Fear and Hunting Cooperation's Impact on the Eco-Epidemiological Model's Dynamics

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INTRODUCTION
Cooperation during hunting is a common habit among several large predators that improves the predator's skill to catch prey and can cause fear, which lowers the prey's birth rate [1].
Understanding the predation connections among species in an ecosystem largely depends on examining prey-predator models [2][3][4].Predators must consume food in order to survive, hence they frequently work to improve their skills to catch and kill prey since it is more beneficial to their long-term survival.Some animals frequently employ the method of cooperative hunting to increase their capacity to acquire and kill prey [5][6][7][8][9][10].Several cooperative hunting tactics are explored [11], including the advantages of living in a social group, and how predators create cooperative groups with individuals who are actively engaged in prey capture.Although animals of the same species typically compete with one another for resources, some predators cooperate with one another and offer assistance to one another, because of a certain hunting process.
The study of the various mechanisms relating to the prey-predator relationship is one of the main subjects in ecology and evolutionary biology, which are considered by many researchers, see for example [12][13][14][15] and the references mentioned therein.Therefore, cooperative hunting and their induced prey's fear during the predation process is the most potent factor in a prey-predator relationship, especially when it comes to changing behavior that can affect both prey and predator features.Understanding these complicated situations has benefited greatly from the use of mathematical models.Since direct predation is very simple to observe in nature, most existing prey-predator models rely on the traditional Lotka-Volterra paradigm [16].It usually presupposes that predators can only affect the prey population by direct killing.However, the presence of a predator may drastically alter the prey's physiological and behavioral characteristics to the point where it may have a greater impact on the prey population than direct predation [17] and the mentioned references therein.There are many research studies using mathematical modeling considered the impact of hunting cooperation [1,[17][18][19][20][21][22][23][24][25] and the impact of fear [16,[26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] in the prey-predator models separately.Numerous prey species alter their behavior when predators are around because of the possibility of being eaten.In fact, the researchers have noted that prey that are afraid of predators have lower reproductive rates [42].
Prey clustering is a common antipredator activity in many prey species, and some studies have looked at how group size changes in response to the risk of predation.When the prey are in a group, the greatest advantage is enhanced predator detection.Predators can sometimes find larger groups easily, though, see [43].
Similar to how Kermack-McKendrick's groundbreaking research on SIRS (susceptible-infectiveremoval-susceptible) epidemiological models has drawn significant interest from experts.
Because infectious diseases are primarily spread through contact between species, there is a high risk of disease transmission between prey species, predator species, or both because of their frequent interactions, particularly when hunting cooperation and fear are present.As a result, eco-epidemiology emerged as a brand-new field of study.Anderson and May [44] were the first to introduce eco-epidemiological models.Numerous ecologists and eco-epidemiologists have been familiar with eco-epidemiology due to the importance of protecting wild creatures [45].
Recent research has merged eco-epidemiological prey-predator models with hunting cooperation and fear.For example, an eco-epidemiological model with disease in the prey population that incorporates the fear effect of predators on prey and cooperative hunting among predators was proposed and researched by Liu et al [23].They demonstrated the occurrence of several limit cycles for low disease transmission rates and predator mortality rates, and they also demonstrated that the system becomes stable at high disease transmission rates and predator mortality rates.An eco-epidemiological model that Fakhry and Naji [24] devised and researched included diseased prey devoured by a predator with a fear cost and hunting cooperation property.It is assumed that the predator couldn't tell the difference between healthy and sick prey, therefore it ate both.They discovered that the persistence of the system and the spread of sickness are both influenced by the presence of fear.The created fear, however, might stop the spread of disease in the event that hunting cooperation rates increase.Finally, the development of a mathematical model by AL-Jubouri and Naji [25] was adopted in order to explain how the interaction between the prey and the predator changes in the presence of infectious disease as well as the predator community's propensity for cooperative hunting, which instills enormous anxiety in the prey community.Additionally, the existence of a period of incubation for the illness delays the spread of the illness from sick predators to healthy predators.They discovered that the dynamics of the system are stabilized by the fear rate and destabilized by the delay.On the other hand, until a specific value is achieved, at which point the infected predator dies, the hunting cooperation rate has a destabilizing effect on the dynamics of the system.
The novelty of the topic investigated in this research, in contrast to the studies previously mentioned, lies in the existence of infectious diseases inside the predator community with the potential for curing the sickness based on treatment.In order to ascertain the dynamic behavior of the suggested epidemic ecology, it is crucial to investigate the effects of predator-hunting cooperation and the fear that is produced in the prey population.Therefore, the goal of the current work is to develop and examine an eco-epidemiological prey-predator model that includes predation fear and cooperative hunting.It is thought that the sickness in the predator is of the SIS kind, meaning that the sick predator can recover and revert to a vulnerable state with the help of medical treatment.We present the mathematical formulation in the section that follows.In Section 3, we look at the characteristics of the model's solution; in Section 4, we look at the viability of equilibria and their stability; in Section 5, we look at persistence; in Section 6, we talk about global stability; and in Section 7, we talk about the occurrence of local bifurcations.In Section 8, a numerical simulation is carried out.The paper concludes properly with a biological conclusion in Section 9.

