Mathematical Analysis of a Prey–Predator System: An Adaptive Back-Stepping Control and Stochastic Approach

: In this paper, stochastic analysis of a diseased prey–predator system involving adaptive back-stepping control is studied. The system was investigated for its dynamical behaviours, such as boundedness and local stability analysis. The global stability of the system was derived using the Lyapunov function. The uniform persistence condition for the system is obtained. The proposed system was studied with adaptive back-stepping control, and it is proved that the system stabilizes to its steady state in nonlinear feedback control. The value of the system is described mostly by the environmental stochasticity in the form of Gaussian white noise. We also established some conditions for oscillations of all positive solutions of the delayed system. Numerical simulations are illustrated, and sustained our analytical ﬁndings. We concluded that controlled harvesting on the susceptible and infected prey is able to control prey infection.


Introduction
Theoretical research and field observations have established the prevalence of various infectious diseases amongst the majority of the ecosystem population. In the ecological system, the impact of such infectious diseases is an important area of research for ecologists and mathematicians. The processes of merging ecology and epidemiology in the past few decades have been challenging and interesting. By nature, species are always dependent on other species for its food and living space. It is responsible for spreading infectious diseases and also competes against and is predated by other species. The dynamical behavior of such systems is analyzed using mathematical models that are described by differential equations. Mathematical epidemic models have gained much attention from researchers after the pioneering work of Kermack and McKendrick [1] on the SIRS (Susceptible-Infective-Removal-Susceptible) system, in which the evolution of a disease which gets transmitted upon contact is described. The influence of epidemics on predation was first studied by Anderson and May [2,3]. Hadler and Freedman [4] considered the prey-predator model in which predation is more likely on infected prey. In their model, they considered that predators only became infected from infected prey by predation. Haque and Venturino [5,6] discussed the models of diseases spreading in symbolic communities. Mukhopadhyay [7] studied the role of harvesting and switching

Mathematical Model
In this section, a continuous-time prey-predator system with susceptible, infected prey and a predator is considered. It is assumed that the susceptible prey population was developed on the basis of logistic law, and that only infected prey are predated. The disease is inherited only from the prey population, and they remain infected and do not recover.
The model becomes: Here, the parameters x(t), y(t), and z(t) denote the susceptible prey, infected prey, and predator populations, respectively. The parameters a, d, b, f , h 1 , and h 2 denote the rate of transmission from the susceptible to infected prey population, death rate of predators, searching efficiency of the predator, conversion-efficiency rate of the predator, and constant harvesting rate of susceptible prey and infected prey, respectively. Now, we will analyze system (1) with the following initial conditions: x(0) ≥ 0, y(0) ≥ 0, z(0) ≥ 0. (2)

Positiveness and Boundedness of the System
In theoretical eco-epidemiology, boundedness of the system implies that the system is biologically well-behaved. The following theorems ensure the positivity and boundedness of the system (1): Theorem 1. All solutions of (x(t), y(t), z(t)) of system (1) with the initial condition (2) are positive for all t ≥ 0.
Theorem 2. All the non-negative solutions of the model system (1) that initiate in 3 + are uniformly bounded.
Proof. Let x(t), y(t), z(t) be any solution of system (1). Since from (1) Substituting Equation (1) in Equation (3), we get where m and µ are positive constants. Applying Lemma on differential inequalities [23], we obtain and, for t → ∞, we have 0 ≤ ξ(x, y, z) ≤ (µ/m). Thus, all solutions of system (1) enter into the region This completes the proof.

