THE DYNAMICS OF A DELAYED ECOLOGICAL MODEL WITH PREDATOR REFUGE AND CANNIBALISM

: This study has contributed to understanding a delayed prey-predator system involving cannibalism. The system is assumed to use the Holling type II functional response to describe the consuming process and incorporates the predator's refuge against the cannibalism process. The characteristics of the solution are discussed. All potential equilibrium points have been identified. All equilibrium points' local stability analyses for all time delay values are investigated. The system exhibits a Hopf bifurcation at the coexistence equilibrium, which is further demonstrated. The center manifold and normal form theorems for functional differential equations are then used to establish the direction of Hopf bifurcation and the stability of the periodic solution. To demonstrate the key findings, various numerical simulations are then run.


INTRODUCTION
Because predators are the main cause of prey death, prey is strongly selected to evolve more effective methods of spotting, avoiding, or fending off predators. The population of predators declines exponentially in the absence of prey species, forcing predators to develop better tracking and killing techniques or to look for new food sources. Since the predator population derives its energy from its species and can thus exist without its lone prey, we develop the model by taking cannibalism into account. In some mantises, fish, carnivore mammals, and spiders, cannibalism, also known as intraspecific predation, refers to devouring offspring or siblings. This occurrence is sometimes referred to as the "lifeboat mechanism" since it prevents a predator community from going extinct. Depending on the cannibalism rate, cannibalism can positively or negatively impact population abundance. This has sparked the interest of many academics in their research and produced a number of notable research findings. Therefore, for certain cannibals, including cannibalism in a stage-structured paradigm makes more sense. [1][2][3][4][5][6][7].
There is a delay in the appearance of some species' behavior in the natural ecosystem. In biological systems, the influence of a population on its condition has frequently been described using time delays. Delay differential equations are frequently more practical than regular differential equations in physics, ecology, biology, and other applications because a time delay could make an equilibrium that was previously stable unstable and induce population fluctuations.
In the current work, the notion that energy obtained by eating food does not occur instantly is included. The functional response of the Holling type frequently displays this as a time delay. In delayed predator-prey models, the time delay may have an impact on the stability or instability of the prey densities as a result of predation. [8][9][10][11]. Due to new ecological evidence and theoretical advancements, there has been a recent increase in interest in various aspects of modeling biological interactions. Holling [12] proposed a more accurate illustration, which has since become one of the most widely used in ecology. It takes into account the hunt for prey and the handling of prey, two different components of the relationship between prey and predators. Functional responses may be influenced by a variety of ecological parameters, including prey habitat structure, predator hunting skills, and prey escape mechanisms [13]. Actually, the Holling type-II functional response, which is based on the traditional Lotka-Volterra model, is a function that increases, concave, smooths out, and saturates at high prey numbers. Numerous writers have examined this kind of functional response in predator-prey models, both with and without time delays even with spatial dependency [11,[14][15] and the references therein.
The occurrence of cannibalism is affected by a number of factors, including temperature, population density, developmental stage, and prey (predator) refuge [16]. You could argue that cannibalized predators have the ability to seek refuge from cannibalistic predators. By lowering the likelihood of extinction as a result of excessive predation, the refuge is an anti-predator behavior that can aid in extending a predator-cannibalism relationship. When a predator detects cannibalism, they prepare for an attack by altering their behavior, including body shape, size, color, and habitat. It becomes particularly intriguing to research the delayed interactions between predator and prey, including predator cannibalism and predator refuge [4,7,17]. The present article modifies an ecological model with logistic growth in the prey to incorporate cannibalism in the predator and refuge with a temporal delay. The remainder of this paper is structured as follows.
The mathematical model is built in section 2. In section 3, the prerequisites for equilibrium points' existence are established. In section 4, it is examined if each possible equilibrium of the system is locally stable. Section 5 studies the occurrence of Hopf bifurcation around the positive point. The direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions are established in Section 6. Finally, Present some numerical findings and explore their biological implications in Sections 7 and 8.

MODEL CONSTRUCTION
The following model, which depicts a Lotka-Volterra prey-predator with predator cannibalism, was put forth by Deng et al. [4].