MODEL FORMULATION
In the following, the adopted assumptions to build the mathematical model that describes the eco-epidemiological system are stated.
1. Let () be the density of the prey biomass at time , and () = () + () be the density biomass of the predator at time , where () is separated into two compartments due to the presence of the disease: susceptible population () and infected population ().
2. In the absence of the predator, the number of prey increases logistically.In the lack of food, the predator decays exponentially.
3. The disease is thought to be of the SIS kind, and it is transmitted only between predator individuals by contact between an infected predator and a healthy predator, rather than genetically.The medicine administered to the infected predator also cures the sickness.
4. Because the predator has a hunting cooperation behavior, it attacks the prey in a group.Fear of predation is generated in a prey population as a result of this.
Accordingly, the stated eco-epidemiological system's dynamic can be represented by the following set of nonlinear first-order differential equations.
where (0) =  0 ≥ 0, (0) =  0 ≥ 0, and (0) =  0 ≥ 0 represent the initial condition of the system (1), and all parameters are assumed nonnegative and can be described in Table 1.The infected predator death rate combined natural and disease death rates The conversion rate of prey biomass to predator biomass The following transformations were applied to remove all units from the system (1).

PROPERTIES OF THE SOLUTION
This section discusses the characteristics of the system (2)'s solution, such as positivity and boundedness, as provided in the following theorems.
Theorem 1: All system (2)'s solutions with the initial conditions belonging to .ℝ + 3 are positively invariant.
Proof.It is derived from the system's first equation ( 2): Integrating the above equation within the range [0, t] yields: Similarly, for the second and third equations, it is obtained This completes the proof.
Theorem 2: All system (2)'s solutions with the initial conditions belonging to ℝ + 3 are uniformly bounded Proof.From the first equation of system (2), it is easy to verify that Then according to the lemma (2.2) [54], it is obtained that 1 +  2 +  3 , then by using system (2) equations, derive W with respect to t gives where  = min{1,  7 ,  8 }.Hence, simple manipulation yields So, according to the lemma (2.1) [54], it is obtained that Therefore, for  → ∞, it is obtained that: That completes the proof.
While the other coefficients   ;  = 1,2, … ,9 were computed using Mathematica software and have complicated forms, so it's omitted.Consequently, equation ( 7) has at least one positive root provided that the following condition is met.
Moreover, the PEP exists in the interior of ℝ + 3 , if in addition to condition (8) the following condition holds.
The following calculated Jacobian matrix (JM) can be used to study the local stability analysis of the aforementioned EPs. where The Jacobian at the TEEP can be written as: Therefore, the eigenvalues of ( 1 ) are given by As one of the eigenvalues is positive and the others are negative, hence,  1 is a saddle point.
The Jacobian at the AEP can be written as: Therefore, the eigenvalues of ( 2 ) are given by Hence, the AEP is locally asymptotically stable (LAS), nonhyperbolic point, and saddle point provided that the following conditions are met respectively.
The Jacobian at the DFEP can be written as: where Direct computation shows that, the eigenvalues  31 and  32 have negative real parts if the following conditions hold.
While the third eigenvalue  33 is negative if the following condition is met.
Therefore, the DFEP is a LAS if the conditions ( 21)-( 23) are satisfied.
Finally, the Jacobian at the PEP can be written as: where The characteristic equation of ( 4 ) can be written as where The characteristic equation (25), according to the Routh-Hurwitz criterion, has three eigenvalues with negative real portions if the following conditions are met:  1 > 0;  3 > 0, and ∆=  1  2 −  3 > 0.Moreover, the Routh-Hurwitz requirements are satisfied if the conditions given in the following theorem hold.

