Hopf bifurcation of the model with terms of two time-delays and delay-dependent parameter based on the theory of crossing curves

Abstract

A three-layer prey–predator model with two time-delays and one delay-dependent parameter is established in this paper. To begin, the paper calculates the conditions for each population in the model to maintain the quantity stable and Hopf bifurcation when [Formula: see text], [Formula: see text], [Formula: see text] is in the stable interval, and [Formula: see text]. The crossing curves, their type, and the direction of the crossing curves are then obtained using the crossing curve method, which is composed of the threshold values of the dynamic behavior change on the two time-delays plane when [Formula: see text]. The real data from the forage grass– Ochotona curzoniae–Buteo hemilasius food chain is used to conduct an empirical study of the model. When [Formula: see text], the feasible region of the crossing curves is open-ended, and the model's crossing curves on the [Formula: see text] plane are truncated. This indicates that the model's threshold distribution of dynamic behavior change is a regular curve made of several curves. The simulation using the time-delay value on the crossing curves shows that the model produces different dynamic behaviors such as stability, bifurcation, and chaos depending on the time-delay value on both sides of the curves. The critical values of dynamic behavior change are time-delay values on the crossing curves. The empirical study shows that increasing Ochotona curzoniae’s environmental capacity can easily cause Hopf bifurcation of the system. At this time, the number of each population in the Plateau ecosystem constantly fluctuates, and Ochotona curzoniae is vulnerable to extinction.