Infected Intermediate Predator and Harvest in Food Chain

In this paper, a mathematical model consisting of the food chain model with disease in intermediate predator is proposed and discussed. The food chain model consists of four types: prey, intermediate predator, infected intermediate predator, and top predator. We studied the solutions for the original model and positive and bounded solutions in the sub models. Also found equilibrium points with sufficient and necessary conditions. By using Jacobian matrix and Lyapunov function to provide local and global stability. Can use the harvesting to control the disease and it can be used as tool to prevent disease transformation into an epidemic. Finally, some results were illustrated in numerical simulations.

group can survive on its own alone, researchers and scientists have presented numerous studies that have been studied in describing population interaction.Researchers first describe Lautka and Volterra in modern mathematical sciences, describing the competition between predators and prey.
But most models involve injury of one type of population originated from the classic movement of Kermack and McKendrick [1].After these two wonderful works, the door become open for researchers to offer many studies in epidemiology and environmental science theory.Even in the last few decades, these models have become important tools for analyzing and understanding the spread of infectious diseases and controlling them.One of the most important studies of the predator model was the study presented by researchers [2].Many researchers [3,4,5,6] also studied these models with diseases.Also studied [10,4,8,7] the role of disease in destabilizing the system.The atmosphere of the regions is important for biological activities primarily responsible for environmental changes.The coexistence of interacting biological species has been very important in the past few decades, and has been extensively studied using mathematical models by many researchers [11][12][13][14][15][16][17][18][19][20] .As a result, many species are extinct and many other extinctions due to the presence of many external forces and influences such as over-exploitation.From this basis, we study the predatory prey, where the predator is at risk of disease and harvest.[12] Krivan proposed a mathematical model and research on the effects of antimicrobial behavior on the predator system.Anti-social behavior has been shown to lead to constant fluctuations and low population density.
Chattobadai et al. [13] prey -predator model with some cover on the prey species.There is the observation that the stability of the global system around the positive balance does not necessarily mean continuity of the system.More recently, Kar [21] has proposed a model for the study of predatory prey and independent harvesting on any species.Has shown that the use of harvest and control efforts, it is possible to break the periodic behavior of the system.In the above investigations, the dynamics of the predator living in the unprotected area with prey are also not explicitly studied.
The protected area plays a vital and important role in the aquatic environment to protect resources and fisheries from overexploitation [22][23][24][25][26].In particular, Dubey et al [22] add a proposal from the mathematical model to analyze the dynamics of the fisheries resource system in a two-zone aquatic environment, the first free fishing zone and the second protected area where strict fishing is prohibited.He pointed out that even if the fisheries are continuously exploited in an unprotected area, fish are at an appropriate level of habitat balance.Moreover, harvest is one of the best and most important means of combating and eradicating the disease and the epidemic and preventing its spread among the population.Using this method requires great care and care.Very severe because any misuse may expose species of extinction.There are several important studies that have been studied on this important subject, and the harvest [9] in the predator model has been studied to create a controlled environment while ensuring species survival and continuous harvesting.
Continuous harvesting in prey predator model in [5,29].This is paper divide into seven sections, section two is the described and developed of the model, section three contained nature of the solutions , section four use a Dynamic of Subsystem to study subsystems of system (1), section five is Existence of Equilibrium points and Stability in the model, section six Numerical Simulation, for Dynamical.Finally, section seven contained Conclusions.

The Food Chain Model with Holling Type II and Harvest
The following model describes the relation between food chain function with

Nature of Solution
Lemma 1: All the solutions of the system (1) in  and 45   [28], then say  , then we get  

Dynamic of Subsystems
In this section we want to study subsystems of system (1).Subsystems obtained in case of extinction one or two population of system (1).Therefore, there are many subsystems, as system (1) as classical model contains prey and predator only, system (1) contains prey and infected predator, system (1) contain all population without top predator, system (1) without disease, finally all population survive.

