Spatiotemporal Dynamics in Predator–Prey Models Incorporating Time Delay and Harvesting

A three-compartment predator-prey model (prey, middle predator, top predator) is developed by integrating spatial diffusion, discrete time delays (gestation and biomass conversion), and proportional harvesting on the two lower trophic levels. The objectives are to formulate the model, analyze local stability dynamics and Hopf bifurcation due to time delay variations, and determine a sustainable harvesting strategy based on the Maximum Sustainable Yield (MSY) concept. Methods include equilibrium analysis, linearization using a variation matrix, Routh–Hurwitz criterion, and numerical simulations with GNU Octave in one- and two-dimensional spatial domains. The results show that a positive interior coexistence equilibrium exists and is locally asymptotically stable under certain conditions. Single or double time delays without harvesting trigger Hopf bifurcation when exceeding critical values, while pure diffusion does not produce instability on its own. Harvesting at safe rates increases the critical delay values and maintains coexistence, in contrast to the mathematical MSY harvesting that leads to species extinction. In conclusion, the integration of diffusion, time delays, and harvesting yields complex dynamics, with time delays as the main destabilizer and controlled harvesting as a stabilizing agent that delays the onset of Hopf bifurcation.