Lin, Some redults on global stability of a predator prey system., J.Math. Biol. 12 (1981), 115-126.
[25] Yuri A. Kuznetsov, Elements of applied bifurcation theory, Second
Edition, Springer, (1997).
[1] O. Bergh, K.Y. Borsheim and G. Bratbak, High abundance of viruses is
found in aquatic environment, Nature. 340 (1989), pp. 467 - 468.
[2] S.Brenner and L. Scott, The Mathematical Theory of Finite Element
Methods, Applied Mathematics, Springer, New York, 1994.
[3] J. Chattopadhayay, R. Sarkar and S. Mandal, Toxin producing phytoplankton
may act as biology control for planktonic bloom eld study and mathematical modelling. J.Theor.Biol. 215 (2002), pp. 333 - 344.
[4] P. Ciarlet, The Finite Element Method for Elliptic Problems., Studies in
Mathematics and its Applications, North-Holland, Amsterdam, 1979.
[5] J. Dhar and A. K. Sharma, The role of viral infection in phytoplankton
dynamics with the inclusion of incubation class, Nonlinear Analysis:
Hybrid Systems. 4 (2010), pp. 9 - 15 .
[6] J.A. Furman, Marine viruses and their biogeochemical and ecological
effects, Nature. 399 (1990), pp. 541 - 548.
[7] W. Gentleman, A. Leising, B. Frost and S. Strom, J. Murray, Functional
responses for zooplankton feeding on multiple resources: A review of
assumptions and biological dynamics, Deep Sea Res. 50 (2003), pp.
2847 - 2875.
[8] C. Holling, Some characteristics of simple types of predation and
parasitism, Can. Entomol. 91 (1959), pp. 385 - 398.
[9] E. Holmes, M. Lewis, J. Banks and R. Veit, Partial differential equations
in ecology: Spatial interactions and population dynamics, Ecology. 75
(1994), no. 1, pp. 17 - 29.
[10] M. Horst, F.M. Hilker, V. Sergei and S.V. Petrovskii, Oscilation and
waves in a virally infected plankton system, Ecological complexity1.
(3)(2004), pp. 211 - 223.
[11] J. Jeschke, M. Kopp and R. Tollrian, Predator functional responses:
Discriminating between handling and digesting prey, Ecol. Monogr. 72
(2002), no. 1, pp. 95 - 112.
[12] A. Medvinsky, S. Petrovskii, I. Tikhonova and H. Malchow, Spatiotemporal
complexity of plankton and fish dynamics,SIAM Rev. 44 (2002),
no. 3, pp. 311 - 370.
[13] J. Murray, Mathematical Biology, Biomathematics Texts. Springer,
Berlin, 1993.
[14] S. Petrovskii and H. Malchow, A minimal model of pattern formation
in a prey-predator system, Math. Comput. Model. 29 (1999), pp. 49 -
63.
[15] V. Rai and G. Jayaraman, Is diffusion-induced chaos robust?, Curr. Sci.
India. 84 (2003), no. 7, pp. 925 - 929.
[16] G. Skalski and J.F. Gilliam, Functional responses with predator interference:
Viable alternatives to the Holling type II model, Ecology. 82(2001), no. 11, pp. 3083 - 3092.
[17] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren
der mathematischen Wissenschaften, Springer-Verlag, New York, 1983.
[18] C.A.Suttle, A.M. Chan and M.T. Cottrell, Infection of phytoplankton by
viruses and reduction of primary productivity, Nature. 347 (1990), pp.467 - 469.
[19] C. Suttle and A.M. Chan, Marine Cyanophages infecting oceanic and
coastal strain of Synechococcus: Abundance , morphology, cross infectivity
and growth characteristic, Mar Eco Prog.Ser. 92 (1993), pp. 99 -109.
[20] J.L. Vanetten, L.C. Lane and R.H. Meints, Viruses and virus like particles of eukaryotic algae, Microbiol Rev. 55 (1991), pp. 586 - 620.
[21] S. S. Riaz, R. Sharma, S. P. Bhattacharya and D. S. Ray, Instability and
pattern formation in reaction-diffusion systems: a higher order analysis., J. Chem. Phys.,(2007), 126, 064503.
[22] Platt, T., Local phytoplankton abundance and turbulence., Deep-Sea Res. 19 (1972), 183-187.
[23] J. Thomas, Numerical Partial Differential Equations: Finite Difference
Methods., Texts in Applied Mathematics. Springer, New York (1995).
[24] K.S. Cheng, S.B. Hsu and S. S. Lin, Some redults on global stability of a predator prey system., J.Math. Biol. 12 (1981), 115-126.
[25] Yuri A. Kuznetsov, Elements of applied bifurcation theory, Second
Edition, Springer, (1997).
@article{"International Journal of Engineering, Mathematical and Physical Sciences:61376", author = "Randhir Singh Baghel and Joydip Dhar and Renu Jain", title = "Analysis of a Spatiotemporal Phytoplankton Dynamics: Higher Order Stability and Pattern Formation", abstract = "In this paper, for the understanding of the phytoplankton dynamics in marine ecosystem, a susceptible and an infected class of phytoplankton population is considered in spatiotemporal domain.
Here, the susceptible phytoplankton is growing logistically and the
growth of infected phytoplankton is due to the instantaneous Holling
type-II infection response function. The dynamics are studied in terms of the local and global stabilities for the system and further
explore the possibility of Hopf -bifurcation, taking the half saturation period as (i.e., ) the bifurcation parameter in temporal domain.
It is also observe that the reaction diffusion system exhibits spatiotemporal
chaos and pattern formation in phytoplankton dynamics,
which is particularly important role play for the spatially extended phytoplankton system. Also the effect of the diffusion coefficient
on the spatial system for both one and two dimensional case is obtained. Furthermore, we explore the higher-order stability analysis
of the spatial phytoplankton system for both linear and no-linear system. Finally, few numerical simulations are carried out for pattern
formation.", keywords = "Phytoplankton dynamics, Reaction-diffusion system, Local stability, Hopf-bifurcation, Global stability, Chaos, Pattern Formation, Higher-order stability analysis.", volume = "5", number = "12", pages = "2064-7", }
