Dynamics of the Stage Structured Population Model, Predator Accompanied by Michaelis Menten Holling Type Functional Responseand Delay and Prey Taking Refuge

Abstract: The current work considers predator prey system, prey taking refuge, predator reckoned with time delay and Michaelis Menten Holling type II response function undergoing two stages: juvenile and mature. From the characteristic equation, we derive conditions for the local stability of the system at the equilibrium points. Also, at the coexistence equilibrium point, the system is analyzed for the occurrence of Hopf bifurcation. Lyapunov function provides sufficient conditions for the global stability of the system. Numerical simulations are given to support the theory.

Mortoja et al. proposed a model with stage structure in both prey and predator with prey flaunting anti predator activity and group defense mechanism with different type of Holling type response function. Hopf bifurcation is analysed by considering transition rate as bifurcation parameter [17]. Using MATCONT software bifurcation and stability of a ratio-based model with noncontant predator harvesting rate were computed. Their analysis revealed that the system exhibits various form of bifurcation including bifurcations of Fold, Cusp and Bogdanov Takens [18]. Jost et al. in 1999 [19] studied the behavior of the predator prey system particularly at the origin with ratio dependent functional response.
Here N and P are prey abundance and predator abundance. They showed that the system exhibits different behaviors for different parameter values at the point (0, 0) and at this point the system has well defined dynamics. The above system has been consideredwith self-diffusion, cross diffusion and prey taking refuge for its spatial patterns [20]. Triggered by the research work of Xiao et al. [21], and keeping the basic model from Jost [19] and Sambath [20] we come up with a following population model. Here the predator is divided into two subgroups as juvenile and mature. Only the mature predator is capable of hunting and the immature predators depend solely on their elders for their food. Time delay due to gestation of mature predator and Michaelis Menten Holling type II functional response have been considered. The model takes the form with initial conditions ( ) = 1 ( ), 1 ( ) = 2 ( ), 2 ( ) = 3 ( ) (4) where ( 1 ( ), 2 ( ), 3 ( )) ∈ ([− ,0), ℝ + 3 ), the continuous functions in Banach space mapping the interval [− , 0) into ℝ + 3 = { 1 , 2 , 3 : ≥ 0, = 1,2,3}. Here ( ), 1 ( ), 2 ( ) denotes size of prey, juvenile and adult predator respectively. , , 1 represents the prey growth rate, environmental carrying capacity, predators benefit rate from cofeeding.λ is the proportion of prey taking refuge and [0,1)   ; , , 1 , 2 are the capture rate, conversion coefficiency, death rate of immature and mature predators respectively. τ is the time lag due to gestation of mature predators. This paper is sectioned as follows. In section 2 we show that the model system is bounded. We find equilibrium points and conditions for local stability at these equilibrium points. Hopf bifurcation at the positive equilibrium point is discussed in section 3. Permanence and global stability of the system at the coexistence equilibrium point are investigated in section 4. Numerical simulations are given in section 5. Section 6 is the conclusion.
Taking the time derivative of the above along the nonnegative solution of (1-3), Hence all the solutions of the considered system are uniformly bounded. Hence the theorem.

Local Stability Theorem 2.2:
The zero equilibrium 0 (0,0,0) is unstable for the system (1-3). Proof: Working out the characteristic equation from the variational matrix and substituting the equilibrium point 0 (0,0,0) we get, We observe that one of the roots is positive in (9) and hence the equilibrium 0 is always unstable. Hence the theorem.
Separating the real and imaginary part, which gives, If < 1 and (C1) holds then, ℎ 0 2 − 0 2 > 0 which shows that (20) has no positive roots. Therefore by a theorem [22], for all ≥ 0, all the eigen values of (20) have negative real parts. Hence at the positive equilibrium ( , 1 , 2 ) the system is locally asymptotically stable for all ≥ 0.

Global Stability
Here we use Lyapunov functional and Laselle invariance principal to study the global attractivity of the coexistence equilibrium of the system.
From the above we get,

. Conclusion
The mathematical model of the prey predator system considered for analysis is uniformly bounded which implies that the model is well behaved biologically. The boundary equilibrium is asymptotically stable under appropriate conditions. The conditions for the coexistence equilibrium to be locally stable is obtained. Also, it is found that the delay in time can make the stable equilibrium to be unstable, as the time lag crosses a critical value and making way for Hopf bifurcation. By suitable construction of the Lyapunov functional global stability at the positive equilibrium point is established.

Conflicts of Interest
No conflict of interest was declared by the authors.