Existence and Global Stability of Positive Periodic Solutions to the Food Chain System of Three Species with Time Delays

In order to verify the existence and global stability of positive periodic solutions to the food chain system of three species with time delays in a periodic environment, this article constructed a suitable Lyapunov function to obtain a set of sufficient conditions for the global stability of the system by using Gaines and Mawhin's continuation theorem of coincidence degree theory. The author considered three species (y₁, y₂, y₃; y₁ is the prey to y₂; y₂ is the prey to y₃), growth rate (of y₁ primarily) and mortality rate (of y₂ and y₃ primarily) of independently existed species, as well as the interspecies pillage and sustentation capabilities. The prior bounds should be estimated for three species separately; furthermore, the upper and lower bonds should be estimated more accurately.