THE DYNAMICS OF A STAGE-STRUCTURE PREY-PREDATOR MODEL WITH HUNTING COOPERATION AND ANTI-PREDATOR BEHAVIOR

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INTRODUCTION
Ecology is the study of the dispersion, resources, connections, and interactions of organisms with their surroundings. Ecological research aims to investigate every factor that influences how 2 DAHLIA KHALED BAHLOOL individual organisms interact with one another and their surroundings to produce findings supporting the ecosystem's continued sustainability. An important area of study in ecological systems has centered on the prey-predator relationship [1]. Lotka and Volterra propose and examine the earliest model of the prey-predator interaction, which includes two first-order, nonlinear differential equations. Following that, multiple attempts have been made to expand the groundbreaking work, see [2][3][4][5][6][7][8].
The rate at which prey is consumed by the predator population or the predator's functional response is a key element in the dynamics of prey-predator interaction. It aids in more accurate prey-predator dynamics analysis. Predation rates vary depending on several factors including age category, corpulence, environment characteristics, interference, and cooperation among members of a specific species. A linear function called the Holling type I functional response consumes the prey if it is weak, tiny, juvenile, or readily available. Functional responses come in various categories: prey-dependent, prey-predator-dependent, and ratio-dependent. In the literature, numerous investigations of prey-predator systems with various categories have been conducted, see [5][6][9][10][11][12][13]. Most biological species depend on individuals' age or stage of development to determine their survival and rate of reproduction. Adult and juvenile stages of a species' life cycle can be distinguished from one another. For adult versus juvenile prey, a predator's nature is entirely different. While adult prey has greater potential to escape than juvenile prey, the predator has a strong attraction to juvenile prey. Therefore, it makes sense to incorporate the impact of a species' past life to get more accurate results. Many academics analyze stagestructured models in an effort to overcome the drawbacks of traditional Lotka-Volterra models [14][15][16][17][18]. Even while biologists categorize animals as either predators or prey, the ecological function of an individual is frequently unclear. There are numerous instances of predators and prey switching roles, where an adult prey attacks juvenile, weak predators, see [19] and the references therein. This suggests that young prey can grow up, escape from predators, and subsequently become a threat to weak predators. Anti-predator adaptations are biological defenses created by evolution to aid prey creatures in their ongoing conflict with predators. For every stage of this conflict, adaptations have developed throughout the animal kingdom. Recently, some researchers have proposed and studied prey-predator models with anti-predator properties; see for example [20][21]. 3

THE DYNAMICS OF A STAGE-STRUCTURE PREY-PREDATOR MODEL
On the other hand, some ecosystem predators engage in cooperative behavior when hunting and frightening their prey. During hunting, wolves participate as a potential keystone species, and they indirectly impact their prey [22]. Numerous studies examined the function of hunting cooperation in predator-prey systems [23][24]. To our knowledge, no research has been done on the combined impact of hunting cooperation, stage structure, and harvest in a prey-predator system.
The current study aims to simultaneously examine the effects of cooperation and stage structure in a harvested prey-predator scenario.

MATHEMATICAL MODEL CONSTRUCTION
In this section, an ecological model consisting of a prey-predator system has been constructed mathematically. Different biological factors are included in this system according to the following assumptions.
1. The prey population is a stage structure species consisting of the juvenile prey population and the adult prey population, which are denoted to their population densities at time by 1 ( ) and 2 ( ) so that the total prey population density is 1 ( ) + 2 ( ). On the other hand, 3 ( ) is represented the predator population density at time .
2. The prey population grows logistically in the absence of the predation process, while the predator decays exponentially in the absence of their prey.
3. The adult prey has anti-predator defensive property so that it can kill the attacked predator.
However, the predator cooperates in hunting the juvenile prey according to the Lotka-Volterra type of functional response.
4. An external force harvests the prey population only.
According to the above assumptions, the dynamics of the described prey-predator system can be simulated mathematically using the following set of non-linear first-order differential equations.
where 1 (0) ≥ 0, 2 (0) ≥ 0, and 3 (0) ≥ 0 are the initial values of the populations. Moreover, all the system parameters are positive and described in Table 1. The grown-up rate of juvenile prey to adult prey The attack rate of the predator to the juvenile prey.
The hunting cooperative rate of a predator on the juvenile prey.
The harvesting rate of the juvenile prey.
The harvesting rate of adult prey. 1 The death rate of adult prey 2 The death rate of predator The conversion rate of juvenile prey biomass to predator biomass.
The anti-predator rate by the adult prey To simplify the study of a system (1) and reduce the number of its parameters, the following nondimensional system is examined instead of the corresponding system (1).
Now it is easy to prove that, the solution of system (2) has the following properties.  Proof. Assume that 1 ( ) = 1 + 2 is the total prey density, then from the system (2), it is easy to verify that 1 ( ) ≤ , it is obtained: Then according to the lemma (2.1) [25], it is obtained that Therefore, for → ∞, it is obtained that: That completes the proof.

