Phase turbulence in population models with diffusion and delay

Predator-prey type models are commonly used to describe the dynamic interplay of two interacting species, the predator and the prey. Contrary to earlier studies where delay terms represent the lag between birth and maturity of the particular species, we here analyze a delayed predator-prey interaction. Delayed interaction can be associated with the occurrence of an additional intermediate species, generated from the prey and consumed by the predator. In an epidemiological context, this scenario can be connected with a compartment model, where the susceptible compartment plays the role of the prey, the intermediate compartment corresponds to the latency (incubation) period, and the infected and infectious state is identified with the predator. We consider this setting with and without spatial diffusion. The diffusion corresponds to random motions of the microscopic agents. In the case without diffusion and for a sharp delay time (sojourn time in the intermediate state), a Hopf instability emerges when the delay time exceeds a critical threshold. A weakly nonlinear expansion around this critical point leads to a Hopf normal form allowing to determine a stable long-time limit cycle in an explicit form. Including diffusion into the model, we derive the Ginzburg-Landau normal form, which leads to an approximate phase diffusion equation with stable turbulent phase solutions. We derive analytically the coefficients of the Ginzburg-Landau equation and compute the parameter regions where phase turbulence and weak chaos are expected. As new features, defect mediated patterns such as spirals, but also weakly turbulent structures emerge, resulting from the complex interplay of delay and diffusion. In addition to mathematical biology, our model has various interdisciplinary applications in the kinetics of certain chemical reactions, social science, finance, economy, and others.