CONTRIBUTION OF HUNTING COOPERATION AND ANTIPREDATOR BEHAVIOR TO THE DYNAMICS OF THE HARVESTED PREY-PREDATOR SYSTEM

: Fear, harvesting, hunting cooperation, and antipredator behavior are all important subjects in ecology. As a result, a modified Leslie-Gower prey-predator model containing these biological aspects is mathematically constructed, when the predation processes are described using the Beddington-DeAngelis type of functional response. The solution's positivity and boundedness are studied. The qualitative characteristics of the model are explored, including stability, persistence


INTRODUCTION
One of the prominent themes in Mathematical Ecology, and particularly in Population Dynamics, has been and remains to be the dynamic interplay between predators and their prey.This is due to its universality, as well as the fact that a more comprehensive understanding of this relationship allows for a better understanding of the movement of food chains or trophic webs.
The first prey-predator model, defined by an autonomous nonlinear ODE system, was proposed by the Italian scientist Vito Volterra in 1926.This model corresponded to a two-dimensional model for biological interactions published previously by American scientist Alfred J. Lotka; for the aforementioned, the ODE system is known as the Lotka-Volterra model [1][2].The primary dynamic feature of this first prey-predator model is that the single point of positive equilibrium is a center, implying that all pathways are concentric closed orbits around that point [3][4].This indicates that given any beginning state, the density size of predators and their prey would continually swing about that point.This behavior of the system solutions was fiercely questioned when they were developed because no prey-predator interactions with these features were seen in nature.
The model developed by British scientist Leslie in 1948 offers a new alternative that does not suit the Lotka-Volterra model scheme, which is based on a notion of mass or energy transfer, the Leslie model distinguishes itself because the predator growth equation, like the prey growth equation, is of the logistic type.Leslie assumed that the predators' traditional ecological carrying capacity relates to the abundance of prey () = , where  denotes prey density [5].When a predator is a generalist and no preferred prey exists, the predator may shift to another food source.
In this scenario, () =  +  , where  > 0 denotes the quantity of other nourishment available to predators or predator carrying capacity in the absence of the prey.As a result, an improved Leslie-Gower system or a Leslie-Gower strategy is produced [6].Subsequently, these systems' applications began to expand.New population dynamics applications have been developed, and these systems had been used to simulate a range of other natural phenomena, see [7][8][9][10] and the references therein.RAID KAMEL NAJI developed in the next section.Section 3 focuses on the system's stability.Section 4 investigates the system's uniform persistence.Section 5 determines the conditions for the occurrence of local bifurcation.We performed various numerical simulations to demonstrate our theoretical findings, which are described in Section 6.Finally, Section 7 discusses the study's findings.

CONSTRUCTION OF THE MODEL
In this section, a prey-predator model with a generalist predator is formulated, indicating it can live without the model's prey population.Hence, it has an alternate food source.This suggests that the per capita growth rate function will be zero at some positive density.The simplest situation is when we describe the dynamics of a predator population using logistic growth in the absence of prey.Taking the simplest version of the predator population's growth rate, the Leslie-Gower preypredator [5] with logistic growth in both prey and predator and general functional response is described as [32] In this case, () > 0 and () > 0 are utilized to represent the magnitude of the prey and predator populations at time .With carrying capacity  and intrinsic growth rate , the prey population grows logistically.The predator's growth is also logistic, with an intrinsic growth rate .Nonetheless, carrying capacity is prey-dependent, with ℎ indicating the importance of the prey as food for the predator.The term  ℎ is called the Leslie-Gower term.
On the other hand, predators can consume other populations when food is short, but their expansion will be limited because their preferred prey is rare.To address this issue, Aziz-Alaoui and Okiye [6] proposed a modified Leslie-Gower model in which a constant  is introduced into the denominator of the Leslie-Gower term that assesses ecological safeguards for the predator in order to avoid singularities when  = 0, so that system (1) becomes Many researchers have now investigated the modified Leslie-Gower models incorporating many different kinds of functional responses, harvesting, the Allee effect [8,[33][34][35][36], and so on.
The modified Leslie-Gower prey-predator model (2) with the Sarkar and Khajanchi fear function [37] that influences the prey's birth rate and the quadratic fixed effort harvesting with the Beddington-DeAngelis type of functional response is proposed and investigated by Jamil and Naji, [9] in the following form: Because of the significance of the prey's refuge, prey refugees are assumed to minimize predatorprey fluctuations and prevent prey extinction [10].A review of the real-world proof suggests that refuges can perform the former purpose.As a result, in the aforementioned dynamical model, the overall amount of prey refuge is dependent on both species.Assume that the amount of prey refuge is  [38], where  is the refuge coefficient.Therefore, the predators prey on the remaining ( − ) prey species, where 0 <  < 1. Accordingly, the dynamics of the above-described model can be written as [27].
Keeping the above in view, this paper considers the influence of hunting cooperation on the model (3) instead of predator-dependent refuge with antipredator behavior, which can be seen in realworld life between wild buffalo and lions.Consequently, the modified Leslie-Gower preypredator system that has hunting cooperation and antipredator behavior can be represented using the following set of differential equations.
where all the parameters are nonnegative and described in Table 1.The birth rate of the prey population and predator population, respectively.

