![]() Part of the time-delay differential equations will alternately appear from stable to unstable and then to stable as the singularity of the time-delay changes. Moreover, the stability of the singularity will change as the time delay changes so that Hopf branches will occur near some critical values. Since the characteristic equation of a linear system is a function of time delay, the characteristic root is also a function of time delay. For time-delay differential systems with finite time delays, we hope to understand the stability of the system and the Hopf bifurcation based on the characteristic roots of the linear system. However, Hall's theory is difficult to apply to practical problems. He studied the existence of the central flow pattern of time-delay differential systems and the Hopf bifurcation theorem. Hall was the first to study the local bifurcation of time-delay differential systems. Compared with ordinary differential systems, there is relatively little research on branch theory. Until the 1960s, the research on delayed differential systems mainly focused on stability, boundedness, asymptotics and equilibrium, periodic solutions, and oscillations of almost periodic solutions. Some scholars have conducted Hopf bifurcation research on Volterra predator–predator system with time delay. Some scholars have studied the gene regulation model of the time lag effect. Moreover, simple time-delay differential systems often contain a wealth of complex dynamic behaviours. Time lag can correspond to the incubation period, delivery delay and response delay of the disease. ![]() The time lag effect is common in real problems. Another way is to consider the time lag effect. One method increases the equation's dimensionality continuously, but the cost is that it is difficult to estimate the parameters multiplied by actual data. To this end, the model needs to be continuously improved. However, an overly simple model is difficult to accurately reflect the observed complex dynamic behaviours, such as periodic solutions. These models were once used to discover and understand various biological phenomena and social problems. Differential equation modelling can be traced back to the early population model of Malthus and the predator–prey model of Lotka and Volterra.
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