Hopf bifurcation and stability analysis of delayed Gauss-type Plant–Herbivore model with Holling type-III functional response

In order to account for the realistic situation where herbivory increases with plant density, we present a novel delayed Gauss-type plant–herbivore model in this study that incorporates a Holling type-III functional response. The Allee effect, which greatly influences the growth of plant populations at low densities, is also taken into consideration in the model. One important development in plant growth regulation is the addition of a time delay parameter ([Formula: see text], which significantly changes system dynamics and gives rise to intricate behaviors such as Hopf bifurcation, limit cycles and periodic oscillations. Using the delay parameter as a bifurcation trigger, we analytically determine the feasible nontrivial equilibrium point and examine its stability. Our analysis shows that, as [Formula: see text] varies, the system behavior undergoes a critical transition from global stability to asymptotic stability and, ultimately, to instability via Hopf bifurcation at the interior equilibrium [Formula: see text]. These shifts provide insights into how ecological interactions and time delays influence population dynamics, and they are both analytically verified and graphically depicted. MATLAB-based numerical simulations clearly show how changes in system parameters affect stability and produce complex dynamic patterns.