THREE-SPECIES FOOD CHAIN MODEL WITH CANNIBALISM IN THE SECOND LEVEL

: This article suggests and explores a three-species food chain model that includes fear effects, refuges depending on predators, and cannibalism at the second level. The Holling type II functional response determines food consumption between stages of the food chain. This study examined the long-term behavior and impacts of the suggested model's essential elements. The model's solution properties were studied. The existence and stability of every probable equilibrium point were examined. The persistence needs of the system have been determined. It was discovered what conditions could lead to local bifurcation at equilibrium points. Appropriate Lyapunov functions are utilized to investigate the overall dynamics of the system. To support the analytical conclusions, numerical simulations were done to validate the model's inferred long-term behavior and to comprehend the implications of the model's significant parameters


INTRODUCTION
Food chains are significant environmental phenomena in several academic fields, including ecological science, applied mathematics, engineering, and economics. In a food chain model species, energy and resources flow in a single direction; however, food webs are complex because they comprise multiple food chains [1]. In a feeding chain, various trophic levels have been seen.
Many types of organisms, including producers, consumers, and decomposers, can be found in the stimulation phases. On the other hand, a formation-wise lattice architecture is used in a food web.
To describe the food chain as a system of differential equations, mathematical analysis, and modeling techniques could be employed. A food web is a conglomeration of food chains, although food chains are referred to as "food chains" in ecology [2][3].
Another intriguing aspect of the prey-predator relationship is cannibalism. When an animal consumes members of its own species, this behavior is known as cannibalism or intraspecific predation [4]. There has been a lot of discussion on how cannibalism affects environmental strategy for decades [5]. Cannibalism is influenced by a number of crucial variables, including population density, temperature, population size, developmental stage, and more [6]. Some researchers have looked into the mathematical representation of cannibalism, see for example [7][8][9]. It is intriguing to explore a prey-predator model with cannibalism because many animals in nature exhibit cannibalistic behaviors. Cannibalism has been observed in a wide range of animal species, including carnivore mammals, frogs, monkeys, spiders, fish, and insects, see [10]- [14].
In addition to cannibalism, the ecological term for the behavior of prey that hides after being trapped and attacked by predators is a refuge. Many prey species use the refuge strategy to ward off predators. Sea urchins conceal their young from crab predators in articulated coralline algae, while Daphnia hides its young from crab predators in shallow lakes in the Mediterranean [15][16].
In addition to prey's natural behavior, humans can help prey by creating conservation forests [17], natural areas, wildlife reserves, or even basic security. The mathematical model of prey-predator with prey refuge has also been the subject of many investigations [18][19][20][21].
Recent studies have shown that predators affect refuge prey populations in ways other than just killing the prey; they also instill fear in the prey, which reduces the prey birth rate [22][23][24].
Predator-induced fear keeps prey animals out of open settings, denying them the freedom to carry 3 THREE-SPECIES FOOD CHAIN MODEL WITH CANNIBALISM out regular activities like mating. As a result, their capacity for reproduction is decreased by their fear of predators. It is critical to consider the price of anxiety as a decrease in reproduction. Wang et al. published a prey-predator model that took into account the effect of fear on prey reproduction [22]. Additionally, it was explained how a high level of fear may stabilize the system by ruling out the possibility of periodic fixes. Panday et al. [23] also looked into the impacts of fear using a Holling type-II functional response and a tri-trophic food chain model. Since the system displays chaotic behavior for smaller values of both of these variables, they came to the conclusion that chaotic oscillations may be controlled by increasing the fear parameters. A prey refuge is a great way to reduce the possibility that predators may use their victim's biomass excessively. But Abdulghafour and Naji [24] constructed and investigated a mathematical model of a prey-predator system including infectious diseases in the prey population. They believed that prey serves as a constant refuge from predators' exploitation and hunting as a defense mechanism.
This research proposes and investigates a three-species food chain model with cannabilism at the second level while considering the aforementioned. The next section contains the model formulation. Section 3 addresses the solution's characteristics, nevertheless. The analysis of stability and persistence is examined in Section 4. Section 5 examines local bifurcation, while Section 6 provides a numerical simulation analysis of the system. Finally, the last section provided the conclusions.

