DELAY IN ECO-EPIDEMIOLOGICAL PREY-PREDATOR MODEL WITH PREDATION FEAR AND HUNTING COOPERATION

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INTRODUCTION
Although epidemiology is a significant research subject in its own right, a recent tendency has emerged to combine it with ecology to better understand species interactions in ecosystems under epidemiological variables.As a result, it gave birth to a new field: eco-epidemiology.
Studying the effect of disease in the context of interspecies interactions is more realistic than studying it in isolation because no species lives in isolation in nature but interacts with a huge number of other species, either directly or indirectly.The spirit of this new route is the desire to learn about the effects of disease in prey-predator models.Anderson and May [1] were among the first to pioneer eco-epidemiological modeling.They investigated a prey-predator paradigm in which both the prey and the predator were infected.Following their pioneering work, a large number of scholars examined eco-epidemiological systems, see [2][3][4][5][6][7][8][9][10][11], which is still ongoing.
The incubation period of a disease is very important in infectious disease modeling because it determines the dynamics of disease transmission and predicts the future evolution of disease outbreaks.As a result, researchers are working to create mathematical models that more precisely reflect reality by modeling time delay.Maiti et al [12] investigated a Crowely-Martin prey-predator model in which the prey population was afflicted with illness.The nonlinear incidence rate at which vulnerable prey are infected is taken into account.Due to the existence of an incubation period, a time lag is added to convert the vulnerable prey to the infected one.Samanta et al [13] suggested and investigated a nonautonomous prey-predator model with disease in prey and a discrete-time delay for disease transmission incubation.While Hussien and Naji [14] studied the impact of the incubation time delay on disease transmission in a prey-predator system with a SI type of disease in the prey population using a nonlinear incidence rate.A modified Holling type II functional response was employed to illustrate the predation process.Further studies regarding the DELAY IN ECO-EPIDEMIOLOGICAL PREY-PREDATOR MODEL impact of incubation time delay on the dynamics of eco-epidemiological systems may be found here [15][16] and the references therein.
The importance of incorporating the influence of a predator's hunting cooperation techniques in destabilizing the prey population by creating fear is one of the fundamental insights in the biological models.This is due to the fact that psychological influences can have a greater impact than direct physical predator attacks, causing the prey to migrate from more susceptible regions to less risky ones, disperse, and lose attention to key biological tasks that are part of their daily routine.
For example, certain studies, such as those in [17][18][19], found that fear can reduce prey populations' reproductive rate and bioactivity, lowering disease transmission rates.However, it is observed that memory dependence, fear, hunting cooperation, and therapy have an impact on the fractional-order eco-epidemiological model's qualitative behavior, see [18].On the other hand, the hunting cooperation technique affects the stability of a three-species food chain model and increases anxiety in both prey and mid-level predators, forcing them to seek refuge, see [19].Fear of predation has many additional causes, and its prevalence is not restricted to the presence of hunting cooperation.Therefore, numerous studies have been conducted to investigate the impact of fear on the qualitative behavior and dynamics of prey and predator systems, see [20][21][22][23][24] and the references therein.
Based on the preceding, the study of eco-epidemiological models where there is a delay in disease transmission due to the presence of an incubation period has become a vital necessity to understand and preserve system stability by preventing the spread of infectious illness.In this study, we created a mathematical model to mimic the prey-predator system where there is an infectious sickness in the predator community with the attribute of hunting cooperation, which causes fear in the prey community as a defense against predation.The following is how this paper was structured: In Section 2, an eco-epidemiological model was established that considered infection transmission delays, hunting cooperation, and the cost of fear.Section 3 discusses solution positivity and boundedness.Section 4 addresses the presence of the model's equilibrium points.Section 5 addressed the stability of the delay model, whereas Section 6 created favorable conditions for Hopf bifurcations.Section 7 looks into the physical properties of Hopf bifurcation limit cycles.Section 8 contains numerical simulations to validate the analytical results obtained in this paper.With a quick conclusion, the work is completed.

MODEL FORMULATION
This section considers the following biological hypotheses to mathematically simulate the realworld eco-epidemiological system.
• The prey population () has the potential to reproduce and increase naturally because of the food provided by the environment, according to the logistic growth with intrinsic growth rate  −  1 and carrying capacity • The predator can cooperate to pursue their prey, and the predation process follows the Lotka-Volterra type of functional response.
• The severity of predation causes fear in prey individuals, resulting in a decline in birth and biological functioning.• The disease is thought to be spread through contact between () and (), but it is not inherited genetically.Furthermore, an incubation period results in some lag in infectiousness at the time period .
• The disease leads to death in addition to the natural death of predators.
In light of the aforementioned biological hypotheses, Figure 1 The biological meanings for all of the positive parameters in system (1) are shown in Table 1.
Table 1.Biological description of the system (1) parameters.

