Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 21, pp. 1–12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu BIFURCATION AND SPATIAL PATTERN FORMATION IN SPREADING OF DISEASE WITH INCUBATION PERIOD IN A PHYTOPLANKTON DYNAMICS RANDHIR SINGH BAGHEL, JOYDIP DHAR, RENU JAIN Abstract. In this article, we propose a three dimensional mathematical model of phytoplankton dynamics with the help of reaction-diffusion equations that studies the bifurcation and pattern formation mechanism. We provide an ana- lytical explanation for understanding phytoplankton dynamics with three pop- ulation classes: susceptible, incubated, and infected. This model has a Holling type II response function for the population transformation from susceptible to incubated class in an aquatic ecosystem. Our main goal is to provide a qual- itative analysis of Hopf bifurcation mechanisms, taking death rate of infected phytoplankton as bifurcation parameter, and to study further spatial patterns formation due to spatial diffusion. Here analytical findings are supported by the results of numerical experiments. It is observed that the coexistence of all classes of population depends on the rate of diffusion. Also we obtained the time evaluation pattern formation of the spatial system. 1. Introduction It is well known that the phytoplankton and zooplankton are single cell or- ganisms that drift with the currents on the surface of open oceans. Further, the phytoplanktons are the staple items for the food web and they are the recycler of most of the energy that flows through the ocean ecosystem. It has a major role in stabilizing the environment and survival of living population as it consumes half of the universal carbon-dioxide and releases oxygen. So far, there is a number of studies which show the presence of pathogenic viruses in the plankton community [1, 11]. A good review of the nature of marine viruses and their ecological as well as their biological effects is given in [12]. Some researchers have shown using an electronic microscope that these viral diseases can affect bacteria and phytoplank- ton in coastal area and viruses are held responsible for the collapse of Emiliania huxleyi bloom in Mesocosms [2, 14]. Marine viruses infect not only plankton but cultivated stocks of Crabs, Oysters, Mussels, Clams shrimp, Salmon and Catfish, etc., are all susceptible to various kinds of viruses. We know that the viruses are nonliving organisms, in the sense, they have no metabolism when out side the host and they can reproduce only by infecting the living organisms. Viral infection of 2000 Mathematics Subject Classification. 34C11, 34C23, 34D08, 34D20, 35Q92, 92B05, 92D40. Key words and phrases. Phytoplankton dynamics; reaction-diffusion equation; local stability; Hopf-bifurcation; diffusion-driven instability; spatial pattern formation. c 2012 Texas State University - San Marcos. Submitted July 25, 2011. Published February 2, 2012. 1