Theorem 3:
The PEP of the system (2) is LAS if and only if the following sufficient conditions are met.
Proof.Direct application of the Routh-Hurwitz criterion with the given condition the proof follows.

PERSISTENCE
The persistence and extinction properties of an eco-epidemiological model including fear and hunting cooperation are investigated, in this section.The goal is to examine how fear and hunting cooperation affect the persistence and extinction of system species in a sick prey-predator system.
The system's border levels' dynamics must be understood to understand the conditions that assure the continued existence of all species.Clearly, system (2) has a subsystem representing the infected predator's absence, which can be written as: It is noted that, subsystem (31) has three equilibrium points given by  11 = (0,0),  12 = (1,0), and , which are coincide with the projection of TEEP, AEP, and DFEP on the  1  2 −plane respectively.As a result, they have the exact prerequisites for local stability.The Dulac-Bendixon criterion is now used to evaluate the possibility of not existence of periodic dynamics in the interior of positive quadrants corresponding to subsystems (31).
Theorem 4: There are no periodic dynamics that fall entirely in the interior of a positive quadrant of  1  2 −plane provided that Proof.Consider the continuous differential function ( 1 ,  2 ) = = − 1  2 (1+ 1  2 ) +  3 .It's clear that ∆ has the same sign and does not equal zero under the condition (32).Therefore, as a result of the Dulac-Bendixon criterion, a subsystem (31) lacks periodic dynamics in the interior of the positive quadrant of the  1  2 −plane.
Theorem 5: Assume that there is no periodic dynamics in the boundary planes of ℝ + 3 .Let the conditions ( 17), (21), and ( 22) with the next condition are met, then the system (2) is uniformly persistent.
Proof: Assume  is a point in the interior of ℝ + 3 and () is the orbit through it.Let Ω() represent the ⎼limit set of ().It is worth noting that because the system (2) is bounded, so is Ω().To demonstrate that  1 ∉ Ω(), the opposite is assumed first.Because  1 is a saddle point, it cannot be the only point in Ω(), and hence there must be at least one other point  such that  ∈   ( 1 ) ∩ Ω(), where   ( 1 ) is the stable manifold of  1 , see Butler-McGhee lemma [55].
Given that the stable manifold of  1 is given by the  2  3 −plane, it is included in Ω().As a result, if  lies on the boundary axes of the  2  3 −plane, then the positive particular axis (containing ) is contained in Ω(), which contradicts its boundedness.
Let  now belong to the inside of the  2  3 −plane.Because there is no other equilibrium points in the interior of the  2  3 −plane, the orbit through  included in Ω() must be infinite.Giving a contradiction also demonstrates that  1 ∉ Ω().
To demonstrate that  2 ∉ Ω(), the opposite is also assumed.Because  2 is saddle under condition (17) with stable manifold given by  1  3 −plane, the evidence is the same as in the proof of the first point  1 .As a result,  2 ∉ Ω() is found.
Because there are no periodic dynamics in the boundary planes of ℝ + 3 , and the above points  1 ,  2 , and  3 are the only potential attractive points for the solutions of system (2), system (2) uniformly persists.

GLOBAL STABILITY ANALYSIS
In this section, we study the global asymptotic stability (GAS) of all single-existence equilibrium points using the Lyapanov function whenever it exists as described in the following theorems Theorem 5: Assume that condition (15) holds, then the AEP is a GAS provided that the following condition holds.
Proof: Consider the following positive definite real-valued function around the AEP Then, we have , and  2 =  3 = 1 with use of the theorem (2), it is obtained that Clearly, condition (34) gives that is negative definite.Moreover, since  1 is radially unbounded function, then AEP is a GAS.Theorem 6: assume that the DFEP denoted by  3 = (̅ 1 , ̅ 2 , 0) = ( 1 ,  2 , 0) is a LAS, then it is a GAS provided that the following sufficient conditions hold.
Proof: consider the following positive definite real valued function around  3 Hence, we have . By using  1 =  3 ,  2 =  3 =  2 it is obtained that: Then we have is negative definite.Moreover, since  2 is radially unbounded function the DFEP is a GAS.