MODEL FORMULATION
Recently, Andulghafour and Naji [20] proposed and studied a mathematical model of preypredator incorporating fear cost, predator-dependent refuge, and cannibalism given by where ( ) and ( ) are the population densities of the prey and the predator at the time respectively. Since the environment contains many species that interact with each other in a food web and food chain forms. Therefore, in this section, system (1) will be extended so that it contains a top predator that represents their population density at time by ( ) consumed the predator species in the system (1) according to Holling type II functional response. Hence the modified 4 AHMED SAMI ABDULGHAFOUR, RAID KAMEL NAJI model that represents a food chain can be written as: where (0) ≥ 0, (0) ≥ 0, and (0) ≥ 0, and all the coefficients are non-negative constants and can be described in Table (1). The middle predator's half-saturation constant.
The prey's refuge rate; hence the refuge amount is , which leaves (1 − ) of the prey available to be hunted by the predator 2 The conversion rate of prey biomass into middle predator biomass. 3 The conversion rate of cannibalism into middle predator birth 2 The middle predator's natural death rate The cannibalism rate in the middle predator.

2
The half-saturation constant of cannibalism The middle predator's refuge rate 4 The middle predator's attack rate. 3 The top predator's natural death rate. 3 The top predator's half-saturation constant. 5 The conversion rate of middle predator's biomass into top predator biomass.
Hence, using a contradiction will yield that.
Therefore, all the eigenvalues are negative and 0 is stable node provided that the following conditions are met.
Clearly, when the equilibrium point 0 is stable node the equilibrium point 1 dose not exist. Now, the Jacobian matrix at the equilibrium point 2 is determined as.
Then the eigenvalues of ( 2 ) are given by: Consequently, the eigenvalues of ( 2 ) are negative and hence 2 is a stable node point provided that the following conditions are statisfied.
(1− ) 2 ̿ 2 The Jacobian matrix at the equilibrium point 3 is computed as: The characteristic equation can be written as follows: Therefore, all the eigenvalues of ( 3 ) have negative real parts and hence 3 is a stable point if the following conditions are satisfied.
Now, the Jacobian matrix at the equilibrium point 4 is witten as: The characteristic equation of ( 4 ) can be written as Consequently, all the eigenvalues of the ( 4 ) will have negative real parts and makes 4 a stable point if the following conditions are met. 33 = 0. Therefore, the characteristic equation of ( 5 ) can be written as Accordingly, the stability conditions of 5 can be determined through the following theorem.

Theorem 3:
The positive equilibrium point 5 of the system (2) is locally asymptotically stable provided the following sufficient conditions are met. (41)

THREE-SPECIES FOOD CHAIN MODEL WITH CANNIBALISM
Proof. According to the Routh-Hurwitz criterion the proof follows if and only if 1 > 0, 3 > 0, and ∆> 0 are met. Direct computation shows that these requirements are satisfied under the given conditions, and hence the proof is done.
While stable coexistence is the capacity of species to coexist forever in the absence of external perturbations, persistence can be defined as the length of time that a species remains in a community before local extinction takes place. It follows mathematically that there are no boundary attractors in the solution's omega limit set. Therefore, an investigation of the boundary plane dynamics is carried out in the following.
It is clear that system (2) has two subsystems that fall in the positive quadrant of the −plane and −plane respectively. At the same time, there is no subsystem in the −plane. These subsystems can be described respectively: Straightforward computation shows that the subsystem (44) Then the exprations ∆( , ) and ∆( , ) do not identically zero in the . ℝ + 2 of the −plane and −plane and they do not change sign under the following conditions: According to the Dulac-Bendixson criterion [28], for all trajectories meeting conditions (46)-(47), there is no closed curve lying in the . ℝ + 2 of the −plane and −plane. Moreover, the unique equilibrium points in the . ℝ + 2 of the −plane and −plane that is determined by 12 and 22 will therefore be globally asymptotically stable whenever they are locally asymptotically stable, according to the Poincare-Bendixon theorem [28].
Theorem 4: Assume that conditions (46)-(47) are satisfied and the following conditions are met then system (2) is uniformly persistent.
Clearly, by using the given conditions with suitable choice of the positive constants it is obtained that Ω( ) > 0 for all = 0,1, … ,4. Hence the proof is complete.
In the following theorems, the global stability of the above mentioned equilibrium points is studied. Direct computation shows that: Biologically, it is well known that 1 − 2 > 0, and 4 − 5 > 0, hence it is obtained that Therefore, under the local stability conditions (20)-(21), 0 is negative definite. Hence, the vanishing equilibrium point is globally asymptotically stable.

