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In the present work, we have developed a plankton-fish model consisting time delay using functional response Holling type-II and linear harvesting. Two different additional foods are supplied to both the species zooplankton and fish and study which one additional food more influences positively to the system than other. We derive some stipulations for existence of equilibria and its stability. Stipulations for local and global stability, positivity and boundedness of the system are investigated. Stability and instability analysis have been performed through the time series, phase space and bifurcation diagrams with respect to various ecological factors. In both cases, presence of delay and absence of delay, the main observation of the study indicates that the low additional food supply to zooplankton has an extinction risk for both the species fish and themselves, although the certain additional food is provided to the fish. It is beneficial not only for the zooplankton but also for the fish in both cases. It is also observed that enhancing the additional food supply of the zooplankton in presence of delay is more advantageable than in absence of delay, because of, delay enhances that quantity of additional food supply to a certain level without changing the stable nature of the system for which the system exhibits a stable dynamic in absence of delay. The complex behaviour (chaos) which is appeared due to conversion rate and small harvesting of zooplankton that can be completely controlled by the delay. Double Hopf bifurcation behaviour has been inspected due to delay. A small gestation delay has destabilizing effect, whereas a relatively large gestation delay has stabilizing effect.

This article deals with a singularly perturbed delay differential equation involving two small parameters. First, an upwind scheme is proposed on the fitted Shishkin mesh. The method is shown to be first order convergent. Then a hybrid scheme consisting of the midpoint upwind scheme in the outer region and the central difference scheme in the inner region is proposed. The error analysis is carried out for both the schemes and the accuracy is increased the from first order to second order. The performance of both schemes are assessed on some test problems and the results are in accord with the theoretical findings.