Analysis On Turing Pattern In Three Components Reaction Diffusion System

The reaction-diffusion system demonstrates a variety of dynamical behaviors,and has become a standard model for explaining complex Turing patterns.Here,in this dissertation an analytical and computational study of Turing's reaction diffusion models has been done.Turing reaction diffusion model is a set of coupled partial differential equations illustrating the reaction and diffusion behavior of chemical species.Under certain conditions,such systems are capable of producing stationary as well as oscillating patterns of numerous characteristic wave lengths,even if the system is given some random initial configuration.The characteristics of the resulting patterns are calculated intrinsically by the diffusion and the reaction rates of the chemicals.In natural systems Turing patterns have been shown to have counterparts and hence Turing systems could provide a feasible way to model the mechanisms of biological developmental processes.The main factor give rise to Turing patterns is called diffusion-driven instability as a result of the minute perturbation about the homogeneous steady state of the model under unstable conditions.Turing systems have been analyzed using various techniques including mathematical tools,chemical experiments,and numerical simulations.We have done analytical and numerical calculation of various two and three components reaction diffusion models.The set of equations are highly nonlinear which is linearized by using Taylor's expansion.The model was then studied using a standard linear stability analysis,which discloses the sets of parameters correlating to Turing instability and the required unstable wave modes.The linear stability is achieved by obtaining the roots of fourth order characteristic polynomial which is quadratic in x2.The typical dispersion relation curve has been plotted which displays the value of Turing bifurcation parameter and numerical values of different parameters for numerical simulation.The analytical predictions have been confirmed by using numerical simulations.For different sets of parameters,different patterns have been obtained through numerical simulations such as strips,spots,and labyrinthine.It has been found that Turing patterns were highly sensitive to the initial conditions(robust).Starting with slightly different initial conditions,we ended up entirely with different results.Similarly,in case of three components reaction diffusion system,we performed the analytical analysis of the three components Gray-Scott reaction-diffusion system.In this system we obtained a characteristic polynomial,which is of the order sixth.Using Routh-Hurwitz criterial the analytical conditions for Turing instability about the homogeneous steady state and the Turing bifiurcation parameter has been derived.The linear stability has been theoretically discussed in the main text.In this thesis,the main analytical results presented are the analysis of the Turing pattern selection in case of three components reaction diffusion system which depends on the initial configuration of the system.In order to get further information about the dynamics of the Pattern we deduced the amplitude equation by using multiple scale analysis or weakly nonlinear analysis.The amplitude equation provided the parameter values for each type of spatial pattern and enumerates about the rich dynamical behavior of the Gray-Scott model such as spot-,strip-and hexagon-patterns.