𝑛
The level of fear.

𝑑 1
The natural death rate of the prey.

𝑏
Decay rate due to intraspecific competition.

𝑎
The attack rate. ℎ The Hunting cooperation rate.

𝑐 2
A level of interference between the individuals of a predator.
The catchability coefficients of the prey and predator, respectively.

𝐸
The effort level for harvesting the prey and predator.

𝐾
The carrying capacity of the predator in the absence of its prey.
According to the interaction functions (, ) and (, ), the right-hand side functions of the system (5) are continuous and have continuous partial derivatives, therefore these functions are Lipschitzain.Consequently, depending on the fundamental theorem of existence and uniqueness for the solution of the initial value problems, system (5) with the initial condition (0) ≥ 0, and (0) ≥ 0 have a unique solution.
Proof.The form of System (5) indicates that the system is a Kolmogorov system, with (, ) and (, ) being continuously differentiable functions reflecting the prey and predator growth rates, respectively.Therefore, we can solve (5) using the positive conditions ((0), (0)) to obtain: As a result, of the exponential function's definition, any solution in the .ℝ + 2 = {(, ) ∈ ℝ 2 : () > 0, () > 0} that begins with positive starting conditions ((0), (0)) remains there eternally, due to the previous two equations.HUNTING COOPERATION AND ANTIPREDATOR BEHAVIOR IN PREY-PREDATOR Theorem 2. In the region, All of the solutions to system (5) are uniformly bounded, where  1 ,  2 ,  1 ,  1 ,  2 , , and  are positive constants that satisfy  1 −  1 > 0, which reflects the prey species' survival condition in the absence of the predator.
Proof.From the first equation of System (5), it is obtained that By solving the above differential inequality it is obtained that: =  > 0 , because the survivor species' reproduction rate is naturally bigger than its mortality rate.Now from the subsequent equation of system (5), it is inferred that: Similarly, solving the last differential inequality gives: . Consequently, the total solution of system (5) will be a uniformly bounded solution, hence the proof is complete.
Straightforward computation shows that these two nullclines intersect uniquely at  3 in the region Υ if and only if the following set of sufficient conditions is met, see Figure (1a) using a selected set of data.
Now the Jacobian matrix of system (5) at the point (, ) can be written as: where Therefore, the Jacobian matrix at the entire extinction point  0 becomes: Consequently, the eigenvalues are given by Obviously,  0 is a saddle point if the following condition is satisfied: While it is an unstable node when The Jacobian matrix (7) at predator-free equilibrium point  1 becomes where ). ).
Hence the eigenvalues are given by: Therefore, the equilibrium point  1 is a stable node if and only if the following conditions hold.
It is a saddle point if only one condition of the conditions ( 14)-( 15) holds, while it is an unstable node when both conditions ( 14)-( 15) are reflected.It is a non-hyperbolic point when one condition of ( 14)-( 15) holds while equality occurs at the other condition.
The Jacobian matrix (7) at prey-free equilibrium point  2 turns into: where Therefore, the eigenvalues can be written as: Direct computation shows that, the equilibrium point  2 is a stable node if the following condition is met.
It is a saddle point if condition (18) is reflected, while it becomes a non-hyperbolic point when equality occurs.
Theorem 3. The co-existing point  3 is a sink if and only if the following sufficient conditions are satisfied.
Proof.The characteristic polynomial of the Jacobian matrix ( 19) can be written in the form where   =  ̂11 +  ̂22 and   =  ̂11  ̂22 −  ̂12  ̂21 .According to Routh-Hurwitz criterion, the equation ( 23) have two roots with negative real parts if and only if   < 0, and   > 0. Direct calculations indicate that the above conditions ( 20)-( 22) satisfy the requirements of the Routh-Hurwitz criterion.Therefore, the co-existing point is a sink.
As a result of theorem (4), using the Poincare-Bendixson theorem, the unique co-existing equilibrium point of the system (5) in the interior of the first quadrant is a globally asymptotically stable point.HUNTING COOPERATION AND ANTIPREDATOR BEHAVIOR IN PREY-PREDATOR

UNIFORMLY PERSISTENCE
Mathematically, uniform persistence refers to the presence of an area in the phase plane at a positive distance from the border where population species arrive and must eventually lie, guaranteeing the ongoing existence of species in a biological sense.Uniform persistence is defined analytically as follows.
Theorem 5.The system ( 5) is uniformly persistent provided that Proof.From the first equation of system (1), for  >  1 it is obtained Therefore, due to lemma (2.2) of [40], the following is obtained.
Similarly, from the second equation of system (5), it is observed that Again, using lemma (2.2) of [40], the following is obtained Thus, for arbitrary  > 0, so that  = min{  1 ,  2 }, it is obtained that The proof is done.

