EFFECT OF PREDATORS’ BEHAVIOR ON PREY-PREDATOR INTERACTION

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we study the effect of the behavior of predators on prey-predator interaction. We assume that there is one prey population that suffers from the fear of predators which could force it to move in to the refuge; and three are two predator populations, one of them is aggressive in its attack and the other one using sit-and-wait procedure, which is less aggressive. All the model’s equilibrium points have been found and their stability was established. The possibility of transcritical and Hopf bifurcation was also investigated and numerical simulations were given. The effect of prey refuge and fear also are detected. The cost of them is to allow the model to reach to double transcritical point. The effect of the competition between prey populations is to convert the model from the stable limit cycle to a spiral stable equilibrium point of afraid prey with predator. When it becomes large it converts the model to the stable trivial solution.


PRELIMINARIES
Theory predicts that organisms should modulate behavior to maximize their expected fitness [1]. Since death is the most definite negative effect on fitness, it is very important for individuals to avoid predation and therefore a strong selection for proper anti-predator response is to be expected. Behavioral adjustments to predator presence are very widespread responses used by organisms in many taxa [10]. The optimal behaviorally mediated anti-predator response is of course highly dependent on what kind of predator an organism is encountering. For example, [11] showed that damselfly larvae that swim to escape fish predators have a very low chance of survival, while the same escape response in the presence of invertebrate predators resulted in much higher chances of survival. Their study illustrates that a good anti-predator response to one type of predator can be maladaptive in the presence of another predator type. It is therefore vital for the prey to be able to identify and correctly assess the risk linked to a specific predator. Since many organisms encounter several different types of predators, plasticity in the behavioral response to predators is expected to be beneficial for prey populations [4,13]. In order to study the effects of preys' behavior changes due to the fear of predators and the effect of predator populations, we developed a mathematical model with one prey population and two predator populations. It is assumed that the prey population fears the predator and each predator follows a different type of attack; one of them is very aggressive and the other one follows the sit-andwait strategy which considered to be less aggressive in its nature [2,3,5].

MATHEMATICAL MODE BUILDING AND ANALYSIS
To develop our model we assume that there is one prey population and two predator populations. It is assumed that the prey population suffers from fear of predator which affect its reproduction and cause it to move inside a refuge for sometimes which affects the availability of food and causes the rise of inter-species competition [7], therefore the reproduction term of the prey population takes the form bN 1 + e 1 P 1 + e 2 P 2 − dN − sN 2 Where the term f (e; P 1 , P 2 ) = 1 1 + e 1 P 1 + e 2 P 2 represents the fear of the prey from the predator. It is also assumed that one predator is aggressive in nature and therefore its attack follows Holling type-II functional response, and the other one is less aggressive in its behavior and therefore it follows a modified Holling type-II functional response in its attack [8]. Taking these assumption into consideration, our model is represented by the following set of differential equations (1) where N represents the prey which afraid and stay in refuges, P 1 and P 2 represent bold and aggressive predator respectively. The next table is demonstrated the meaning of each parameter: 2.1. Equilibrium Points. The system (1) has the following equilibrium points: (i) E 0 (0, 0, 0), the trivial equilibrium point.
, 0, 0), representing the existence of prey which stay in refuge.
, P * 1 , 0), representing the existence of afraid pray and first predator. Where P * 1 is the root of equations , representing the existence of afraid pray and second predator which is faster than first predator. Where N * represents the solution of equation N * = AX 3 + . Consequently we can choose all factors to be positive or negative. We will consider the equation N * = AX * 3 2 + BX * 2 2 + CX * 2 + D = 0. Consequently by using Descartes's Sign Rule it can be chosen D > 0 with B < 0 and C > 0 or D < 0 with B > 0 and C > 0 to get positive solution and, it can be taken D > 0, with any sign of b and c except B > 0 and C < 0 to get a unique positive solution.
Parameters The meaning b The birth rate of prey.
Theorem 1. The stability of the system (1)is given by: , is locally asymptotically stable by using Qualitative Matrix Stability method.

