The Local Bifurcation of an Eco-Epidemiological Model in the Presence of Stage- Structured with Refuge

In this paper, we establish the conditions of the occurrence of the local bifurcations, such as saddle node, transcritical and pitchfork, of all equilibrium points of an eco-epidemiological model consisting of a prey-predator model with SI (susceptible-infected) epidemic diseases in prey population only and a refuge-stage structure in the predators. It is observed that there is a transcritical bifurcation near the axial and free predator equilibrium points, near disease-free equilibrium point is a saddle-node bifurcation and near positive (coexistence) equilibrium point is a saddle-node bifurcation, a transcritical bifurcation and a pitchfork bifurcation. Further investigations for Hopf bifurcation near coexistence equilibrium point are carried out. Finally, numerical simulations are used to illustrate the occurrence of the local bifurcations of this model.


Introduction
Bifurcation theory is the mathematical study of changes in the qualitative asymptotic structure of a dynamical system [1,2]. It attempts to explain0 various phenomena that have been discovered in the natural sciences over the centuries. Performing a bifurcation analysis is often a powerful way to analyze the properties of such systems. The prey and predator model is an important topic at present, as it is used to solve many problems in ecology, nature and other sciences. The prey system includes several interactions, including competition co-existence and stage-structured [3,4]. The ecological models of the age stage are more logical than models that do not contain1 phase structure. In addition, there are several factors that affect the system, for example, refuge, disease, shelter and others. Sometimes, differences in any parameter in the system can lead to complex behaviors that lead to system instability, causing a bifurcation that is the main qualitative change in the behavior of a dynamic system as a result of changing one of its coefficients. The bifurcation is divided into two principal classes, local 4 and global. Local1 bifurcation can be analyzed through changes in the local stability properties of equilibria0 or periodic orbits. While global bifurcation occurs when periodic orbits collide with equilibria. This causes changes in the topology of the trajectories in phase space which cannot be confined to a small neighborhood, as is the case with local1 bifurcation. These bifurcations happen when one varies a single parameter [5][6][7][8][9]. On the other hand, Perko [10] established the conditions of the occurrence of local bifurcation, such as saddle-node, transcritical, pitchfork, period1-doubling, and Hopf bifurcation near coexistence equilibrium point. The Hopf bifurcation is a local bifurcation in which the equilibrium1 point of a dynamical system loses stability,1as a pair of complex conjugate eigenvalues of the linearization around the equilibrium point cross the imaginary axis of complex plane. This type is also known as Poincare Andronov Hopf bifurcation. In this paper, an application of Sotomayor's theorem [11] for local bifurcation is used to study the occurrence of local bifurcation near the equilibria. Furthermore, Hopf bifurcation near positive equilibrium point conditions1 is established for a mathematical model previously proposed by Kafi and Majeed [12]. 2 .The mathematical models [12] In this section, an eco-epidemiological model is proposed for study. The model consists of a prey, whose total population density at time is denoted by , interacting with a stag-structured predator . It is assumed that the prey population is infected by an infectious disease with the prey refuge. Now, the following assumptions are adopted in formulating the basic eco-epidemiology model:

1.
There is an epidemic disease in the prey population which divides the prey population into two classes, namely that represents the density of susceptible prey at time and which represents the density of infected prey at time . Therefore, at any time T, we have . The predator population is divided into two classes, namely that represents the density of immature predator at time and which represents the density of mature predator at time .