LOCAL BIFURCATION
Changes in the qualitative structure of a collection of curves, such as the integral curves of a set of vector fields or the solutions of a set of differential equations, are investigated by bifurcation theory.A bifurcation happens when a slight smooth change in the parameter values of a system results in a major significant shift in its behavior.It is most frequently employed in the mathematical analysis of dynamical systems.Bifurcation might take place in two ways.Local bifurcations are visible when parameters pass through vital thresholds by observing alterations in the regional stability features of equilibria, periodic orbits, or other invariant sets; global bifurcations take place when the system's larger consistent sets clash with each other or with the system's equilibria.They cannot be discovered just by looking at the stability of the equilibria.The where  ∈ ℝ be the parameter.
Where all the new symbols are given in the proof.

SIMULATION ANALYSIS
It is well known that, the natural environment's interaction between prey and predator is one of mutual constraint and control.To further understand the dynamic connection between prey and predator, numerical simulations of the model ( 5) will be run to demonstrate some complicated dynamic behaviors.For simplicity, we set the parameters values as follows: for the data set (34) with  1 = 0.1.Now, to understand the role of  1 on the dynamics of the system (5), the numerical solution was obtained for different values of  1 .It is observed that for  1 ≤ 0.1 there are only two boundary points  0 and  2 = (0,1.53),which are source and sink (stable node) respectively.For, 0.1 <  1 ≤ 1.88 the system has three boundary points  0 ,  1 , and  2 = (0,1.53),which are source, saddle, and sink respectively.However, for 1.88 <  1 ≤ 2.89, system (5) has three boundary points  0 ,  1 and  2 = (0,1.53)with two co-existing points  31 and  32 so that the system exhibits bistable case between  2 and  32 while the rest of points are unstable.For 2.89 <  1 ≤ 3.63, there is a unique co-existing point with three boundary points and the system exhibits bistable between  2 and  3 up to  1 = 3.13 after that  3 becomes a worldwide sink.Finally, for  1 > 3.63 the system (5) has three boundary equilibrium point only  0 ,  1 , and  2 = (0,1.53)behave as source, worldwide sink, and saddle point.To explain the above-obtained results, Figure (2) is obtained using a numerical solution of the system (5) depending on parameters (34) with the selected values of  1 .The role of  1 on the dynamic of system (5) in studied numerically and the obtained results give the following.For  1 ≤ 0.18 system (5) has two co-existing points with three boundary points and undergoes a bistable dynamic between  2 and  32 while the other points are unstable, see  It is observed that, the parameter  1 has similar influence of the dynamic of system (5) with different bifurcation positions as that obtained with  1 , see Table (2) below.
The impact of varying the parameter  on the dynamics of the system (5) is studied numerically and the results show that, for  ∈ (0,0.38], ∈ (0.38,0.92), and  ≥ 0.92 the system (5) has a unique co-existing equilibrium point  3 , two co-existing equilibrium points  31 with  32 , and not co-existing equilibrium points, respectively.Moreover, the systems have, respectively, a worldwide sink at  3 , a bistable case between  2 and  32 , and worldwide sink at  2 .Some typical results are presented in Figure (7).It is observed that, the parameter ℎ and  have similar influence of the dynamic of system (5) with different bifurcation positions as that obtained with , see Table   Similar behaviors have been obtained, as those shown in the case of varying  1 in the system (5) when the parameters  2 ,  2 , and  are varying with different bifurcating positions, see Table (2).
A study using numerical simulation for the influence of the parameter  2 on the dynamic of the system (5) has been carried out.It is obtained that, for the  2 ≤ 0.  Finally, the role of varying  2 on the dynamic of system ( 5) is investigated numerically and the following results are obtained.For the range  2 ≤ 0.02 the system have three boundary

( 2 )
HUNTING COOPERATION AND ANTIPREDATOR BEHAVIOR IN PREY-PREDATOR

Figure
Figure (2c)-(2(d)when  1 = 0.1.However, for 0.18 <  1 < 1, the co-existing points disappear from the system and only three boundary points are there, where  2 is a worldwide sink.Figure(6) shows these results at a specific value of  1 .
35 the system has only three boundary equilibrium points  0 ,  1 , and  2 with  1 being the worldwide sink.For 0.35 <  2 ≤ RAID KAMEL NAJI 0.54 a unique co-existing equilibrium point  3 is born but the system (5) still has a worldwide sink at  1 .However, for the range 0.54 <  2 ≤ 1.11 two co-existing equilibrium points  31 ,  32 have appeared in addition to the three boundary points and the system (5) has a bistable case, see Figures(2c)-(2d) for explain.Finally, for  2 > 1.11 the system (5) has only three boundary equilibrium points with a worldwide sink at  2 .To explain the obtained results, Figure (9) is drawn for some selected values of  2 .
Therefore, ∆ 41 ≠ 0 provided that condition (29) is satisfied.Thus from the values of ∆ 11 , ∆ 21 , ∆ 31 , and ∆ 41 , the Sotomayor theorem of local bifurcation leads to the fact that system (5) enters into pitchfork bifurcation around the predator-free equilibrium point and that completes the proof.