Equilibrium Points and Criteria for Existence
The possible steady states for system (1) and their existence conditions for each of them are as follows: 1. The vanishing equilibrium point, E 0 = (0, 0, 0), always exists. 2. The disease and predator free equilibrium point, exists.
However, x * is a positive root of (5) where

Stability Analysis
In this section, we analyzed the local stability of system (1) that is examined by constructing the Jacobian matrix relating to every equilibrium point. The Jacobian matrix of the system at any point (x, y, z) is given by 1. The variational matrix for the equilibrium point at E 0 (0, 0, 0) is Hence, E 0 is locally asymptotically stable in the x-y-z direction. 2. The variational matrix for the equilibrium point at Hence, E 1 is asymptotically stable in the x-y-z direction. 3. The variational matrix for the equilibrium point at E 2 (x,ŷ, 0) is . Hence, the equilibrium point E 2 is locally asymptotically stable in the x-y-z direction.
Theorem 3. The co-existent equilibrium point E 3 (x * , y * , z * ) of system (1) exists. Then, E 3 is locally asymptotically stable if the following conditions satisfy The characteristic equations of Jacobian matrix J(E 3 ) is given by The sufficient conditions for A 1 A 2 − A 3 > 0 are a 11 ≤ 0, a 22 ≤ 0, which implies

Global Stability Analysis
In this section, we investigated the global stability behavior of the system (1) at the interior equilibrium E 3 (x * , y * , z * ) by using the Lyapunov stability theorem.
Proof. Let us define where P, Q are positive constants to be chosen later. Differentiating (6) along the solution of the system (1) with respect to t, we get Choosing P = 1, Q = b/ f , and then simplified to Then, dV dt is negative definite. Consequently, V is a Lyapunov function with respect to all solutions in the interior of the positive octant.

Permanence of the System
In this section, our main intuition is that the long time survival of species in an ecological system. Many notions and terms are identified in the literature to discuss and analyze the long-term survival of populations. Out of such, permanence and persistence are the ones to better analyze the system. From an ecological point of view, permanence of a system means that the long-term survival of all populations of the system. Definition 1. The system (1) is said to be permanent if ∃ M ≥ m > 0, such that for any solution of (x(t), y(t), z(t)) of system (1), (x(0), y(0), z(0)) > 0, Now, we show that system (1) is uniformly persistent. The survival of all populations of the system in the future time is nothing but persistence in the view of ecology.
In the mathematical point of view, persistence of a system means that a strictly positive solution does not have omega limit points on the boundary of the non-negative cone.
Proof. We will prove this theorem by the method of Lyapunov average function. Let the average Lyapunov function for the system (1) be σ(X) = x p y q z r , where p, q, r are positive constants. Clearly, σ(X) is non-negative function defined in D of 3 Then, we have Ψ(X) =σ (X) Furthermore, there are no periodic orbits in the interior of positive quadrant of x-y plane. Thus, to prove the uniform persistence of the system, it is enough to show that Ψ(X) > 0 in 3 + for a suitable choice of p, q, r > 0 : We noticed that, by increasing the value of p, while (10) holds. If the inequality in Equation (7) holds, then (11) is satisfied.

Introduction of Adaptive Back-Stepping Control in a Prey-Predator System
Adaptive back-stepping method is the back-stepping control and adaptive laws. The back-stepping design is initiated with the first state equation whose state variable has the highest integration order from the control input. The second state variable is considered as the virtual control and the stabilizing function is replaced by it. This stabilizing function can stabilize the first state variables and we set the error between the virtual control and stabilizing function as η. Then, for the second state equation, we will design a new stabilizing law to replace the third state variable for the second order system, and step back to control the signal. From the steps above, we can see that the term back-stepping means that we use the latter state as a virtual control to stabilize the previous one. The Lyapunov direct method is utilized from the stabilization method for the error between virtual control and stabilizing function. The control Lyapunov function is to be used which will be a positive definite and includes the quadratic form of the errors. In this section, the system with susceptible prey, infected prey and a predator population controlled by back-stepping using a nonlinear feedback control approach is studied. We initiate the study by assuming that system (1) can be written in the suitable form by introducing nonlinear feedback control inputs u 1 , u 2 , u 3 into the system to better analyze the prey-predator interactions: where u 1 , u 2 , u 3 are back-stepping nonlinear feedback controllers that are the functions of state variables and will be suitable choices to make the trajectories of the whole system (12)- (14). As long as this feedback stabilizes the system (1), lim t→∞ ||x(t)|| = 0 converges.

