Bifurcation and Global Dynamics of Continuous and Discrete Competitive Models for Genetic Toggle Switches
We investigate the asymptotic behavior of a proposed ordinary differential equation (ODE) model for Genetic Toggle switches from Gardner et. al. and I. Rajapakse and S. Smale: dxdt=a1+ym−x and dydt=b1+xn−y where a,b,m,n>0 and x(t),y(t)≥0. We also investigate the asymptotic behavior of the Euler discretization of this system: xn+1=a1xn+b11+ynm=f(xn,yn) and yn+1=a2yn+b21+xnn=g(xn,yn), where 1−h=a1, 1−k=a2, ah=b1 and bk=b2, a1,a2∈(0,1) and h,k>0 are steps of discretizations. Here, x and y represent protein concentrations at a particular time in both genes and a,b,m,n>0, respectively, above. We will apply the theory of competitive maps to find the basins of attractions of different equilibrium points and period-two solutions of systems of difference equations.