STABILITY ANALYSIS
In this section, the existence and stability analysis of all possible equilibrium points are investigated. System (2) has at most three nonnegative equilibrium points given by: The vanishing equilibrium point is represented by 0 = (0,0,0) always exists.
The predator-free equilibrium point is represented by 1 = ( ̅ 1 , ̅ 2 , 0) where Clearly, 1 exists if and only if the following condition holds.
The Jacobian matrix of the system (2) at the point ( 1 , 2 , 3 ) can be written as: Accordingly, the Jacobian matrix at the point 0 can be written as: The characteristic equation of ( 0 ) can be written as Direct computation shows that the eigenvalues of ( 0 ) are given by: Obviously, the Jacobian matrix ( 0 ) has three negative real parts eigenvalues if and only if the following condition is met.
Therefore, the point 0 is locally asymptotically stable. However, if condition (4) holds (that is the predator-free equilibrium point exists) then the vanishing point becomes a saddle point.
The Jacobian matrix at the point 1 can be written as: with Γ = 1 − ( 1 + 3 )( 4 + 5 ) > 0 due to existence condition. The characteristic equation of ( 1 ) can be written as Direct computation shows that the eigenvalues of ( 1 ) are given by: Obviously, the above eigenvalues have negative real parts provided that the following condition is met.
Therefore, condition (17) guarantees the local stability of the predator-free equilibrium point. 8 DAHLIA KHALED BAHLOOL Now, the Jacobian matrix of the system (2) at the coexistence equilibrium point is computed by: Then the characteristic equation of ( 2 ) can be written by: Theorem 3. The coexistence equilibrium point 2 is locally asymptotically stable provided that the following set of sufficient conditions are satisfied.

PERSISTENCE AND GLOBAL STABILITY
In order to discuss the persistence the possibility of the existence of periodic dynamics in the boundary planes is discussed using a Bendixson criterion [26] that provides a sufficient condition for the non-existence of periodic solutions within simply connected domains in the phase plane.
Since system (2) has only one subsystem falling in the 1 2 −plane and is defined by: Direct computation on the system (24) shows that: Clearly, is not identically zero and does not change the sign over any subdomain of ⊆ ℝ + 2 . Thus the only possible attractor in the boundary planes of the system (2) is the equilibrium point 1 .
In the following theorem, the following Butler-McGhee lemma, which is stated in Freedman and Waltman [27], is used in the proof. Let Ω( ) stand for the omega limit set of an orbit, and let   (2) is globally stable provided that the following condition is satisfied.
Proof. Consider the following real-valued function 0 = 1 + 2 + 3 6 . Clearly, 0 (0,0,0) = 0, Therefore, using the above-given conditions gives that Clearly, 1 is a negative definite and hence the predator-free equilibrium point is an asymptotically stable point. Since, the Lyapunov function 1 is a radially unbounded function in the ℝ + 3 , hence it's a globally asymptotically stable point.   Therefore, by using the conditions (31)-(33), it is obtained that Clearly, 2 is a negative definite under the condition (34) and hence the coexistence equilibrium point is asymptotically stable point and has a basin of attraction satisfies the given conditions.

BIFURCATION ANALYSIS
This section examines the potential that altering a parameter could lead to a change in quality. The Sotomayor theorem [26] causes the system (2) to face a transcritical bifurcation at the equilibrium point 0 as the parameter 1 swings through 1 * .
Therefore, pitchfork bifurcation takes place and the proof is done.
Therefore condition (41) guarantees that system (2) faces a saddle-node bifurcation at the equilibrium point 2 as the parameter 5 swings through 5 * .

NUMERICAL SIMULATION
To verify our theoretical findings and comprehend the impact of changing the parameter values on the system's dynamics, we run some numerical simulations of the system (2). The following fictitious parameters are used for the simulation that follows.

CONCLUSIONS
In this study, a prey-predator system has been constructed mathematically. The prey is thought to be a species with a stage structure that includes juveniles and adults. While the adult prey species possesses antipredator abilities against the predator, the predator cooperates in pursuing juvenile prey. Additionally, the prey was believed to be under the influence of harvest. System (2) was shown to have three nonnegative equilibrium points. The system's persistence as well as local and global analyses of stability were investigated. The Sotomayor theorem is used to describe local bifurcation. It is determined that system (2) has saddle-node and transcritical bifurcation as its two types of bifurcation. Finally, using a fictitious set of parameter values, the following findings are derived numerically. The persistence and stability of the system at the positive equilibrium point are positively influenced by the cooperative hunting rate and the conversion rate of the hunted prey biomass to predator biomass. All other system parameters, on the other hand, have a negative impact on the system's persistence and stability at the positive equilibrium point.