𝑟
The birth rate of the prey population

POSITIVITY AND BOUNDEDNESS
Before proceeding with the study, confirm that the proposed model is biologically sound.
Consequently, in this section, we present the following theorem, which explores the positivity and boundedness of the system.
The positivity of predator species is now demonstrated, from the susceptible equation of system (1), we have  Hence, all the solutions of system (1) are bounded.

THE SYSTEM'S (1) EQUILIBRIA
It is worth noting that system (1) has no more than four non-negative equilibrium points.The followings are the equilibrium points and their existence conditions: The total extinction equilibrium point (TEE),  0 = (0,0,0), always exists.
The positive equilibrium (PE),  * = ( * ,  * ,  * ), where while  * is the positive root of the following equation: where It is worth noting that equation ( 6) has a unique positive root; hence, the PE exists uniquely if the DELAY IN ECO-EPIDEMIOLOGICAL PREY-PREDATOR MODEL following conditions are met.  > 0 ;  = 5,6. (7)

STABILITY ANALYSIS
One of the strategies to analyze the nature of the stability of the equilibrium is through linearization.Accordingly, define  ̃= (  ̃,  ̃,  ̃) as an arbitrary equilibrium point of the system (1).Further, let () =  1 () +  ̃, () =  2 () +  ̃, and () =  3 () +  ̃.Then, the linearized system at  ̃ can be expressed as follows: where Consequently, the characteristic equation of the system (1) at  ̃ can be determined from the following characteristic equation: Next, the behavior of the each equilibrium point of system (1) for all  ≥ 0, is analyzed in the following theorems: Theorem 2. The TEE of the system (1) is unstable saddle point when  1 < .
Theorem 3. The AE of the system (1) is locally asymptotically stable if and only if the following condition is met.
Otherwise, the AE is unstable saddle point.
Proof: For the equilibrium point AE that given by  1 , the characteristic equation ( 9) can be expressed as: Clearly, the equation ( 12) have three real roots given by Thus, condition (11) guarantees that, AE is locally asymptotically stable.Otherwise,  1 is unstable saddle point due to existence of negative eigenvalues.
Theorem 4. Assume that the following condition holds.
Then, the IPFE of the system (1) is locally asymptotically stable for all  ≥ 0 if and only if the following condition holds.
It is unstable saddle point for all 0 ≤  <   = is replaced by the following condition Where all the new symbols are defined in the proof.DELAY IN ECO-EPIDEMIOLOGICAL PREY-PREDATOR MODEL Proof: For IPFE that given by  2 , the characteristic equation ( 9) become which gives where Consequently, due to Routh-Hurwitz criterion, a quadratic factor in equation ( 16), has two eigenvalues with negative real parts if the condition ( 13) is met.While, the transcendental factor in equation ( 16) is subject to proposition (1) [26], which gives that: If condition ( 14) holds, then all the roots of the transcendental equation have negative real parts for any  ≥ 0. Thus, the IPFE that given by  2 is locally asymptotically stable.
However, if condition (15) holds then the transcendental equation have positive real parts roots for 0 ≤  <   .Hence, the IPFE is a saddle point.This is complete the proof.
Then,  * is locally asymptotically stable for  = 0 and becomes unstable for  ≥  ^ for a critical value  ^> 0 if in addition to the set (17) the following requirement is met where   's are specified in the proof.
Proof: For the PE that given by  * , the characteristic equation ( 9) will be written as: Direct computation leads to have the following characteristic equation of the system (1) at  * : where () =  4  2 +  5  +  6 . With () = () + () represents the general eigenvalue of equation (19).Then substituting this eigenvalue in the equation ( 19), isolating real and imaginary parts yields: Now, since the stability of the point  * is changed when the real part of the eigenvalues () crosses the imaginary axis from the left to the right at a specific value of  =  ^ with ( ^) > 0 to be exist.Therefore, substituting  = 0 in the equations ( 21)-( 22), yields: By squaring and adding ( 23) and (24), it is arrived the following algebraic equation of : Let =  2 , then the equation ( 25) is transform to: where Obviously, condition (18) guarantees that equation (26), and hence (25), has at least one positive root ( ^) = √.Thus, for  ^≤ , the equation ( 19) has roots with the positive real part that makes  * unstable.
Moreover, it is well known that the equation ( 26) possesses three roots represented by  1 ,  2 , and  3 .At least one of them, say   ;  = 1,  2,  3, will be positive provided that conditions ( 17)-( 18) are satisfied.Hence, the equation ( 25) will have a positive root  0 = √  .Hence, based on the equations ( 23) -( 24), if   > 0, the corresponding  ^> 0, can be determined such that ) where  ^= min   ^.In order to complete the proof criteria for  =  ^, we will explore this in the next section.

