Influence of Boundary Conditions on the Dynamic Properties of the Logistic Equation with Delay and Diffusion

The logistic equation with delay and diffusion, which is important in mathematical ecology, is considered. It is assumed that the boundary conditions at either end of the interval [0,1] contain parameters. The problem of local dynamics, in a neighborhood of the equilibrium state, of the corresponding boundary value problem is investigated for all values of the boundary condition parameters. Critical cases are identified in the problem of stability of the equilibrium state and normal forms are constructed, which are scalar complex ordinary differential equations of the first order. Their nonlocal dynamics determines the behavior of solutions of the original problem in a small neighborhood of the equilibrium state. The problem of the role of asymptotically small values of the diffusion coefficient in the dynamics of the boundary value problems under consideration is studied separately. In particular, it is shown that boundary layer functions may arise when constructing asymptotic solutions in a neighborhood of the boundary points 0 and 1.

DOI 10.1134/S106192082404006X