A generalization of the three-dimensional Bernfeld–Haddock conjecture and its proof

Consider the following system of delay differential equations

where r₁, r₂ and r₃ are positive constants, F, GC(R¹), and F is nondecreasing on R¹. These systems have important practical applications and also are a three-dimensional generalization of the Bernfeld–Haddock conjecture. In this paper, by using comparative technique, we obtain the asymptotic behavior of solutions that each bounded solution of the systems tends to a constant vector under a desirable condition.