=
(1 − ) − 1 , where prey and predator density are denoted by u ≥ 0 and v ≥ 0. The parameters r, K, a 1 , a 2 , b 1 , c, b 2 , and β are positive constants that, respectively, represent the intrinsic growth rate of the prey, the carrying capacity of the prey in the environment, the rate of predation, the rate at which prey biomass is converted into predator birth, the rate at which cannibalism is converted into predator birth, the rate at which predators die, the rate at which cannibalism occurs within predator individuals, and the cannibalism half-saturation constant. The final term and the second term in the second equation of the system represent the cannibalism phenomena (1) A model explaining the prey-predator interaction that incorporates predator cannibalism and refuge was recently proposed by Rayungsari et al [7], and then the proposed model underwent a dynamic study. By employing the Holling type II functional response rather than a Lotka-Volterra type of functional response and assuming that there is predator refuge from cannibalization, the suggested Rayungsari's model (2) is an improvement on Deng's model. The saturation predation mechanism is essentially described by the Holling type II functional response.
where β 1 , β 2 , and m are positive constants that, respectively, represent the predation halfsaturation constant, the cannibalism half-saturation constant, and the predator refuge constant.
While the other parameters are described above.
In actuality, although hunted victims are transformed into predator growth, there is a delay in predator biomass production due to the gestation period. Therefore, model (2) is altered by assuming that a continuous temporal lag known as gestation delay τ ∈ ℝ + governs the reproduction of predator populations. Consequently, the new system is expressed as follows: where the system's (3) initial conditions are Set c 1 = a 2 and c 2 = a 1 , then For any positive Μ, such that Since ′′ ( ( − )) = − 2 2 < 0 then ( ( − )) has maximum value given by 2 ( + ) 2 4 . This implies ( , ) is bounded and so are ( ) and ( ). Hence the proof is done.

THE EXISTENCE OF EQUILIBRIUM POINTS
The non-negative equilibrium points of the system (3) are determined, and the following four equilibrium points are obtained: The trivial equilibrium point 0 = (0,0) and predator-free point 1 = ( 1 , 0), where 1 = , exist unconditionally in ℝ + 2 .
In other words, even if the prey is extinct, the predator will continue to exist as long as the rate of cannibalism conversion is higher than the rate of the predator's death and lower than the total rates of cannibalism and conversion.
The coexistence equilibrium point 3 = ( * , * ), while * represents the positive root of the following third-order polynomial equation: where Clearly, according to Descartes' rule of signs, equation (7) has a unique positive root and hence system (3) has a unique coexistence equilibrium point if and only if the following sufficient condition is met.
with one set of the following sets of conditions.

LOCAL STABILITY
To study the local stability behavior of the equilibrium points, a linearization technique around an arbitrary equilibrium point is applied, resulting in the Jacobian matrix, which is then computed at each equilibrium point.
. Then the characteristic equation of the system (3) at = ( ̅, ̅ ) can be determined by The local stability around the equilibrium points of the system (3) is determined by the eigenvalues of the characteristic equation (12) and the result is obtained in the following theorem.
Then, if 1 > then 0 is a source, and if 1 < then 0 is a saddle point. So, 0 is always unstable for all ≥ 0.
Now before the next theorems are given, the following proposition that is given in [10] is presented.

Theorem 4:
The predator-free equilibrium point 1 = ( , 0) of the system (3) is asymptotically stable for ≥ 0 provided that the following condition is met While it is unstable point for all that condition (14) is reflected.
Proof: Substituting 1 = ( , 0) in the general characteristic equation (12) gives that So the eigenvalues are 11 = − < 0, while the other eigenvalues are given by the roots of the following transcendental equation: Clearly, for = 0, it is obtained 12 = 2 1 + + 1 − , which is a negative provided condition (14) holds, however, 12 will be positive when the condition (14) is reflected. This makes 1 a saddle point.
On the other hand, for > 0, due to proposition (3) all the roots of equation (16) have negative real parts roots or positive real parts roots for < = when the condition (14) is satisfied or reflected respectively.
Consequently, the predator-free equilibrium point is locally asymptotically stable for all > 0 if the condition (14) is met, while it is an unstable point for 0 < < when the condition (14) is reflected. This is complete the proof.

Theorem 5:
The prey-free equilibrium point 2 = (0, 1 ) of the system (3) is locally asymptotically stable for all ≥ 0 if the following condition is met.
Proof: Substituting 2 = (0, 1 ) in the general characteristic equation (10) gives that Therefore, the eigenvalues of the system (3) at 2 are given by: Since 22 is negative under the existing condition of 2 , which is given by equation (5). Then, 2 is locally asymptotically stable if and only if condition (17) is met. While it is saddle point if that condition is reflected.

Theorem 6:
The coexistence equilibrium point 3 = ( * , * ) of the system (3) is locally asymptotically stable for all 0 ≤ < 0 provided that the following condition is met.
However, it is unstable point for 0 ≤ provided that 2 − 2 < 0 (20) where all the new symbols are given in the proof.

HOPF BIFURCATION ANALYSIS
In this section, the possibility of occurrence of Hopf bifurcation around the coexistence equilibrium point is discissed. From the above, it is proved that the characteristic equation (22)

Theorem 7. System (3) undergoes a Hopf bifurcation around the coexistence equilibrium point at
= 0 provided that the following condition is met.
where all the new symbols are given in the proof.

