Feedback control for a diffusive delay logistic equation: Semi-analytical solutions
Semi-analytical solutions are considered for a delay logistic equation with non-smooth feedback control, in a one dimensional reaction-diffusion domain. The feedback mechanism involves varying the population density in the boundary region, in response to the population density in the centre of the domain. The effect of the two sources of delay, from the logistic equation itself and the feedback term, is explored. The Galerkin method, which assumes a spatial structure for the solution, is used to approximate the governing partial differential equation by a system of ordinary differential equations. The form of feedback is chosen to leave the steady-state solution unchanged and guarantee positive population densities at the boundary. Whilst physically realistic, the feedback is non-smooth as it has discontinuous derivatives. A local stability analysis, of the four smooth parts of the full system, allows a band of parameter space, in which Hopf bifurcations occur, to be found. A precise estimate of the Hopf bifurcation parameter space, for the non-smooth system, is obtained using a hybrid stability condition. This is found by considering the dominant eigenvalues of the smooth parts of the system. Examples of bifurcation diagrams, stable solutions and limit cycles are shown in detail. Comparisons of the semi-analytical and numerical solutions show that the semi-analytical solutions are highly accurate.