Dynamics of Beddington De-Angelis Type Eco-Epidemiological Model with Prey Refuge and Prey Harvesting

: Analysing the prey-predator model is the purpose of this paper. In interactions of the Beddington De-Angelis type, the predator consumes its prey. Researchers ﬁrst examine the existence and local stability of potential unbalanced equilibrium boundaries for the model. In addition,For the suggested model incorporating the prey refuge, we investigate the Hopf-bifurcation inquiry. To emphasise our key analytical conclusions, we show some numerical simulation results at the end.


Introduction
In prey predator models are two type one is an ecological model and another one is an epidemiological model.In ecological model interactions between organisms, including humans, and their physical environment.In epidemiological models are used to study diseces in animals and humans.Also, the above study of ecology and epidemiology is called eco-epidemiology.In 1949, Solomon first used the term 'functional response'.In the late 1950s, C. S. (Buzz) Holling conducted experiments to investigate how predators capture prey.In the resulting series of influential articles, Holling established three main functional response types, which he referred to as Holling types 1, 2, and 3.The Holling type I functional response g(X,Y) = aX, where a > 0, is based on the principle of mass action and depends on the prey.Therefore, in the event of a superabundant supply of food, predators will feed at the highest rate possible for each individual predator, and a subsequent rise in food supply will not be able to increase the eating rate further.Because of this, it is given in the form g(X, Y) = bX w+X , which is bounded as well as non-linear (the Michaelis-Menten function or the Holling type II function).Except at low prey density, the Holling type 3 is similar to the type 2, but the Holling type III prey capture rate accelerates.The Holling type III functional response is of the form g(X, Y) = cX 2 w+X 2 , which is bounded as well as non-linear [6].Up to a certain range, the Holling type II functional response accurately describes feeding rate; however, there may be circumstances in which an increase in predator density indicates a decrease in feeding rate because of mutual interference between individual predators.For this reason, we transform the Holling type II functional response into the Beddington-DeAngelis functional response, g(X, Y) = bX w 1 +Y+w 2 X [3].DeAngelis proposed the Beddington-DeAngelis functional response to solve the apparent problems with the predator-prey model.For describing parasite-host interaction independently, Beddington offered the same kind of functional response.It accurately represents the majority of the qualitative features of the ratio-dependent model while avoiding the "low density problem," which is typically contentious [2][3][4][5].The prey refuge and harvesting are incorporated into the eco-epidemiological model using Holling Eng.Proc.2023, 1, 0. https://doi.org/10.3390/0https://www.mdpi.com/journal/engproctype II behaviour, which has been studied by many authors [1].In this paper, We analyze the Beddington De-Angelis type eco-epidemiology model's behaviour towards the prey refuge and prey harvesting [7].This piece is structured as follows: The prey-predator system's past is described as section 1.In section 2, the model formation is presented.Section 3 shows some mathematicals results like positivity, positive invariance and boundedness.The existence of equilibrium points is described in section 4. Local stability analyses in section 5.The Global stability and Hope-Bifurcation Analysis is found in Section 6 and 7.
Results are presented numerically in section 8. Finally, this paper concludes with a few observations about the suggested system in secton 9.

Model Formation
The non-linear differential equation are: and the positive values are W > 0,S > 0 and I > 0.

Positive Invariance
Note the function f i (s, i, w), i = 1, 2, 3 are defined for s > 0, i > 0, w > 0. lim (s,i,w)→(0,0,0) we can extend the domain and conclude that the functions f i (s, i, w), i = 1, 2, 3 is locally Lipschitzian and continuous on R 3 + ={(s, i, w) : s ≥ 0, i ≥ 0, w ≥ 0}.Hence, the soluation of equation ( 2) with non-negative initial condition exists and is unique.It can be show that these solution exists for t > 0 and stay non-negative.Hence, the region R 3 + is invariant for the system (2).

Positivity of Solutions
Theorem 1.The solutions of (2) are positive in the R 3 + .

