We can study the system of equation with single delay using "Ruan Lemma":
S. Ruan, Absolute stability, conditional stability and bifurcation in
kolmogorov-type predator-prey systems with discrete delays, Quarterly of Applied Mathematics 59 (1) (2001) 159–174.
I am looking for results with two non-zero delay. Please suggest any recently published manuscript with two or more delay.