Theorem 6.
A diseased prey-predator system (12)- (14) is globally asymptotically stable provided the following adaptive nonlinear controls with the errors Proof. Consider the parameter estimators Considering Equation (12), the Lyapunov function of x is given by By using the derivative of (20), Differentiating (21) with respect to t, we havė Considering y as a virtual controller, then Choosing ν 1 = 0 and using the controller (15), Ref. (22) becomeṡV The updated law by the unknown parameteṙâ Substituting (24) in (22), we getV 1 is the negative definite function, where Differentiating (25), we getη Now, Equation (13) along with Equation (26), we get thaṫ Considering the Lyapunov function of (x, η 1 ), Differentiating (28) with respect to t, we geṫ Again, considering a new virtual controller z = ν 2 (x, y) where ν 2 = 0, and using this in (24), we havė Now, choosing the controller (16) along with (30), we geṫ The updated law for the unknown parameter˙b and˙ĥ 2 iṡb Substituting (32) in (31), we getV which is again a negative definite function where Differentiating (34) with respect to t, we haveη 2 =ż. Now,η where the controller (17) along with (35) giveṡ Now, considering the Lyapunov function of (x, η 1 , η 2 ), Differentiating (38) with respect to t giveṡ The unknown parametersȧ,˙b,˙f ,˙ĥ 2 are updated bẏâ Substituting the updated parameters along with choosing the controller (17) and by updating law d is the negative definite function. Thus, by the Lyapunov stability theorem, systems (10)-(12) is globally asymptotically stable with adaptive back-stepping controllers.

Stochastic Analysis
All usual occurrences explicitly in the ecosystem are continuously under random fluctuations of the environment. The stochastic examination of any ecosystem gives an enhanced vision on the dynamic forces of the populace by means of population variances. In a stochastic model, the model parameters oscillate about their average values [24][25][26][27]. Therefore, the steady state which we anticipated as permanent will now oscillate around the mean state. The method to measure the mean-square fluctuations of population is proposed by [24] and it was applied by [28] nicely. Furthermore, many researchers like Samanta [29], Maiti, Jana and Samanta [30] have investigated critically the stochastic analysis to interpret local as well as global stability using mean-square fluctuations on population variances. Now, this segment is meant for the extension of the deterministic model (1), which is formed by adding a noisy term. There are several ways in which environmental noise may be incorporated in the model system (1). External noise may arise from random fluctuations of a finite number of parameters around some known mean values of the population densities around some fixed values. Since the aquatic ecosystem always has unsystematic fluctuations of the environment, it is difficult to define the usual phenomenon as a deterministic ideal. The stochastic investigation enables us to get extra intuition about the continuous changing aspects of any ecological unit. The deterministic model (1) with the effect of random noise of the environment results in a stochastic system (41)-(43) given in the following discussion: where α 1 , α 2 , α 3 are the real constants and is the Kronecker delta function; δ is the Dirac delta function. Let The linear parts of (41), (42) and (43) are (using (44) and (45)) Taking the Fourier transform on both sides of (46), (47) and (48), we get The matrix form of (49)- (51) is where Equation (52) can also be written intoũ where and Here, If the function Y(t) has a zero mean value, then the fluctuation intensity (variance) of its components in the frequency interval [ω, ω + dω] is S Y (ω)dω. S Y (ω) is the spectral density of Y and is defined as If Y has a zero mean value, the inverse transform of S Y (ω) is the auto covariance function The corresponding variance of fluctuations in Y(t) is given by and the auto-correlation function is the normalized auto-covariance For a Gaussian white noise process, it is From (54), we haveũ From (59), we have where Hence, by (61) and (62), the intensities of fluctuations in the variable u i (i = 1, 2, 3) are given by and by (54), we obtain From (55), (64), (65) and (66), where |M(ω)| = R(ω) + iI(ω).
If we are interested in the dynamics of stochastic process (41)-(69) with either α 1 = 0 or α 2 = 0 or α 3 = 0, the population variances are Equations (67)-(69) give three variations of the inhabitants. The integrations over the real line can be estimated, which gives the variations of the inhabitants.

