DYNAMIC ANALYSIS OF A FRACTIONAL ORDER PHYTOPLANKTON MODEL

The fractional order phytoplankton model (PM) can be written as d Ps dtα = rPs ( 1 − Ps K ) − υPsPi Ps+1 + γ1Pi, dPin dtα = υPsPi Ps+1 − βPin, d Pi dtα = β1Pin − δPi, Ps(ξ) = ρ0, Pi(ξ) = ρ1, Pin(ξ) = ρ2, where Ps and Pi be the population densities of susceptible and infected phytoplankton respectively and Pin 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.


Introduction
Phytoplankton are microscopic plants that live in the ocean. These small plants are very important to the ocean and to the whole planet! They are at the base of the food chain. Many small fish and whales eat them. Then bigger fish eat the little fish, etc. The food chain continues and at some point in time we (people) come into it when we eat the fish. So the energy of plankton becomes our energy too!. 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 [7,13]. A good review of the nature of marine viruses and their ecological as well as their biological effects is given in [14]. Some researchers have shown using an electronic microscope that these viral diseases can affect bacteria and phytoplankton in coastal area and viruses are held responsible for the collapse of Emiliania huxleyi bloom in Mesocosms [10,15]. Baghel et al. [3], proposed a three dimensional mathematical model of phytoplankton dynamics with the help of reaction-diffusion equations that studies the bifurcation and pattern formation mechanism. They provide an analytical explanation for understanding phytoplankton dynamics with three population classes: susceptible, incubated, and infected.
In 2010, Dhar and Sharma [5] investigated the stability of the phytoplankton system where P s , P i are the population densities of susceptible and infected phytoplankton at any instant of time t. r is the intrinsic growth rate of the population of susceptible phytoplankton, K is the carrying capacity of the population of susceptible phytoplankton, υ is the disease contact rate of the disease phytoplankton population, β is the removal rate of the disease phytoplankton population, out of which γ fraction of infected phytoplankton rejoin the susceptible phytoplankton population. Also the same system investigated by applying a frequency domain approach with time delay by Xu [18]. He use the delay as a bifurcation parameter; as it passes through a sequence of critical values, Hopf bifurcation occurs. A family of periodic solutions bifurcate from the equilibrium when the bifurcation parameter exceeds a critical value. Ghosh [9], proposed the interrelationship of latency period in viral infection and overall infection process in host community are of critical importance in context of pest control program. Both of them regulate the overall system stability as they are dynamically linked to predation by natural enemies in the system. In 2010, Dhar and Sharma [5], proposed the role of viral infection in phytoplankton dynamics without and with incubation population class is studied. It is observed that phytoplankton species in the absence of incubated class are unstable around an endemic equilibrium but the presence of delay in the form of incubated class has made it conditionally stable around an endemic equilibrium. The authors of [16] proposed a prey-predator model for the phytoplanktonzooplankton system with the assumption that the viral disease is spreading only among the prey species, and, though the predator feeds on both the susceptible and infected prey, the infected prey is more vulnerable to predation as is seen in nature (see references quoted earlier). The dynamical behaviour of the system is investigated from the point of view of stability and persistence. The model shows that infection can be sustained only above a threshold of force of infection. Gakkhar and Negi [8] investigate the dynamical behaviour of toxin producing phytoplankton (TPP) and zooplankton. The phytoplanktons are divided into two groups, namely susceptible phytoplankton and infected phytoplankton. The conditions for coexistence for the populations are presented. Chattopadhyay et al. [4], deals with the problem of a nutrient-phytoplankton (N-P) populations where phytoplankton population is divided into two groups, namely susceptible phytoplankton and infected phytoplankton. Conditions for coexistence or extinction of populations are derived taking into account general nutrient uptake functions and Holling type-II functional response as an example.

Preliminaries
Definition 2.1. The Riemann-Liouville fractional integral operator of order α > 0, of function f ∈ L 1 (R + ) is defined as where Γ(·) is the Euler gamma function.
Definition 2.2. The Riemann-Liouville and Caputo fractional derivative of order α > 0, n − 1 < α < n, n ∈ N is defined as where the function f (t) have absolutely continuous derivatives up to order (n − 1).

The initial value problem related to Definition 2.2 is
where 0 < α < 1 and D α = D α 0 . Now, some stability theorems on fractional-order systems are introduced.

Mathematical Model
Let P s and P i be the population densities of susceptible and infected phytoplankton respectively. The population of susceptible phytoplankton is assumed to be growing logistically with intrinsic growth rate r and carrying capacity K. Now let P in be the population density of population in incubated class. Here, we will use nonlinear Holling Type II functional responses for disease spreading because the disease conversion rates become saturated as victim densities increase. Let υ be the disease contact rate and it is volume-specific encounter rate between susceptible and infected phytoplankton, which is equivalent to the inverse of the average search time between successful spreading of disease. The coefficients δ and β are the total removal of phytoplankton from the infected and incubated class because of the death (including recovered) from disease and due to natural causes respectively. Again, γ 1 be the fraction of the population recovered from infected phytoplankton population and joined in the susceptible phytoplankton population and β 1 is the fraction of the incubated class population which will move to the infected class. Therefore, quantitatively δ > γ 1 and β > β 1 . Using these assumptions the dynamics of the system can be governed by the following set of differential equations: The Holling type-II the functional response υPsPi Ps+1 is used [11] and many other researchers. In this paper we investigate the following fractional order phytoplankton model with initial population; i.e., P s (ξ) > 0, P in (ξ) > 0, P i (ξ) > 0 and the total population at any instant t is N (t) = P s (t) + P in (t) + P i (t): where the parameters 0 < α ≤ 1, and d α dt α is in the sense of the Caputo fractional derivative defined in (2.2) with the initial time t = ξ.

If the Jacobian matrix of system (3.2) at the equilibrium point O is
with the characteristic equation The eigenvalues corresponding to the equilibrium O are Then we have λ 1 > 0, λ 2 > 0 and λ 3 > 0. Whence it follows that the equilibrium O of system (3.2) is unstable. Thus the stable manifold of the origin W s (O) is onedimensional and the unstable manifold of the origin W u (O) is two-dimensional.
The Jacobian matrix of (3.2) at equilibrium point E 1 = (K, 0, 0) is with the characteristic equation Let D(Q) denote the discriminant of a polynomial Q(λ). Then Using the proposition given in [17], we have the following result by using Routh-Hurwitz conditions.
3 . The Jacobian matrix of (3.2) at equilibrium point E 2 = (P * , Q * , R * ) is with the characteristic equation where Using the proposition given in [17], we have the following result by using Routh-Hurwitz conditions.

The numerical decomposition method
In order to solve (3.2), we shall use a numerical method introduced by Atanackovic and Stankovic [1] to solve the single linear fractional differential equation (FDE). Also the same authors [2] developed the method to solve the nonlinear FDE. In [1] it was shown that for a function f (t), the fractional derivative of order α with 0 < α ≤ 1 may be expressed as where with the following properties We approximate D α f (t) by using M terms in sums appearing in Eq. (5.1) as follows } .

Conclusion
In this work, we analyze the dynamic behavior of the fractional-order phytoplankton model. Firstly, we study the existence of extinction equilibrium and boundary equilibria, furthermore give stability criteria of the model from local point of view by using the fractional Routh-Hurwits criterion. The corresponding results are illustrated by the numerical simulation. Simulation results show the effectiveness of the method. Further, it has been shown that the supply rate α play an important role in shaping the dynamics of the system. A stable limit cycle is observed in Figs.7-9.