2.
It is assumed that only susceptible prey S is capable of reproducing in a logistic growth with a carrying capacity and an intrinsic growth rate constant . The infected prey I is removed before having the possibility of reproducing. However, the infected prey population I still contribute with S to population growth towards the carrying capacity.
3. The disease is transmitted within the same species by contact with an infected individual at infection rates of α 0 for the prey. 4. The mature predator w(T) consumes the susceptible prey S(T) and the infected prey I(T), according to Holling type-II of functional responses with a maximum attack rate of and a half saturation rate for the susceptible prey , as well as a maximum attack rate of and a half saturation rate for the infected prey .
5. The disease may causes mortality with a constant mortality rate for the infected prey.
6. The immature predator depends completely in its feeding on his parents, so that it feeds on the portion of the uptake food by the mature predator from the susceptible and infected prey, with portion rates of associated with uptake rates of 7. There is a type of protection of the prey species from facing predation by the mature predator with refuge rate constants for susceptible and infected prey, respectively.
8. Finally, in the absence of the predator facing death with natural death rates of for immature and mature predators, respectively. Therefore, the dynamics of the above proposed model can be represented by the following set of first order nonlinear differential equations.
( * } . = = m For the simplicity of the above model, it is assumed that Note that the above proposed model has sixteen parameters in all, which makes the analysis difficult. In order to simplify the system, the number of parameters is reduced by using the following dimensionless variables and parameters. α Then dimensional form of system (1) can be written as: It is observed that the number of parameters was reduced from sixteen in system to fourteen in system . Obviously, the interaction functions of system are continuous and have continuous partial derivatives on the following positive four dimensional spaces. ={ Therefore, these functions are Lipschitzian on , and hence the existence and uniqueness of solutions for system are guaranteed. Further, all the solutions for system with non-negative initial conditions are uniformly bounded, as demonstrated in the following theorem. Theorem 1 [12] All the solutions of system are uniformly bounded.

Existence and stability of equilibrium points [12]
System has at most five equilibrium points, which are mentioned in the following: The equilibrium point which is known as the vanishing point that always exists and unstable.

Kafi and Majeed
Iraqi Journal of Science, 2020, Vol. 61, No. 8, pp: 2087-2105 7828 The axial equilibrium point , which exists unconditionally. Also, it is a locally asymptotically stable if the following conditions hold: The free predators' equilibrium point ( , ,0 ,0 ) which exists uniquely in the (Interior of ) of , provided that: where (8) And it is a locally asymptotically stable if the following conditions hold: where: The disease-free equilibrium point ̃ ̃ ̃ exists uniquely in the interior of where, if the following conditions hold: (15) And it is a locally asymptotically stable if the following conditions hold: Finally, the positive (coexistence) equilibrium point ( ) exists in the Int. if the following conditions hold: ), 7827 + Hence will be positive under conditions Therefore, all the eigenvalues of have negative real parts under the given conditions and hence is locally asymptotically stable. However, it is unstable otherwise.

Local bifurcation analysis
In this section, the effects of varying the parameter values on the dynamical behavior of system around each equilibrium point are studied. We recall that the existence of non-hyperbolic equilibrium point of system is the necessary but not sufficient condition for bifurcation to occur. Therefore, in the following theorems, an application to the Sotomayor's theorem for local bifurcation is appropriate. Now, according to Jacobian matrix of system which is given in equation , it is easy to verify that, for any nonzero vector ̇ ( ̇ ̇ ̇ ̇ ) , we have: where: and is any bifurcation parameter. In the following theorems, the local bifurcation conditions near equilibrium points are established.
Now, by substituting ̇ in (29), we get: Hence, it is obtained that: Then system (2)

Local bifurcation analysis near Theorem (5):
Suppose that the following 4conditions are satisfied: ( ) ̈ So, 4according to the 4condition we obtain that: ( ) ̈ Therefore, according to Sotomayor's theorem 4, neither a transcritical nor a pitchfork bifurcation 4 can occur at , while the first condition of a saddle-node 4 bifurcation is satisfied. Moreover, by substituting in (29), we get: Hence, it is 4obtained that: So, 4according to the 4condition we obtain that: Thus, by using Sotomayor's theorem, system (2) has a saddle-node bifurcation at ( ) at ̈ Now, if the condition (37a) is not satisfied, we obtain that: ( ) ̈ Therefore, 4according to Sotomayor4's theorem,4 the saddle4-4node 4bifurcation cannot occur4. While the first 4condition of 4transcritical 4bifurcation is satisfied4. Now, since So, according to the condition (37b) we obtain that: Now, by 4substituting ̇ in (29), we get: Hence, it is 4obtained that: So, according to the condition we obtain that: Now, if the condition (38a) is not satisfied, by 4substituting ̇ in (30), we obtain that: So, 44according to the 44condition we obtain that: ( ) ̈ ( ̇ ̇ ̇ ) Thus, by using Sotomayor's theorem, system (2) has a transcritical bifurcation and pitch fork bifurcation at ( ) by the conditions (37a) and (38a), respectively, which are not satisfied at ̈