LOCAL BIFURCATION
Sotomayor's bifurcation theorem [56] was applied to determine the possibility of local bifurcation near the equilibrium points of the system (2) when the parameter passes through a specific value making the equilibrium point a non-hyperbolic point.The condition that the equilibrium point is non-hyperbolic is a necessary but not sufficient condition for a local bifurcation to occur as is known.The study of the bifurcation of the system (2) is vital because the parameters are not constant in actual reality and are constantly changing according to the conditions of the environment containing the organisms of the system.Now, rewrite the system (2) in the vector form as:   = (, ),  = ( 1 ,  2 ,  3 ) T ,  = ( 1  1 (, ),  2  2 (, ),  3  3 (, )) T , (39) where  ∈ ℝ represents a bifurcation parameter.Hence the second and third directional derivatives for ( 39) can be written respectively as: where  = ( 1 ,  2 ,  3 )  be any vector with ( 6 + 3 ) 3 )}  31 = 2 3 ( 2  4 +  3  5  6 ( 6 + 3 ) 3 ) And where .
Therefore, the eigenvalues of  1 * are given by Therefore,  1    3 ( 2 ,  3 * ) = 0, as a result, the first requirement for TB is met.Moreover, since Therefore: Now using equation ( 40), gives Hence a TB take place near AEP when  2 ≠ 1. Otherwise using equation (41) gives where Accordingly, obtained Therefore, PB takes place near AEP, and the proof is complete.
Theorem 8: Assume that conditions ( 21)-( 22) hold, and the parameter  4 passes through the value , then system (2) undergoes a TB at DFEP provided that the following condition holds Otherwise, PB takes place, where all the new symbols are defined in the proof.
Proof.From the equation (18)  where where all the new symbols are defined in the proof.where Therefore, SNB takes place as the  7 =  7 * .

NUMERICAL SIMULATION
The purpose of this section is to explore the diseased prey-predator interaction contained by the system (2) using numerical approaches.We are particularly interested in the consequences of prey fear, sickness, and predator-hunting cooperation.Unless otherwise specified, parameter values are fixed according to the biologically feasible set listed below.

CONCLUSIONS
This research involved the study of the impact of both fear and hunting cooperation on the eco-epidemiological system of the prey-predator.The mathematical model was first formulated and then the proposed model was studied theoretically using known mathematical methods and it was noted that the model contains at most four equilibrium points.The proposed system's stability, uniform persistence, and local bifurcation analysis were performed, and all of the prerequisites required to acquire these concepts were determined.To confirm the analytical findings and understand the impact of parameters on the dynamic of the system (2), numerical simulation was used.Figure (2) shows that fear has an extinction effect on the infected predator.
Concerning the recovery rate and infected predator mortality rate, similar results have been observed with fear levels.While, as shown in Figure (3), the hunting cooperation level first has a stabilizing influence on the system's (2) dynamic behavior and later, at a critical threshold, becomes an extinction factor for the infected predator.The conversion rate has a beneficial effect on the overall coexistence of the system since it is a stabilizing component at the positive equilibrium point at the start, but when it exceeds a certain point, it has an instability effect and the system switches to cyclic dynamics, see Figure (4).The infection rate (similarly, treatment rate depletion) has a stabilizing influence on the system dynamics at the PEP, see Figure (5).Finally, as shown in Figure (6), healthy predator death rate works as a extinction factor for the predator species.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.

Figure ( 1 )
Figure(1) shows that system (2) has a distinct PEP that is asymptotically stable and dependent on the data collection(45).To investigate the effect of modifying the parameters on the dynamic of the system (2), the numerical solution is calculated using data set (45) by varying one parameter at a time, and the results are then displayed in the form of phase portraits and their time series.It is observed that as  1 ∈ [0,1.3) and  1 ≥ 1.3 the solution of system (2) approaches to PEP and DFEP respectively, see Figure(1) for the first case and Figure(2) for a selected value of  1 in the second case.Similar behavior is obtained as that of  1 when the parameters  5 and  8 varying.

Table 1 :
The parameters description