Proof
From equations (1), the Jacobian matrix of the system is given by: . By evaluating the Jacobian matrix at each equilibrium points, we get: (i) The Jacobian matrix at E 0 (0, 0, 0) is given by: clearly the eigenvalues of the matrix are b − d, −c 1 and −c 2 and it is obvious that clearly this is upper triangular matrix, so the eigenvalues of the matrix are The Jacobian matrix at E 2 (N * , P * 1 , 0) is given by: is one of the eigenvalue of the Jacobian matrix.
The other eigenvalues is gotten from reduced matrix, which is: The characteristic polynomial is: Clearly, this point is locally asymptotically stable if is one of the eigenvalue of the Jacobian matrix. The other eigenvalues is gotten from reduced matrix, which is: The characteristic polynomial is: Clearly, this point is locally asymptotically stable if To prove the stability of the coexistence equilibrium point we will use the Qualitative matrix stability method.
Then: (1) and (2)  Thus from figure there are two predation links between N and P 1 and between N and P 2 .
since these links are connected by node P 1 and P 2 , then the entire signed digraph forms a predation community.

Color test
We color each node with a negative feedback loop with gray and the other nodes by white as in figure . Applying the color test to our model we find that: * The node N is gray and the other nodes P 1 and P 2 are white. * However, condition (ii) is not satisfied because there is no predation link between white nodes. * The gray node N is connected by a predation link to both white nodes P 1 and P 2 .
In conclusion, the signed matrix Q 3 fails the color test. Hence the Jacobian matrix J 4 corresponding to the equilibrium point E 4 (N * , P * 1 , P * 2 ) ,where Q 3 = sign(J 4 ) is qualitative stable. Then the equilibrium is locally asymptotically stable.
Proof By letting N = N * +n, P 1 = P * 1 + p 1 and P 2 = P * 2 + p 2 , where n, p 1 and p 2 are small perturbations about N * , P * 1 and P * 2 respectively, the system of equations (1) is turned into a linear system which is of the formṅ i = J(E i )n i , where J(E i ) is the Jacobian matrix of equations (1). Thus, the linear system of equations (1) is, with respect to time t we get,V (n, p 1 , p 2 ) = nṅ N * + p 1ṗ1 + p 2ṗ2 . By substituting forṅ,ṗ 1 anḋ p 2 in equations of system (3) gives, V (n, p 1 , p 2 ) = −( r κ n 2 +a 2 p)n 2 , which is negative semi-definite. Therefore, E 1 (N * , 0, 0) is globally asymptotically stable.

Hopf bifurcation.
Theorem 3. The system (1) undergoes a Hopf bifurcation at the positive equilibrium: Proof (i) The eigenvalues of the linearized system around the equilibrium point E 2 are: where: where J is the Jacobian of the linearized system at the equilibrium point E 2 .
Therefore from Hopf Theorem the proof is concluded.
(ii) The eigenvalues of the linearized system around the equilibrium point E 3 are: where: where J is the Jacobian of the linearized system at the equilibrium point E 3 .
Therefore from Hopf Theorem the proof is concluded.

