Mathematical study on a dynamical predator-prey model with constant prey harvesting and proportional harvesting in predator

A dynamical predator-prey model with constant prey harvesting, proportional harvesting in predator has been studied. The square root functional response also has been incorporated in the system to describe the prey herd behaviour, assuming the average handling time is zero. The existence and the local stability of equilibria of the system have been discussed. It is examined that, two types of bifurcation occur in the system. The two types of bifurcations have been analyzed, and it has been found by analyzing the saddle-node bifurcation that, there is a maximum sustainable yield. It is observed that if harvesting rate is greater than the maximum sustainable yield, the prey population abolish from the system and then extinction of the predator population happen. But if harvesting rate is lesser than the maximum sustainable yield, the extinction of the prey population can not be possible. By analyzing the Hopf bifurcation, it is obtained that, there exists an unstable limit cycle around the interior equilibrium point. Several numerical simulations are performed to check the results.


Introduction
To describe the interactions between different species in the nature, predatorprey system is one of the most important and useful models. Lotka (1) and Voterra (2) separately offered basic models to describe the interaction between two species and proposed a two dimensional system of ordinary differential equations. They introduced a functional response and used in their model which is proportional to the density of predator. Later, many researchers studied on the classical Lotka-Volterra (1) (2) model and also produced different types of functional responses, e.g., C.S Holling (3) introduced three types of functional responses, Crowley-Martin (4), Beddington-DeAngelis (5) (6) offered different functional responses, etc. In the mentioned studies the authors used these functional responses by assuming that the predator can take any prey for their food i.e., they considered solitary prey behaviour.
But, in nature some preys live in herd behaviour to fight with the predators for their benefits. For that type of herd behaviour, the above mentioned functional responses are not appropriate to describe the interaction. To investigate the model with that type of prey herd behaviour, a model with a new functional response was proposed by Ajraldi et al. (7) and the author states that, it is more appropriate to model the functional response for prey herd behaviour as a square root function of prey density. Later, Braza in his paper (8) introduced a new functional response considering the square root of prey density in the Holling Type II functional response (3) and assuming the average handling time is zero he proposed a model with this functional response. Further many researchers studied that model to investigate the interaction between predator and prey with the functional response (9) (10) (11) (12) (13) (14). In this paper, this type of functional response has been included in the model.
Nowadays, people are very much interested to harvest the species for economic interest which very much affects on the dynamics of any ecological or biological system. For example, we can see in any marine ecosystem people are harvesting fishes for their economic interest which affects on the dynamics of a marine ecosystem. To tackle that type of situation, many researchers worked on a predator-prey system with harvesting effects (15) (16) (17) (18). Basically researchers used three types of harvesting and the classification is constant harvesting (19), proportional harvesting (20) and non-linear harvesting (21). In this paper we consider harvested predator and prey population. Prey populations are harvested by constant rate and predator populations are harvested by proportional rate. The main aim of the paper is to investigate the dynamical behaviour of a predator prey system with constant harvesting in prey, proportional harvesting in predator and also with square root functional response assuming average handling time is zero. The organization of the rest of the paper in this way: In Sec.2 we formulate a mathematical model with constant harvesting in prey, proportional harvesting in predator and also with square root functional response assuming average handling time is zero. The condition for existence of equilibria and their local stability of system (3) have been discussed in Sec.3.
In Sec.4 two types of bifurcations of the system (3) have been analyzed. Sec.5 includes numerical examples. Finally, Sec.6 contains the conclusion.

Mathematical Model Formulation
The basic predator-prey model with the square root functional response (8), constant harvesting in prey and proportional harvesting in predator is as follows: Where the prey population denoted by x and the predator population denoted by y, r is the growth rate of prey, m is the biomass conversion rate, α denotes the predator's search efficiency for prey. N is the environment carrying capacity, h is the constant prey harvesting and β is the predator's natural death rate in the prey free situation and E q is harvesting effort. For the dynamic analysis of the model, the variables and the parameters are scaled as, Excluding the bars of the system (2) we formulate the following system: Where c, d, h, E all are positive.
3 Existence and stability of equilibria

Existence of equilibria
The system (3) has been defined on the following set considering biological background: To identify the equilibrium points of the system (3), the following system has been solved: The above system has three equilibrium points and we get the following theorems regarding their existence. Note that: When 0 < 4h < 1, the system (3) has two axial equilibrium points E i = (x i , 0), (i = 1, 2), which are different. 1. 2. When 4h = 1, two equilibria collide and produce a unique axial equilibrium This implies that prey population will go extinct and then extinction of predator population will happens. It is obvious that in Γ the system (3) has no equilibria. Now for predator free (i.e., y=0) equilibrium points we have solved the following quadratic equation: Two different roots x 1 , x 2 of the equation (6) are mentioned in (6)