= − sin − cos .
Solving the linear algebraic system (32) for , gives that: ≠ 0 under the condition (31) that complete the proof.

DIRECTION AND STABILITY OF HOPF BIFURCATION
The explicit formulas that determine the direction, stability, and period of periodic trajectories bifurcating from equilibrium 3 at critical values of delay 0 are derived as given in the following theorem and are based on the center manifold and normal form theory introduced by Hassard et al. [20]. 1. If 2 > 0 ( 2 < 0), then direction of the Hopf bifurcation is supercritical (sub-critical) and the bifurcating periodic trajectories exist for > 0 ( < 0 ).
Now, since the system (34) undergoes a Hopf bifurcation near equilibrium point 3 . Then system (34) has two pure imaginary eigenvalues ∓ 0 of , which are also eigenvalues of * . Now, by a simple calculation, the eigenvectors of (0) and * associated with the eigenvalues ∓ 0 are computed, respectively, as follows: The characteristics of the bifurcating periodic trajectories of the system (34) can then be explored and evaluated using a method similar to that in [20]. In the following, the coefficients determine the stability and direction of the Hopf bifurcation are listed.
As a result, using Cramer's rule, it is obtained that  The influence of the time delay is studied in Figure (  For the parameter , it is observed that for the ranges ∈ (0,0.37], and ≥ 0.38 system (3) approaches asymptotically to 1 , and 3 respectively as shown in Figure (4) for the typical values of .  For the parameter 1 , it is observed that, for the ranges 1 ∈ (0,1.75], and 1 ≥ 1.76 system (3) approaches asymptotically to a stable limit cycle, and 3 respectively as shown in the Figure   ( Moreover, for the parameter 2 , it is observed that, for the ranges 2 ∈ (0,0.06] , 2 ∈ [0.07,0.34], and 0.35 ≤ 2 ≤ 1 system (3) approaches asymptotically 1 , 3 , and a stable limit cycle respectively as shown in the Figure (7) for the typical values of 2 .    Finally, it is observed that the influence of other parameters on the system's dynamics (3) is similar to some of the above parameters, and Table (1) summarizes them.

CONCLUSIONS
In this paper, a model describing a delayed prey-predator interaction that incorporates predator cannibalism and refuge is investigated. The Holling type II functional response was used to describe the predation and cannibalism processes. All the system solution properties are discussed.
It is obtained that, system (3) has four possible nonnegative equilibrium points. The local stability of them are studied for ≥ 0, and the stability conditions are determined. The existence of Hopf bifurcation as a function of is proved. The stability and direction of the bifurcated periodic dynamics is investigated using the center manifold and normal form theory. Finally, numerical simulation was used to verified the obtained results and understand the influence of varying system's parameters using hypothetical set of parameters values. The following conclusions are obtained numerically depending on the parameters set (54). 1. The system has two types of attractors, stable point or else stable limit cycle depending on the set of parameters used. For set (54), the system approaches asymptotically to the coexistence equilibrium point from different initial points.
2. The system undergoes a Hopf bifurcation as the parameter passing the bifurcation point 3. The system approaches the coexistence equilibrium point with large values of intrinsic growth rates asymptotically. While decreasing this parameter destabilizes the coexistence point and periodic dynamics obtained. It is observed that lowering the intrinsic growth rate further returns the system stability at the coexistence point.
4. Increasing the carrying capacity keeps the system stability at the coexistence point, while, decreasing it below a specific value causing extinction in a predator and the system approaches to predator-free point.
5. Increasing the predation rate above a specific value destabilizes the coexistence equilibrium point and the system approaches periodic dynamics. A similar effect had been shown with increasing the predator refuge constant.
6. Decreasing the predation half-saturation constant below a specific value destabilizes the coexistence equilibrium point and the system approaches periodic dynamics. A similar effect had been shown with decreasing the cannibalism rate.
7. Increasing the conversion rate above a specific value causes destabilizing the coexistence equilibrium point and the system approaches a stable limit cycle. However, decreasing it below a threshold value leads to extinction in predator species and the system comes asymptotically to a predator-free point. A similar effect had been shown with varying the cannibalism half-saturation constant.
8. With the exception of the existence of a threshold value at which the system approaches an asymptotically prey-free equilibrium point, the cannibalism conversion rate has effects on the system dynamics that are comparable to those observed with an intrinsic growth rate.
9. With the exception of the existence of a threshold value at which the system approaches an asymptotically predator-free equilibrium point, the predator death rate has comparable effects on the system's dynamics as those demonstrated with an intrinsic growth rate.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.