DYNAMIC ANALYSIS OF A FRACTIONAL ORDER PHYTOPLANKTON MODEL

Abstract

The fractional order phytoplankton model (PM) can be written as $\frac{d^{\alpha}P_s}{dt^{\alpha}} = rP_s\big(1-\frac{P_s}{K}\big)-\frac{\upsilon P_sP_i}{P_s+1} +\gamma_1 P_i, \frac{d^{\alpha}P_{in}}{dt^{\alpha}}= \frac{\upsilon P_sP_i}{P_s+1} - \beta P_{in}, \frac{d^{\alpha} P_i}{dt^{\alpha}} = \beta_1P_{in} - \delta P_{i}, P_s(\xi)=\varrho_0, \quad P_i(\xi)=\varrho_1, \quad P_{in}(\xi)=\varrho_2,$ where $P_s$ and $P_i$ be the population densities of susceptible and infected phytoplankton respectively and $P_{in}$ be the population density of population in incubated class. In this paper, stability analysis of the phytoplankton model is studied by using the fractional Routh--Hurwitz stability conditions. We have studied the local stability of the equilibrium points of PM. We applied an efficient numerical method based on converting the fractional derivative to integer derivative to solve the PM.