System (1) as Classical Model.
System (1) without top predator and disease known classical model or Lotka Volterra equations.
In this subsystem the interaction between prey and predator only without any external influence.We describe this interaction as follows:

Nature of Solution
Lemma 2: All solutions of subsystem (2) are positive and bounded.Proof: As lemma 1, see Figure 1. .Since carrying capacity of prey is one, then .Jacobian matrix of subsystem ( 2) is The characteristic equation near 3,2 ( , ) and by Routh-Hurwitz criterion it's stability if 1 2 x  . .

Prey and Infected Predator.
When the disease turns into an epidemic and there is no top predator in this case, system (1) becomes follows:    Jacobian matrix of subsystem
By Routh-Hurwitz criterion this point is stability if 1 2 x  .The model in the system (4) has the following equilibrium points:  The trivial equilibrium points   4,0 0, 0, 0 P always exists.

Natural of Solution
 The equilibrium points   The Jacobian matrix is: Also we will only study the positive point 4,2 P , the characteristic equation near this point is

In the Absence of Top Predator
In absence top predator, System (1) becomes prey, susceptible and infected predator.This model known classical model with disease.Interaction between these populations describe as follows:

4.4.1Natural of Solution
Lemma 12: All the solutions of the sub system (5) in 3 are positive and bounded.Proof: As proof in Lemma 1.

4.4.2Existence of Equilibrium Points and Stability
The model in the system (5) has the following equilibrium points:  The trivial equilibrium points   5,0 0, 0, 0 P always exists.
 The equilibrium points   5,1 1, 0, 0 P exists on the boundary of the octant.
 The nontrivial equilibrium points ,, P x y y where . The Jacobian matrix of subsystem (5) near 5,2 P is : A a a a B a a a a a a a a a a a a C a a a a a a a a a a a a a a a a a a By Routh-Hurwitz criterion, and because 0 AB C  , its stability Lemma 13: Equilibrium point   51 ,, P x y y of subsystem ( 5) is globally stability provided that the following conditions hold The Jacobian matrix is The stability of these points as follows: Then the characteristic equation of   2 JE is given by: A J J J     B J J J J J J J J J J J J J J        C J J J J J J J J J J J J J J J J J J J J J J J J

Numerical Simulation
In this section, we want illustration some results by employ Mathematica Programing in the system (1).To discuss the effect of cure rate from the disease and harvesting on the behavior of the solution.Above all, by looking at a number of papers and taking advantage of the conditions discussed in this paper, we have installed parameters as follows: .Two cases that will be discussed: First case, when the model is the kind of SI.In such model, susceptible intermediate predator (S.I.Predator) become infected intermediate predator (I.I.Predator) and not able to become susceptible again.Then we employ the harvest to see its impact on behavior.Figure (9) the behavior of solution of system (1) as SI model without harvesting.Figure (10) the behavior of solution with harvesting.Note in these two cases how employ the harvesting to disease control.

Consolation
In this research, we studied the dynamic system of the food chain model when the intermediate predator is at risk of disease.All system solutions have proven to be positive and limited, including subsystems.Equilibrium points were found in the subsystems and the original system and the necessary and sufficient conditions for their existence were found in addition to conditions of local and global stability.It turns out that the stability conditions of the subsystems are necessary for the stability of the original system.The disease is controlled and prevented from becoming as epidemic by numerical simulation.

Lemma 4 :,
Subsystem (2) has no periodic orbit in 2 Note that ( , ) xy  does not change sign and is not identically zero in 2 ( , ) xy  .Therefore according to Bendixson-Dulic criterion, there subsystem (2) has no periodic solution.with condition 11 d  

Lemma 5 :
Positive equilibrium point 3,2 ( , ) P x y of the subsystem (2) is globally asymptotically stable in

Lemma 6 :
All solutions of sub system (3) are positive and bounded.Proof: As lemma 1, see figure3.

Figure 3 .
Figure 3.All solutions of subsystem (3) are positive and bounded

Lemma 9 :
All the solutions of the sub system (4) in 3  are positive and bounded.Proof: Same proof in Lemma 1. Lemma 10: In subsystem (4) 32 d   Proof: As lemma 6.

4
 are positive and bounded.
As proof in Lemma 4.
Lemma 11: Equilibrium point   4,2 ˆˆ,, P x y z is globally asymptotically stable provided that the following conditions hold ˆŷx xy  and ˆŷz yz  Proof: