HOPF BIFURCATION FOR DELAYED PREY-PREDATOR SYSTEM WITH ALLEE EFFECT

. In this study, we take into account a predator-prey system with two delays, the prey is sea urchins and the predator is crabs. The focus is given to the Allee effect where the prey population undergoes, the poisoning of few predators, and a ﬁshing effect on both species considered as selective for the prey. We aim to analyze the system’s stability around interior equilibrium using the theory of bifurcations and determine stability intervals related to delays. The theory of normal form and the center manifold are used to determine the direction of the bifurcations. Finally, numerical simulations are given by numerical methods in DDE-Biftool Matlab package to illustrate the theoretical results.


INTRODUCTION
Across the seas, marine resources are at risk of extinction due to several factors: predation by strange species, affectation through chemicals or toxic species, and overexploitation. Therefore, marine biodiversity is threatened, requiring the intervention of relevant agencies to ensure the preservation of these resources through several facets, these include limited access and selective fishing based on size and age to protect the juvenile population from early fishing.
Mathematical models stand as a tool allowing the study of biological phenomena and their influence on the dynamics of populations, they identify the interactions between marine species, the variation of their density as well as the effect of toxins.
The exponential growth established by Malthus remains among these mathematical models, it considers that the species density evolves exponentially with a constant growth rate. However, Verhulst criticizes this concept and claims that the environment has a maximum capacity of inhabitants that it cannot exceed when the population size approaches the carrying capacity, competing on the natural resources and causing a decrease in growth rate.
Among the threatened marine species, sea urchins are distinguished as essential invertebrate herbivores in the Mediterranean that undergo the Allee effect with an aggregate behavior. Sea urchins are characterized by external fertilization: reproduction decreases if the density is insufficient to create effective reproductive aggregates. Therefore, there is a positive dependence between the growth rate and size of the sea urchin population.
Our model considers sea urchins as prey of crabs. With this prey-predator model of two exploitable species, we use the equations of Lotka-Volterra while adding the fishing effect.
Some types of sea urchins are toxic; this toxicity affects crabs after a certain time of predation of these Sea urchins.
Sea urchins play a vital role in maintaining the balance of ecosystems. Given the decline in available stocks, action has been taken to regulate catches of this species. A minimum size designating an early age has been set for fishing to protect the juvenile population. Finally, our model is expressed in the following form such that x sets for sea urchin biomass, and y represents crab biomass.
The remainder of this paper is organized as follows. After presenting the model in the introduction, section 1 focuses on the existence and boundedness of the system's solutions. The stability of the interior equilibrium point is given in section 2, in addition to the search of bifurcation points according to the delay parameters values τ 1 and τ 2 . Section 3 discusses the stability and direction of Hopf bifurcation, which is followed in section 4 by the global stability.
Finally, the numerical simulations of theoretical results are provided in section 5.

EXISTENCE AND BOUNDEDNESS OF THE SOLUTION
1.1. Boundedness of solutions. The first equation of system (1) verifies the following in- We consider w(t) = x(t) + y(t) 1.2. Existence and uniqueness of solution. The system (1) can be represented in the follow- The function f is continuous and the partial derivatives of f i are continuous and bounded, then f is a Lipschitzian function. Consequently, the conditions of Cauchy Lipschitz are satisfied.
According to the fundamental theorem of functional differential equations cited in [6], system (1) admits a unique solution.

STABILITY ANALYSIS
2.1. Equilibrium points. To find the positive equilibrium points, we solve the following sys- Then the system (1) admits a unique strictly positive equilibrium point P * (x * , y * ), where To study the stability of the system (1), we must first calculate its characteristic equation which will be expressed as follows The coefficients of equation (2) are represented in the following table

Study of local stability.
Case 1: Without delays. For τ 1 = τ 2 = 0, the characteristic equation becomes as follows According to the Routh Hurwitz criterion, if A + C + E > 0 and B + D + F + G > 0, then the system without delays is locally asymptotically stable around the equilibrium point P * .
Case 2: One delay. For τ 1 = 0 and τ 2 > 0 We assume that iw (w > 0) is a root of (3) and we get the following couple of equations To arrive at the next system, we must square the previous equations and sum them We note the following conditions If (H 1 ) and (H 2 ) hold, then Eq (5) has no positive roots. Hence, all roots of Eq (3) have negative real parts when τ 2 ∈ [0, ∞).

