The typical starting point for multi-species dynamics is Lotka-Volterra equation; see for example https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation . While analytical work with it is still possible, it already has rich class of solutions and has many generalizations (like 3 species interaction).

Why do you want to go to delay equations? In principle there exist papers even on Lotka-Volterra delay equations, see for example http://www.sciencedirect.com/science/article/pii/S0022247X00972482 . But it is more difficult to study them analytically in general case. The reason to move to them is important only if time delay is crucial for your model.

Thank you for your suggestion.

Kavikumar Jacob Of course, as delay is everywhere, e.g., delay in payments, delay in ordering and supplying, population growth is also not instantaneous, but mediated by the delay period due to bio-physilogical mechanism like gestation period and past records etc.... So we can use delay in multi-species dynamics is Lotka-Volterra equation and other admissible population interacting dynamical systems.

Sahabuddin Sarwardi Thank you@