Theorem 6:
The first axial equilibrium point is a global asymptotically stable provided that the following condition is met.
Proof. Consider the following candidate Lyapunov function Direct computation shows that: Then Obviously, condition (52) guarantees that 1 is negative definite. Hence, the first axial equilibrium point is a globally asymptotically stable.

Theorem 7:
The second axial equilibrium point is a global asymptotically stable provided that condition (20) and the following condition are met.
Proof. Consider the following candidate Lyapunov function Direct computation shows that: Obviously, conditions (20) and (53) guarantee that 2 is negative definite. Hence, the second axial equilibrium point is globally asymptotically stable. THREE-SPECIES FOOD CHAIN MODEL WITH CANNIBALISM Theorem 8: The prey-free equilibrium point is a global asymptotically stable provided that condition (20) and the following condition are met. 4 (1− ) 23 where all the new symbols are given in the proof.
Proof. Consider the following candidate Lyapunov function . Direct computation shows that: Then, by choosing 1 = 4 ( 3 +(1− )̂) 5 3 , and maximizing the right-hand side, it is obtained that where represents the upper bound of the .
Obviously, conditions (20) and (54) guarantee that 3 is negative semi definite, which leads to the prey-free equilibrium point is a stable point. Hence, the proof results from equation (55) and Lyapunov-Lasalle's invariance principle [27].

Theorem 9:
The top predator-free equilibrium point is globally asymptotically stable if the following conditions are met.
12 2 < 4 11 22 , where all the new symbols are given in the proof.
Proof. Consider the following candidate Lyapunov function . Using some mathematical mainupolation gives that Then, by choosing 2 = where all the new symbols are given in the proof.
Proof. Consider the following candidate Lyapunov function Obviously, 5 is negative sime definite, which leads to that, the positive equilibrium point is a stable point. Hence, the proof results from equation (62) and Lyapunov-Lasalle's invariance principle [27].

LOCAL BIFURCATION
The present section investigates the influence of the varying parameters on the qualitative dynamics of the system (2) near the non-hyperbolic. An application to the Sotomayor theorem [27] for local bifurcation is performed.
Rewrite the system (2) as the following vector norm where ∈ ℝ is the bifurcation parameter and ( , ) for all = 1,2,3 are given in the system (2). Therefore, for any vector of the form = ( 1 , 2 , 3 ) T , the following expressions can be Then, as the parameter crosses through * , the Sotomayor theorem makes the system (2) undergo a transcritical bifurcation at the equilibrium point 0 .
Theorem 14: Assume that condition (33) is staisfied, then when the parameter 1 passes through ≡ ( 1 * ), the system (2) undergoes a transcritical bifurcation at the prey-free equilibrium point provided that the following condition is met.
Otherwise, a Pitchfork bifurcation takes place. Therefore, by using condition (66), it is obtained that:

)
Therefore, it is obtained that: THREE-SPECIES FOOD CHAIN MODEL WITH CANNIBALISM Therefore, it is obtained that: , Therefore, it is obtained that: Then, as the parameter b passes through * , the Sotomayor theorem makes the system (2) undergo a saddle-node bifurcation at the equilibrium point 5 .