Boundedness of Soluation
Theorem 2. The solutions of (2) are bounded in R 3 + .

Equilibrium Points
• The E 0 (0, 0, 0) represents the essence of trivial equilibrium.• E 1 ( r−h 1 r , 0, 0)is the free of infection and predator free equlibrium its exists for h 1 < r.

Local Stability Analysis
It is necessary to calculate the Jacobian matrix, which is provided by, in order to evaluate the stability of the system.( 2 Theorem 3. If the trivial equilibrium point E 0 (0, 0, 0) is stable, if it is r < h 1 , then it is unstable.

Proof. The Jacobian matrix for
The characteristic equation of Jacobian matrix is rβ+µ(r−h 1 ) < ϕ , the equilibrium point E 1 ( r−h 1 r , 0, 0) within the infected-free and predator-free regions is stable; otherwise, it is unstable.

Proof. The Jacobian matrix at
The characteristic equation of Jacobian matrix is J(E 2 ), λ 3 + Rλ 2 + Qλ + P = 0, here, R = −u 11 − u 33 , Q = −u 21 u 12 + u 33 u 11 , P = u 12 u 21 u 33 .If P,R and RQ-P are positive, According to the Routh-hurwitz criterion, the negative real parts are the root of the above characteristic equation if and only if P,R and RQ-P are positive.RQ − P = u 11 u 33 (−u 11 − u 33 ) + u 11 u 12 u 21 .The sufficient conditions for u 33 to be negative are ϕ > c(ω + γ).hence, , 0) is locally asymptotically stable.Theorem 6. Locally stable and displaying asymptotic stability, the positive equilibrium point E * .
The characteristic equation of Jacobian matrix isJ(E * ), According to the Routh-hurwitz criterion, the negative real parts are the root of the above characteristic equation if and only if G,C and GD-C are positive.hence, E * is locally asymptotically stable.

Global Stability Analysis
Theorem 7. The endemic equilibrium point E * is globally asymptotically stable.
Proof.Consider a Lyapunov function ≤ 0. we conclude that E * is globally asymptotically stable.
Proof.For θ = θ * , (3) is in the form of an attribute equation.

Effect of Varying the Susceptible Prey Predator Rate ω
We should adjust the database variable 2 as θ = 0.2.For the given limitation value, E * is stable at positive equilibrium point ω ≥ 0.3.

Effect of Varying the Prey Refuge θ
We should adjust the database variable 2 as ω = 0.3.For the given limitation value, E * is stable at positive equilibrium point for θ ≥ 0.2.

Conclusions
In this study, we investigated the three-species food web eco-epidemiolodical model with prey refuge in infected prey population and harvesting sffect in both prey populations.Some mathsmatical results like positive invariance, positivity and boundedness analysed in system (2).The local stability is assigned to each biologically feasible equilibrium point of the system.Golbal stability analysed by sutiable lyapunov function.Hofe-bifurcation analysed by bifurcation parameter (θ).Also, Prey refuge (θ) and susceptible prey predator rate (ω) is used as acontrol parameter.According to the analytical and numerical findings, the prey refuge and susceptible prey predator rate has a major impact on the population.If we increase the susceptible prey predator rate and prey refuge in predator populations, the system loses its stability.This study shows the complex behavior of the proposed model.

Figure 1 .
Figure 1.The system's time series solution additionally parametric plot are displayed in the previously mentioned figure.(2) with limitation values in Table 2 except θ = 0.2 and ω = 0.35.

Figure 2 .
Figure 2. The compactness of predator population, infected and susceptible prey for the limitation values in table 2 except θ = 0.2 and ω = 0.30, 0.33 and 0.36.

Figure 3 .
Figure 3.The compactness of predator poulation, infected and susceptible prey for the limitation values in Table 2 except for ω = 0.3 and θ = 0.2, 0.25 and 0.3.

Table 1 .
A physiological meanings of parameters are listed in the below chart.