Delay-Induced Stability Transitions and Hopf Bifurcation in a Model of Cumulative Hot Pollutant Concentration

The present work provides a detailed discussion of the dynamical behavior of the delay-induced model of cumulative concentration of hot pollutants, including the contribution of the time-delay parameter to the system’s stability. Analytical results indicate that time delay is a bifurcation mechanism that leads to a critical threshold, at which a steady state loses asymptotic stability and a Hopf bifurcation occurs. The directional analysis is carried out to further explain the behavior of the system in the neighborhood of this transition, and this offers some understanding of the nature and stability of the resulting periodic solutions, as well as the qualitative evolution. Numerical simulations are done on representative parameter values to support the theoretical results. Comprehensively, the findings reveal the strong dependence of the accumulation processes of pollutants on the effects of time delays and the significance of considering the temporal lags in environmental modeling. The study provides a viable analytical and numerical system of interpreting transitions caused by delays in pollutant concentration systems.