Pattern Formation In Some Reaction-diffusion Models

In this thesis,the reaction-diffusion behavior of physical and biological processes is treated analytically and numerically.Turing patterns have been shown to have counterpaits in natural systems and thus reaction-diffusion systems could provide a plausible way to model the mechanisms of physical and biological growth.The infinitesimal perturbations around the stationary state of the model grows Turning patterns due to the Turing instability and exists under non-equilibrium conditions.Taring systems have been studied using experiments,mathematical tools and numerical simulations.In Chapter 1,we introduce briefly some background and motivation of this dissertation.In Chapter 2,we discuss the Turing instability and pattern formation in the FitzHugh-Nagumo model with super-diffusion in two-dimensional numerical simulations.We also studied the effects of the super-diffusive exponent on pattern formation concluding that with the presence of super-diffusion the stable homogenous steady state becomes unstable.By using the stability analysis of local equilibrium point,we procure the conditions which ensure that the Turing and Hopf bifurcations occur.For pattern selection,the weak nonlinearmulti-scale analysis is used to derive the amplitude equations of the stationary patterns.We then apply amplitude equations and observe that this model has very rich dynamical behaviours,such as stripes,spots and hexagon patterns.The complexity of the dynamics in this system is theoretically discussed and graphically displayed in numerical simulation.The simulation helps us to show the effectiveness of theoretical analysis and patterns which appear numerically.In Chapter 3,we explore the emergence of patterns in a super-cross-diffusion model withBeddington-DeAngelis type functional response.First,we explore the stability of equilibrium points with or without super-cross-diffusion.Instability of equilibria can be induced by cross-diffusion.We perform the linear stability analysis to obtain the constraints for the Turing instability.It is found by theoretical analysis that cross-diffusion is an important mechanism for the appearance of Turing patterns.For the dynamics of pattern,the weakly nonlinear multi-scaling analysis has been performed to obtain the amplitude equations.Finally,we ensure the existence of Turing patterns such as squares,spots,and stripes by using the stability analysis of the amplitude equations.Moreover,with the assistance of numerical simulations,we verify the theoretical results.In Chapter 4,we consider the well-known predator-prey system with self and super-cross-diffusion terms.Firstly,the stability of the equilibrium points is explored.The occurrence of the Turing instability is ensured by the conditions which are procured by using the stability analysis of the local equilibrium points.Amplitude equations are derived by using the weakly nonlinear analysis close to the Turing bifurcation point.Finally,we ensure the existence of Turing patterns such as hexagons,small spots,big spots,squares,str:ipes and labyrinthine by using the stability analysis of the amplitude equations.Moreover,with the assistance of numerical simulations,we verify the theoretical results.The general nature of the obtained results allows investigating the effect of self-super-cross-diffusion on the other predator-prey systems both theoretically and numerically.In Chapter 5,we explore the pattern formation induced by super-cross-diffusion in a three-species ecological symbiosis model with harvesting.Initially,all the possible equilib-rium points are identified and then by using Routh-Hurwitz criteria stability of an interior equilibrium point is explored.From stability analysis of local equilibrium points,conditions for Turing instability are derived.Weakly nonlinear analysis is employed close to the Turing bifurcation point to obtain the amplitude equations.Through the dynamical analysis of the amplitude equations,conditions for the formation of Turing patterns like hexagons,rhombus,spots,squares,strips and waves patterns are identified.Moreover,with the assistance of numerical simulations,we verify the theoretical findings.