Delay-driven instability and ecological control in a food-limited population networked system

The delay and network are incorporated to describe the spatiotemporal behavior of a food-limited population dynamical system. By using the standard approach of upper and lower solutions, we have shown the global existence and uniqueness of solutions to the system. By analyzing eigenvalue spectrum, we show that the delay can cause the long-term behavior of the system from stability to instability, that is, the positive equilibrium is asymptotically stable in the absence of delay, but loses its stability such that the Hopf bifurcation occurs when the time delay increases beyond a threshold. By the norm form and the center manifold theory, we study the stability and direction of the Hopf bifurcation. We propose some formulas to control the stability and period of the bifurcating periodic solutions. Moreover, numerical simulations reveal that the network structure can switch the type of spatiotemporal patterns.