NUMERICAL SIMULATION
In the following, an investigation of the system's dynamics (2)   Now, the influence of varying the parameter on the system's dynamics (2) is studied in two cases, first when the system (2) has a stable coexistence point and second when the system has a stable limit cycle. In the first case, it is observed that, for ∈ [0,1.76], ∈ (1.76,2.51], and > 2.52 the solution of system (2) approaches asymptotically to 5 , 4 , and 2 , respectively. On the other hand, in the second case, it is observed that rising the value of stabilizes the system so that the solution approaches asymptotically the 5 . Moreover, rising the parameter further leads to extinction in top predators first and then extinction in prey species, and then the system's (2) solution stabilized at 2 , see figure (3) for an explanation of the selected values of . It is observed that, when the system indergoes a periodic dynamics as for = 1.5 in figure (2g) increasing in the ranges ∈ (0,0.3), ∈ [0.3,3.4], ∈ [3.4, 5], and > 5 the solution of system (2) approaches to stable limit cycle, 5 , 4 , and 2 , respectively, as shown in figure (3g) for = 0.5. For the parameters 1 , and 1 , they have a similar influence on the system's (2) solution as that obtained for in the first case. Now, the influence of varying the parameter on the system's (2) dynamics is studied in figure (4) below at a selected values. It is obtained that for the ranges ∈ (0,0.13), and > 0.13 the solution approaches a stable limit cycle, and 5 respectively. Note that, a similar impact on the system's (2) dynamics, as shown by the parameter , is obtained when the parameter value varies. For the parameter 1 in the ranges 1 ∈ (0,0.14] , 1 ∈ (0.14,0.53], 1 > 0.53, it is observed that the system's (2) solution approaches asymptotically to 2 , stable limit cycle, and 5 respectively, as shown in figure (5) for the selected parameter values. For the parameter 3 in the ranges 3 ∈ (0,0.26] , and 3 > 0.26 the system's (2) solution approaches asymptotically to 5 and stable limit cycle respectively, as shown in figure (6)     it is observed that, the system (2) approaches asymptotically to 0 = (0,0,0) as shown in figure (10). Clearly, for the data used in figure (10), the conditions (20)- (21) are satisfied and hence the stability of 0 is confirmed.  (2) as a function of the specific parameter with rest of parameters as given in (67) Parameter

Range
The dynamics 41 THREE-SPECIES FOOD CHAIN MODEL WITH CANNIBALISM collection of characteristics was investigated. It is noted that the system has six nonnegative equilibrium points. Each one's stability analysis is looked into locally. The system's persistence requirements have been identified. The transcritical bifurcation of system (2) is demonstrated to occur close to the boundary equilibrium point, with the pitchfork bifurcation occurring possibly also at the prey-free equilibrium point. Saddle-node bifurcation is, nevertheless, discovered close to the positive equilibrium point. Finally, the model is investigated numerically using a hypothetical set of parameter values to confirm the obtained finding and understand the impact of varying the parameters on the system's (2) dynamics. The following results were obtained numerically depending on the parameter values (67).
• The prey birth rate has three bifurcation points. As its value increases, the system (2) loses its stability at the positive equilibrium point and transfers to periodic dynamics through Hopf bifurcation. On the other hand, decreasing its value leads to extinction in the top predator first and then in the prey so that the solution approaches the second axial equilibrium point through the top predator-free equilibrium point.
• The prey's fear level (similarly the prey's natural death rate and the intermediate predator's attack rate) causes extinction in the top predator first and then in the prey when its value exceeds a specific value. On the other hand, when the system undergoes periodic dynamics, it is observed that increasing the prey's fear level stabilizes the system at the positive equilibrium point.
• The prey intraspecific competition (similarly the prey's refuge rate and the middle predator's half-saturation constant) has a stabilizing effect on the system's dynamics.
• The conversion rate of cannibalism into middle predator birth (similarly the half-saturation constant of cannibalism) has a destabilizing effect on the system's dynamics.
• The middle predator's natural death rate causes extinction in the system and the solution ultimately approaches the first axial equilibrium point.
• The cannibalism rate in the middle predator (similarly the middle predator's refuge rate, the top predator's natural death rate, and the top predator's half-saturation constant) has a stabilizing effect on the system's dynamics up to a threshold value and then the persistence of the system (2) is lost through extinction in top predator.