Transcritical bifurcation.
Theorem 4. The system (1) undergoes a Transcritical bifurcation at the positive equilibrium The eigenvalues of the linearized system around the equilibrium point E 1 are: Let us define v = (v 1 , v 2 , v 3 ) T and w = (w 1 , w 2 , w 3 ) T are the right and left eigenvectors of λ 2 = 0. From (4) and J (E 1 , m 1 0 )v = 0 as well as J T (E 1 , m 1 0 )w = 0, then, w 3 , So the left eigenvector is (0, w 2 , 0) T and the right eigenvector is , v 2 , 0 T . Here, w 2 and v 2 are any non-zero real numbers. Now, system (1) can be rewritten as in the following vector form: Taking derivative on f (X) with respect to m 1 , we get Hence, w T f E 1 ,m 1 0 (X) = 0.
Next, taking derivative on f m 1 (X) with respect to X = (N, P 1 , P 2 ) T , then, ) and then subistitute the value of m 1 0 and E 1 , so we end by where, (v, v) is a Kronecker product of (v 1 , v 2 , v 3 ) T . Therefore, according to the Sotomayor's theorem for local bifurcation [9], system (1) has a transcritical bifurcation at steady state E 1 when the parameter m 1 passes through the bifurcation value m 1 0 .
(i)(b) The eigenvalues of the linearized system around the equilibrium point E 1 are: Let us define v = (v 1 , v 2 , v 3 ) T and w = (w 1 , w 2 , w 3 ) T are the right and left eigenvectors of λ 3 = 0. From (5) and J (E 1 , m 2 0 )v = 0 as well as J T (E 1 , m 1 0 )w = 0, then, w 2 , So the left eigenvector is (0, w 2 , 0) T and the right eigenvector is Here, w 2 and v 3 are any non-zero real numbers. Now, system (1) can be rewritten as in the following vector form: Taking derivative on f (X) with respect to m 2 , then, (1+α 1 N * ) = 0. Furthermore, we find D 2 f (X) = D(J(E i )) = D(D( f (X))) and then subistitute the value of m 2 0 and E 1 , so we end by Therefore, according to the Sotomayor's theorem for local bifurcation, system (1) has a transcritical bifurcation at steady state E 1 when the parameter m 2 passes through the bifurcation value m 2 0 . (ii) The eigenvalues of the linearized system around the equilibrium point E 2 are: Let us define v = (v 1 , v 2 , v 3 ) T and w = (w 1 , w 2 , w 3 ) T are the right and left eigenvectors of λ 3 = 0. From (6) and J (E 2 , m 2 0 )v = 0 as well as J T (E 2 , m 2 0 )w = 0, then, So the left eigenvector is (0, 0, w 3 ) T and the right eigenvector is Here, w 3 and v 2 are any non-zero real numbers. Now, system (1) can be rewritten as in the following vector form: Taking derivative on f (X) with respect to m 2 , we get Hence, w T f E 2 ,m 2 0 (X) = 0.
Next, taking derivative on f m 2 (X) with respect to X = (N, P 1 , P 2 ) T , we get, = 0. Furthermore, we find D 2 f (X) = D(J(E i )) = D(D( f (X))) and then subistitute the value of m 2 0 and E 2 , so we end by Therefore, according to the Sotomayor's theorem for local bifurcation, system (1) has a transcritical bifurcation at steady state E 2 when the parameter m 2 passes through the bifurcation value m 2 0 . (iii) The eigenvalues of the linearized system around the equilibrium point E 3 are: Let us define v = (v 1 , v 2 , v 3 ) T and w = (w 1 , w 2 , w 3 ) T are the right and left eigenvectors of λ 2 = 0. From (7) and J (E 3 , m 1 0 )v = 0 as well as J T (E 3 , m 1 0 )w = 0, then, So the left eigenvector is (0, w 2 , 0) T and the right eigenvector is , Here, w 2 and v 1 are any non-zero real numbers. Now, similarly system (1) can be rewritten as in the following Taking derivative on f (X) with respect to m 1 , we get Hence, w T f E 3 ,m 1 0 (X) = 0.
Next, taking derivative on f m 1 (X) with respect to X = (N, P 1 , P 2 ) T . Then, then subistitute the value of m 1 0 and E 3 , so we end by where, (v, v) is a Kronecker product of (v 1 , v 2 , v 3 ) T . Therefore, according to the Sotomayor's theorem for local bifurcation, system (1) has a transcritical bifurcation at steady state E 3 when the parameter m 1 passes through the bifurcation value m 1 0 .

NUMERICAL SOLUTION
In this section we will present some numerical simulations in order to show the theoretically value of m 2 The existence point [0,0.6) (N * , 0, P * 2 ) 0.6 (N * , P * 1 , P * 2 ) (0. 6,1] (N * , P * 1 , 0) As we have see in figure 2, which demonstrate the effect stability of the trivial solution when the inter-species competition is very big and b − d < 0, i.e. when the death rate is greater than the birth rate. In Figure 2 (i) s = 0.0009 and in (ii) s = 0.8 with b > d . While in Figure   2(iii) s = 0.0009, b < d. Figure 3 illustrate the stability of the existence of prey only (N * , 0, 0) under some conditions. The stability of the solution which reach the point (N * , P * 1 , 0), which represents the existence of prey and the first predator, is shown in figure 4. Figure 5 shows the stability of the solutions when it reach the point (N * , 0, P * 2 ) which represents the existence of prey and second predator. The stability of the solutions when it reaches the coexistence point is illustrated in figure 6. The limit cycles are illustrated in figures 7,8 and 9 for the solutions of existence of prey with first predator, with the second predator and when it reaches the coexistence point, respectively. Figure

CONCLUSION
In this paper we formulated model of one prey with two predators population; aggressive predator, bold predator and prey which is afraid and stay in refuge. We use holing type II functional response and Beddington-DeAngeiis functional response [12] . We proved the locally and globally stabilities of equilibria. In addition we hold a hopf and double transcritical bifurcations of some parameters. The cost of fear and prey refuge is allow model to reach to double transcritical. we notice that if fear is good when it is small. The effect of competition of prey population is to convert the model from the stable limit cycle to a spiral stable equilibrium point of prey with predator. When it becomes large it converts model to stable trivial solution.