Hopf bifurcation analysis
In this section, the possibility of the occurrence of a Hopf bifurcation near the positive equilibrium point of the system is investigated, as shown in the following theorem. Theorem 5: Suppose that the local conditions with the following conditions are satisfied:

, -
where: , bifurcation near the point has a Hopf system Then at the parameter value Proof: Consider the characteristic equation of system at which is given by eq. , then by using the Hopf bifurcation theorem, for n=4, we need to find a parameter, say ( ), to verify that the necessary and sufficient conditions for the Hopf bifurcation are satisfied, that is: where represents the coefficients of the characteristic eq. Straightforward computation gives that: On the other hand, it is observed that gives that: By a straight forward computation, we get: where: where which are mentioned in the text of the theorem. Clearly, provided that in addition to the conditions the conditions holds. Note that, by using Descartes rule of sign, eq. has a unique positive root Now, at the characteristic equation given by eq. . can be written as: Clearly, at there are two pure imaginary eigenvalues ( and ) and two eigenvalues which are real and negative. Now for all values of in the neighborhood of , the roots are in general of the following form: Clearly, ( ) ( ) which implies that the first condition of the necessary and sufficient conditions for Hopf bifurcation are satisfied at . Now, according to the verification of the transversality condition, we must prove that: are given in lemma (1) of a previous work [11]. Note that for we have and √ , thus it gives the following simplifications: Then , we get that: Now, according to condition we have: So, we obtain that the Hopf bifurcation occurs around the equilibrium point at the parameter . .

Numerical simulation
In this section, the dynamical behavior of system is studied numerically for a set of parameters and different sets of initial points. The objectives of this part are: 1-Investigating the effect of varying the value of each parameter on the dynamical behavior of system . 2-Confirming the obtained analytical results. It is observed that, for the following set of hypothetical parameters that satisfies stability conditions of the positive equilibrium point, system has a globally asymptotically stable positive equilibrium point, as shown in Figure   Clearly, igure shows that system is globally asymptotically stable as the solution of system approaches asymptotically to the positive equilibrium point starting from three different initial points, which confirms our obtained analytical results. Now, in order to discuss the effects of the parameters' values of system 2 on the dynamical behavior of the system, the system is solved numerically for the data given in eq with varying one parameter at each time, The obtained results are given in Table-1, while more details are provided elsewhere [4]. The effects of varying the predation rate on susceptible prey in the range of while keeping the other parameters as the data given in eq.(6.1), causes extinction in the infected prey and the system will approach to the infected prey free equilibrium point , as shown in Figure-(2) , for a typical value of . In the range of it is observed that the solution of system approaches asymptotically to the positive equilibrium point , as shown in Figure-  , For the data given in with By varying of the parameter which represents the conversion rate from the susceptible prey to the immature predator in the rang , and keeping the rest of parameters values as in the data given in eq. ( 6.1 ), the solution of system (2) still approaches asymptotically to the positive equilibrium point , as shown in Figure-(3) (a), for a typical value of = 0.08. However, by increasing this parameter further to it is observed that system (2) still approaches the infected prey free equilibrium point , as shown in Figure (3) (b), for a typical value of = 0.12.

Fig
Time series of the solution of system for the data given by with which approaches to . (b) Time series of the solution of system for the data given by with which approaches to .

Conclusions and discussion
In this paper, we proposed and analyzed an eco-epidemiological mathematical model consisting of a prey-predator model involving an SI infectious disease in a prey-stage structured predator species with a prey refuge . Further, in this model, we used Holling type II of functional responses for the predation of susceptible and infected preys which are outside refuge, as well as a linear incidence rate for describing the transition of disease. Our aim is to study the role of infectious diseases on the dynamics. Also, system (2) was solved numerically for different sets of initial points and different sets of parameters, starting with the hypothetical set of data given by eq. System (6.1). The following observations were obtained.  System (2) has only one type of attractor in Int.
approaches to a globally stable point.  or the set of hypothetical parameters' values given in eq. ( . ), system (2) approaches asymptotically to a globally stable positive point . Susceptiple prey Infected prey Immature predator mature predator