HOPF BIFURCATION DYNAMICS
In this section, the dynamics behavior of the system (1) around the PE at  =  ^ is investigated.
Theorem 6.The system (1) will lose its stability and undergo a Hopf bifurcation at  * when  =  ^, if the following condition is met where , , , and  are established in proof.
Proof: According to the theorem (5), system (1) losses its stability at  * when  =  ^, and the system has a complex conjugate eigenvalues () = () ± (), where  in the neighborhood of  ^ with ( ^ ) = ± 0 , and  0 > 0. Hence the first requirements of the occurrence of Hopf bifurcation around  * is satisfied.Therefore, the system (1) is said to be undergo Hopf bifurcation if the transversality criterion where ≠ 0, provided that condition (28) is satisfied.As a result, a Hopf Bifurcation occurs at  =  ^.This ends the proof.
Therefore, according to equation ( 43) the values of  20 () and  11 () can be estimated.As a result, determining the coefficients   become possible.Finally, the following biological expressions will be generated based on the values of   .
Furthermore, the following theorem will be used to explore the dynamical features of cyclic solutions caused at  =  ^ using the preceding formulas [14].
Theorem 7. The following outcomes are achieved for the system (1) at  =  ^.
ii.The bifurcating cyclic solutions on the center manifold are stable (unstable) if iii.Period of the bifurcating cyclic solutions increases (decreases) if  2 > 0 (Τ 2 < 0).

NUMERICAL SIMULATION
The key results are numerically shown in this section using the biologically plausible hypothetical set of values listed below.The goal is to validate the theoretically generated results and understand the parameters' influence on the system dynamics (1).It is obtained that for the ranges (0,0.45),[0.45,0.88),and [0.88,1] the solution of system (1) goes asymptotically to  * ,  2 , and  1 respectively as in Figure (5a), (5b), and (5c) at selected values.
The behavior of system (1) dynamics as a function of the parameter  is studied numerically and the obtained results are explained in Figure (9) at some selected values.when  3 = 0.9.
On the other hand, the influence of delay on the system (1) dynamics is investigated numerically using the following set of data.

CONCLUSIONS
The impact of fear and hunting cooperation on the dynamics of a delayed prey-predator system with predator sickness is theoretically stated and then examined analytically and numerically in this work.The Lotka-Volterra functional response depicts the change of food from prey to predator.
Due to the incubation period, it is thought that there is a time lag between becoming infected after interaction between susceptible and infected predator individuals.The suggested system contains at most four nonnegative equilibrium points, as observed.The stability analysis around them is examined in two cases: when  = 0 and when  > 0. It is explored whether Hopf bifurcation can occur around the inner positive equilibrium point.Furthermore, the center manifold theorem was used to examine the direction and stability of the bifurcation periodic dynamics.Finally, a thorough numerical analysis was performed to comprehend the impact of factors on the system's dynamics.
The following conclusions are drawn from the numerical simulation.The fear rate has a stabilizing effect on the suggested system's dynamics.The delay, on the other hand, has a DELAY IN ECO-EPIDEMIOLOGICAL PREY-PREDATOR MODEL destabilizing influence on the dynamics of the suggested system.The hunting cooperation rate has a destabilizing influence on the dynamics of the system (1) until a certain value is reached, at which point the infected predator dies.All of the system's mortality rates (1) have an extinction effect on the system.The growing intrinsic growth rate of the prey species has a coexistence impact on the system (1) dynamics, and the system becomes stable at the positive equilibrium point up to a vital value before losing stability and undergoing a Hopf bifurcation.In contrast, the rising intraspecific competition rate stabilizes the system at the coexistence equilibrium point up to a vital value, and then it is causing the system species to gradually extinction.Finally, the conversion rate of prey biomass into predator biomass (similar to the infection rate) generates system persistence, and the solution is stable at the positive equilibrium point.

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The development of virus resistance and the unprotected connection among ecosystem individuals generates disease infection within the predator population, dividing it into two compartments () = () + () , where () represents the susceptible individuals and () represents the infected individuals at the time .
Fig. 1.Block diagram of the model

Figure ( 2 )
Figure(2) shows that system (1) has four bifurcation points falling in the  range, confirming the theoretical results.