A population model with time-dependent delay

We present analytical and computational results concerning the linear stability and instability of the uniform steady-state solution of a system of reaction-diffusion equations where a parameter in the kinetic terms is periodic in time. Under suitable assumptions the system is equivalent to a scalar equation with a periodically varying delay. Such a varying delay can model the seasonal fluctuations to the regeneration time of a resource. We study the effect such a varying delay can have on the stability of the spatially uniform steady-state. Analytical results reveal that instability can set in if the delays are large, while computational methods of analysing the stability equations reveal the precise shape of the instability boundary. The nonlinear stability of the uniform state is also examined using ladder methods.