The Bifurcation analysis of Prey-Predator Model in The Presence of Stage Structured with Harvesting and Toxicity

For a mathematical model the local bifurcation like pitchfork, transcritical and saddle node occurrence condition is defined in this paper. With the existing of toxicity and harvesting in predator and prey it consist of stage-structured. Near the positive equilibrium point of mathematical model on the Hopf bifurcation with particular emphasis it established. Near the equilibrium point E0 the transcritical bifurcation occurs it is described with analysis. And it shown that at equilibrium points E1 and E2 happened the occurrence of saddle-node bifurcation. At each point the pitch fork bifurcation occurrence is not happened. For the occurrence of local bifurcation illustration there used some numerical simulation.


Introduction
The prey and predator model is an important topic at present, as it is used to solve many problems in the ecology, nature and other sciences. The prey system includes several interactions, including interactions, competition [10], co-existence, food chain [7] and age stage [9]. The ecological models of age stage are more logical than models that do not contain phase structure. In addition, there are several factors that affect the system, for example, harvesting, disease, toxicity, shelter and others. Sometimes, differences in any parameter in the system can lead to complex behavior that leads to system instability, Causing a bifurcation that is main the qualitative change in the behavior of a dynamic system as a result of changing one of its coefficients.
For defining the ordinary nonlinear differential equations the oscillatory solutions of a system and the stable state the bifurcation theory is consider as mathematical tool. More complex features like exotic attractors, the emergence and disappearance of equilibrium and periodic orbits are the examples. To understand the nonlinear dynamic systems results of the bifurcation theory and model used as fundamental. To identify the complex model controllers it also help..
The bifurcations separated into two chief classes: nearby bifurcations and worldwide bifurcations. Nearby bifurcations, which can be broke down altogether through changes in the neighborhood security properties of equilibria, occasional circle or other invariant sets as parameters cross through basic edges, for example, saddle hub, transcritical, pitchfork, period-multiplying (flip), Hopf and Neimark (auxiliary Hopf) bifurcation. Worldwide bifurcations happen when bigger invariant sets, for example, intermittent circles, slam into equilibria. This causes changes in the topology of the directions in stage space which can't be limited to a little neighborhood.
Moreover, a Hopf bifurcation implies the appearance or the vanishing of an occasional circle over a neighborhood change in the properties of the dependability around a fixedpoint. More accurately,it is a neighborhood bifurcation where a fixed purpose of a dynamical framework loses steadiness, as a couple of complex conjugate eigenvalues (of the linearization around the fixed point) cross the unpredictable In the past few years, this theory has evolved considerably through the use of new ideas and methods and their introduction into the theory of dynamic systems. For an ecological system having of a stage structured prey a predator in [6] Majeed research the occurrence of Hopf bifurcation and local bifurcation. In prey population with a refuge-stage structure in [8] Majeed and Naji described near each of the equilibrium points of a prey-predator model the existance of local bifurcation. In [3] Naji, Majeed and Kadhim introduced with refuge near each of the equilibrium points of a stage structured prey food web model the Hopf bifurcation and local bifurcation. With two functional responses and refuge the Hopf bifurcation near the positive point of a stage structured prey food chain model and near each of the equilibrium points the local bifurcation represented by Ali and Majeed in [5]. With the non-refugees prey it show the connection between the two predators.
Finally, in this paper, a set of basic results and methods in local bifurcation theory around all equilibrium points and a Hopf bifurcation theory around the positive equilibrium point for a system include age stage with harvesting and toxicity which depends on a single parameter is presented and discussed .

Model formulation
In this area, the model comprises of two species prey and predator, every specie isolated into two classes: one is youthful and other is experienced, which are meant to their populationsize at time T by X(T), Y(T), Z(T)and W(T)for juvenile prey, develop prey, youthful predator and develop predator separately. Presently, it is referenced in [2] so as to define the elements of such framework, the accompanying presumptions are viewed: • with grown up rates 1 and 2 respectively the predator and immature prey grown up to be mature. • The immature prey depends completely in its feeding on mature prey that growth logistically with an intrinsic growth rate and carrying capacity > 0 in absence of mature predator. Also the immature predator depends completely in its feeding on mature predator that consumes the immature and mature prey with the classical Lotka-Volterra functional response with consumption rates 1 and 2 , respectively, therefore the predator species growth due to attack by mature predator on immature and mature prey with conversion rates 0 < 1 < 1and 0 < 2 < 1. • It shows mature predator, mature prey, immature predator and immature prey's toxicity coefficients and catchability coefficients through and =1,2,3,4.
According above assumptions, the model is formulated as follows: Theorem 1 [2]: It is uniformly bounded complete solutions of system (2).

Equilibrium points of system[ ] stability analysis and existence
The system (2) has maximum 3 points of nonnegative equilibrium that are as follows: • The equilibrium point is locally asymptotically stable using the given constraint. And it is denoted by 0 = ( 0 ,0 ,0 ,0 ) that is always there.

Analysis of Local bifurcation
For local bifurcation Sotomayor's theorem application is suitable in the given theorems.
So, if the condition (20) is satisfied, we obtain that: .

Analysis of Hopf bifurcation
near the positive equilibrium point of the system(2) the happening of Hopf bifurcation is shown in the theorem given below:  6   ) .

System Numerical Analysis
This segment descibed, the dynamical conduct of framework (2) is read numerically for one lot of parameters and various arrangements of starting focuses. The destinations of this examination are explore the impact of fluctuating the estimation of every parameter on the dynamical conduct of framework (2) and affirm our acquired expository outcomes. It is seen that, for the accompanying arrangement of theoretical parameters that fulfills solidness states of the positive harmony point.