![]() ![]() It is mostly used to study bifurcation and chaotic behaviour in predator-prey interactions. Rosenzweig-MacArthur predator-prey model is one of such model that presents the advantage of being simple and yet exhibits very rich dynamics. Examining bifurcation, especially the supercritical ones, is very common in population dynamics, as one can determine a set of periodic solutions that may lead to system stabilization or to chaos 1, 2, 3, 4. For such systems, it is necessary to analyze stability of some given nonhyperbolic trajectories around equilibrium points and determine whether these systems exhibit rich dynamic or not. Where P( x, t) and Q( x, t) represent the population densities of the prey and predator at any point ( x, t) respectively.In applied engineering and complex system sciences, mathematical models that display deterministic chaotic dynamical behaviour are of interest. considered the following ratio-dependent modelĪnd homogeneous Neumann boundary conditions. ![]() There are two main predictions made by the ratio-dependent prey-predator models:(i) along a gradient of enrichment, equilibrium abundances are positively linked 7 (ii)the “paradox of enrichment” 8 either entirely extinct or enrichment is connected to stability. In terms of geometry, the prey-dependent model has a vertical predator isocline, while the ratio-dependent model has a tilted one 6. There are two types of prey-predator models, one is prey-dependent, and the second is a ratio-dependent model. Various researchers worked on the different aspects of the prey-predator model and added some terms in the models that explain the versatility of the models as compared with the Lotka–Volterra prey-predator model. The first prey-predator interaction was derived independently by Lotka 4 and Volterra 5 and it is known as the Lotka-Volterra prey-predator model. Predation can alter the population of the prey to extinction and then the extinction of predators. The prey-predator interaction has gained the most attention among the ecosystem interactions and predator has a severe effect on the prey-predator population. The population dynamics produce a lot of tough issues for the researchers 1, 2, 3. When population dynamics are simply observed over time, they are referred to as temporal dynamics and when they are observed over both time and space, they are referred to as spatiotemporal dynamics. There is constant interaction between and among ecosystem components and it is an interesting phenomenon how ecosystem population dynamics evolve. The life of man has links with living organisms and nonliving things and ecosystems are made of such organisms and substances. So, that they can understand the phenomena and decide for the betterment of human life. They used numerical and analytical methods for the solutions of such models. The scientists tried different techniques to understand these phenomena and developed various physical models by using a set of equations. ![]() Our numerical results will help the researchers to consider the effect of the noise on the prey-predator model. The graphical behavior of a test problem for different values of the parameters is shown which depicts the efficacy of the schemes. The proposed stochastic non-standard finite difference scheme is unconditionally stable and consistent with the system of the equations. The proposed stochastic forward Euler scheme is conditionally stable and consistent with the system of the equations. The computation of the underlying model is carried out by two schemes. The existence of the solutions is guaranteed by using Schauder’s fixed point theorem. We are considering a model where the prey-predator interaction is influenced by both space and time and the need for a coupled nonlinear partial differential equation with the effect of the random behavior of the environment. The initial prey-predator models only depend on the time and a couple of the differential equations. It relates to the population densities of the prey and predator in an ecological system. In this article, the ratio-dependent prey-predator system perturbed with time noise is numerically investigated. ![]()
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