Application Of Backstep Control In Diseased Prey-Predator System

Application of the Generalized Backstepping Control Method for Lotka-Volterra Prey-Predator System with Constant Time Delay

“…Such diseases play an important role in controlling the specie's population growth. The models are defined by differential equations, and ecologists and mathematicians are particularly interested in the prey-predator model [13].…”

“…For the sake of solving system (13), which is a linear system with nonconstant coefficients which is difficult to integrate, so the coefficient functions are approximated using Lagrange interpolation polynomial of degree 5, and hence system (13) will be reduced to: Then the solution is: 𝑥 1 (𝑡) = 3.176 𝑒 − 0.00292542𝑡 6 + 0.0969738 𝑡 5 −0.70755𝑡 4 +2.10439 𝑡 3 −2.49617𝑡 2 −2.49617t 𝑥 2 (𝑡) = 5.7217 𝑒 − 0.00478621 𝑡 6 + 0.0893614𝑡 5 − 0.558928𝑡 4 +1.44325𝑡 3 −0.798714𝑡 2 −3.91471t } The initial conditions for the third time-step interval [2,3] are supposed to be the previous step solutions which are given by equations ( 14), and similar approach is used as it is followed over the time-step interval [1,2], to get the next solutions: Similarly, we can proceed to evaluate the solutions and the control functions over the time-step intervals [3,4] and [4,5] which are sketched in Figures 7-10, respectively. Finally, the solution for the original system (7) for all conducted time step intervals are collected and sketched in Figure 11, as well as, the controller functions 𝑢 1 and 𝑢 2 are also collected in Figure 12.…”

“…Backstepping method is a control theory approach used for controlling and stabilizing non-linear dynamical systems. This approach is achieved by recursively stabilizing the system origin in which the control process ends when the final external control is evaluated [3]. Moreover, the backstepping method is a typical strategy for achieving control laws by defining virtually error variables and a Lyapunov function for each subsystem of the original system of ordinary differential equations (ODEs) to assure system stability [10].…”

“…A contemporary approach is to investigate ecological systems using backstepping controllers. To improve the system's stability analysis, the controllability requirements for chaotic dynamics control may also be defined utilizing backstepping controllers [3].…”

“…Incorporating time delays into models is frequently required to depict the qualitative behaviours of the models that are dependent on the system's history [1]. Madhusudanan et al [3] employed the backstepping control design, in which the controllability conditions are framed for a three-disease biological system with two preys, namely the infected and susceptible prey-predator.…”