Stability of equilibria
This subsection includes the dynamic analysis of the system (3) around the equilibrium points. The following matrix J is the Jacobian matrix of the system (3): Now, the local stability of the system (3) are analyzed in the neighbourhood of E 1 and E 2 . And regarding this we have the following theorems.
Theorem 2 When 4h < 1 the system (3) has two different equilibria E 1 and Proof. It can be seen that from Theorem (1) that, the system (3) has two different equilibria E 1 and E 2 . Now the following matrix J(E 1 ) is the Jacobian matrix at E 1 of the system (3) The eigenvalues λ 1 , λ 2 of the Jacobian matrix J(E 1 ) are as follows: Proof. From Theorem (1) it can be observed that, the system (3) has two different equilibria E 1 and E 2 . The following matrix J(E 2 ) is the Jacobian matrix at E 2 of the system (3) The eigenvalues µ 1 , µ 2 of the Jacobian matrix J(E 2 ) are as follows: Hence, E 1 is hyperbolic saddle. Now, the local stability of the unique interior equilibrium point E * are analyzed and the following theorem is obtained.

2. Theorem 4 When
exists a unique interior equilibrium point E * = (x * , y * ) of the system (3). The following matrix J(E * ) is the Jacobian matrix at E * of the system (3) Now, E * is center when h = h 1 , since the Trace of the Jacobian matrix (10) is zero at h = h 1 . E * is source when h > h 1 > 0, since the Trace of the Jacobian matrix (10) is greater than zero at h > h 1 > 0. E * is sink when h < h 1 , since the Trace of the Jacobian matrix (10) is less than zero at h < h 1

Bifurcation Analysis
Form Theorem (1) and Theorem (4), it can be seen that the system (3) may exhibits saddle-node and Hopf bifurcation. So, in this section, the conditions for two different types of bifurcations have been analyzed.

Saddle-node bifurcation
From Theorem (1), it is observed that two equilibria of the system (3) collide and produce a unique equilibria point E 0 = 1 2 , 0 . Now, the following matrix J(E 0 ) is the Jacobian matrix at E 0 of the system (3 Easily it can be seen that the above Jacobian matrix (11) has a zero eigenvalue. This means that, at E 0 the stability analysis of the system (3) is not possible by linearization technique. Therefore, the system (3) exhibits a saddle-node bifurcation in the neighbourhood of E 0 , as two equilibria collide at the bifurcation parameter h = 1 4 . The following theorem describe the saddle-node bifurcation.
Proof. As J(E 0 ) contains a zero eigenvalue, according to Sotomayor's theorem (22) the necessary condition for saddle-node bifurcation around E 0 at h = 1 4 satisfies. We consider U, V as the eigenvectors of the matrix J(E 0 ) and J(E 0 ) T corresponding to the zero eigenvalues, respectively.
. It is obvious that, U and V satisfy the transversality conditions: and if h > h M SY , the prey population will go extinct and it will causes the extinction of predator population. But if 0 < h < 1 4 , the prey population do not go extinct.

Hopf bifurcation
Theorem (1) shows that, Hopf bifurcation may occur in the system (3). Now we get d dh Trace(J(E * ))= 1 2x * = 0 which implies that the transversality condition satisfies for occurring Hopf bifurcation. Hence, Hopf bifurcation occur in the system (3) around E * at h = h 1 (22). Now, the equilibrium point E * is translated to origin to find the Hopf bifurcation direction. Using the transformation (X, Y ) = (x − x * , y − y * ) and expanding the system (3) in a power series around origin, we get dX dt = a 10 X + a 01 Y + a 11 XY + a 20 X 2 + a 02 Y 2 + a 21 X 2 Y + a 12 XY 2 + a 30 X 3 + . Now, we calculate first Lypunov number (22): 0. Therefore, from the above expression of ∆ and N it can be conclude that l > 0, that means the Hopf bifurcation is sub-critical at h = h 1 . And a unstable limit cycle present in the system (3) around E * (22).

Numerical Examples
Some numerical examples to verify the analytical results have been discussed in this section. E * = (0.3600, 0.3360) is center (Figure 2a).

Conclusions
A dynamical predator-prey model with the square root functional response to describe the prey herd behaviour assuming the average handling time is zero has been studied. Also we have incorporated constant prey harvesting, proportional harvesting in predator in the model. Conditions for the existence of the equilibria of the system (3) has been discussed. The local stability of two different predator free equilibria and a unique equilibrium point the system (3) has been analyzed. It has been observed that, two types of bifurcations occur. The saddle-node and Hopf bifurcation has been analyzed and it has been found that, there is a maximum sustainable yield h M SY = 1 4 . At h > h M SY , the prey population will go extinct and it will be the cause for extinction of the predator population. But if 0 < h < 1 4 , the prey population do not go extinct. An unstable limit cycle exists around the interior equilibrium point. It has been examined that if harvesting rate is chosen at h ≤ h 1 , both population will coexist and will maintained ecological balance. The calculation of first Lypunov number cleared that the Hopf bifurcation is super-critical. To enhance a good knowledge about the interaction between two species prey and predator, the results of this paper may help.