Mathematical Model with Delay
In this section, we establish some conditions for oscillations of all positive solutions of the delay system dx Here, the parameter τ ∈ R + is the delay. This proposed system is concerned not only with the present number of predator and prey but also with the number of predator and prey in the past. If t is the present time, then (t-τ) is the past.
In [32], Kubiaczyk and Saker studied the oscillatory behavior of the delay differential equation where p, q, r, τ ∈ R + . Using similar methods to liberalize each equation of the delay system, we will establish conditions for oscillations of all positive solutions of the system. Now, we will analyze the system of (70)-(72) with the following initial conditions: Using the same arguments that we got in Theorem 1, we can establish the following theorem: Theorem 7. All solutions of (x(t), y(t), z(t)) of systems (70)-(72) with the initial condition (73) are positive for all t ≥ 0.
Easily, we can see that the equilibrium point remains the same when we have the delay system. However, it is important to know the oscillatory behavior of the solutions around the equilibrium points.

Numerical Simulations
Analytical studies can never complete without numerical verification of the results. Moreover, it may be noted that, as the parameters of the model are not based on the real world observation, the main features described by the simulations presented in this section should be considered from a qualitative rather than a quantitative point of view. We choose the parameters of system (1) as a = 1.5, b = 0.3, f = 0.65, h 1 = 0.5, h 2 = 0.3, d = 0.2, K = 8 with the initial densities x(0) = 2, y(0) = 1.8, z(0) = 1 and observe the dynamical behaviour of system (1). Figure 1a shows that the equilibrium point E 1 is locally asymptotically stable and the corresponding phase-portrait is shown in Figure 1b. Figure 2a shows that the equilibrium point E 2 is locally asymptotically stable and the corresponding phase-portrait is shown in Figure 2b. Figure 3a shows that the co-existence equilibrium point E 3 is locally asymptotically stable and the corresponding phase portrait is shown in Figure 3b. From Figure  4a-4d, if all other parameters are fixed and varying transmission rate a = 1.5 to a = 2, we observe that oscillation settles down into a stable situation for all three of the species. Stability around this state implies extinction of the infected prey. This study interestingly suggests that the harvesting of both prey prevent limit cycle oscillations and the combined effect of both harvests also prevent disease propagation in the system. We also conclude that the inclusion of stochastic perturbation creates a significant change in the intensity of populations because change of responsive noise parameters causes chaotic dynamics with low, medium and high variances of oscillations (see Figures 5-7).      se portrait between prey and predator species respectively with the parameters a , f = 0.65, h 1 = 0.5, h 2 = 0.3, d = 0.2, K = 8.

Concluding Remarks
In this paper we have studied stability of disease model of susceptible, infected prey and predator around interior steady state. The positivity of the solutions and boundedness of the system together with stability analysis of boundary equilibrium provided all necessary information to establish persistence of the system. The deterministic situation theoretical epidemiologist are usually guided by an implicit assumption that most epidemic models, we observe in nature correspond to stable equilibrium of models. In theorem 3, we given condition for stable co-existence in theorem 4 we prove the global stability by using Lyapunov function. Also the condition for which all three species will persist worked out. The controllability conditions and the conditions for global asymptotic stability have been obtained by using the adaptive backstop control approach by using suitable Lyapunov function. We also conclude that the inclusion of stochastic perturbation creates a significant change in the in-

Conclusions
In this paper, we have studied stability of a diseased model of susceptible, infected prey and predators around an interior steady state. The positivity of the solutions and boundedness of the system together with stability analysis of boundary equilibrium providing all the necessary information to establish persistence of the system. The deterministic situation theoretical epidemiologists are usually guided by an implicit assumption that most epidemic models (we observe in nature) correspond to stable equilibrium of models. In Theorem 3, we gave the condition for stable co-existence. In Theorem 4, we proved the global stability by using a Lyapunov function. We also worked out the condition for which all three species will persist. The controllability conditions and the conditions for global asymptotic stability have been obtained by using the adaptive back-stepping control approach by using a suitable Lyapunov function. We also studied the stochastic perturbation of model (1), which generates a significant change in the intensity of populations due to low, medium and high variances of oscillations.