STABILITY AND DIRECTION OF HOPF BIFURCATION
The objective of this section is to determine the direction of Hopf bifurcation and analyze the stability of periodic solutions. To achieve this objective, we will apply the theory of normal form and the center manifold theorem to our system. First, we start by linearizing the system (1) by changing variables u = x − x * , v = y − y * and we get the following system  (8) Under the loss of generalities, we assume that: τ * 2 < τ 0 1 . We pose τ 2 = τ 0 2 + η and ϕ t (θ ) = ϕ(t + θ ) ∈ C. We express the system (8) as the following functional differential system in C([−τ * 1 , 0], R 2 ) Riesz's representation ensure the existence of a second-order matrix g(θ , η) of bounded variation for θ ∈ [−τ * 1 , 0], such as We can also express our system as followṡ The adjoint operator of M is written as follows We use the following bilinear form in For θ = 0, we have Similarly, the eigenvector of M * associated with −iω 0 is written as ρ * (s) = D(1, α * )e iω 0 s , with Using the fact that < ρ * (s), ρ(s) >=1, we have The following table gives the expression of K i j that is important to determine the direction of Hopf bifurcation and the stability of periodic solutions. This result is obtained by Hassard's algorithm and the calculation steps used in [5].

GLOBAL STABILITY
We choose the following Lyapunov function where α 1 and α 2 are positive constants. The derivation of this function will be expressed as followṡ we have g 2 −g 3 > 0, then we can choose α 1 and α 2 such as α 2 (g 2 − g 3 ) = α 1 g 1 . If the condition is verified, thenV (t) 0. Therefore, system (1) is globally asymptotically stable at the interior equilibrium point P * .

DISCUSSION
In this section, numerical simulations will be performed to illustrate the theoretical results obtained in the previous sections. The simulations are performed by DDE-Biftool; a Matlab package designed for the numerical continuation and bifurcation analysis of the system. Its recent version DDE-Biftool V3.1.1 is conducted by J. Sieber in [2]. This numerical tool provides the course of the study with steady-state continuation, and periodic orbit solutions but also a bifurcation continuation in two parameters in which we are interested. = 0.02, g 1 = 0.6, g 2 = 0.04, m = 0.1, g 3 = 0.01, E 1 = 0.3 and E 2 = 0.18. The system admits a single point of equilibrium strictly positive P*=(9.3333, 3.3505). In the following, we vary the two delays τ 1 and τ 2 between 0 and 15 to notice the impact of this variation on the equilibrium as well as its stability.
Then the 2 figures below show the variation of the equilibrium branch x according to the delay parameters τ 1 and τ 2 , the two colors used to represent the nature of stability such that red shows the unstable part of the branch, green presents the stable part of the branch and asterisks are used to determine the points of Hopf bifurcation. The following two figures represent the bifurcation diagrams which are obtained by the maximum and minimum amplitude of x, in which we notice that our system undergoes the Hopf bifurcation for τ 1 =1.309 and τ 2 =3.284, moreover the nature of this bifurcation which is supercritical for both parameters. In the following, we will choose some values for the delay parameters and plot the variation of the system solution (1) in time around the equilibrium P*. The following figure contains six graphs. In the three graphs on the left, we have taken values of τ 1 and τ 2 located in the stability intervals of these two parameters and we notice that the solution converges toward the equilibrium point P * , which is not the case in the three graphs on the right where chosen values of τ 1 and τ 2 are outside the stability interval.

CONCLUSION
Our study focuses on the stability analysis of a prey-predator model consisting of sea urchins and crabs, taking into account the Allee effect in the sea urchin population, a mortality rate in crabs due to the sea urchins toxicity, and the fishing effect in both species. In addition, the fishery is considered selective for sea urchins, to conserve the juvenile population and preserve marine biodiversity. The stability analysis is established by the search for points of bifurcation and intervals of stability linked to delays. In the next work, we aim to add the diffusion effect for a more concretization of our research.

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