Long cycles in subgraphs of (pseudo)random directed graphs

We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every 0≤γ≤1/2 we find a constant c = c(γ) such that the following holds. Let G = (V, E) be a (pseudo)random directed graph on n vertices and with at least a linear number of edges, and let G′ be a subgraph of G with (1/2 + γ)|E| edges. Then G′ contains a directed cycle of length at least (c - o(1))n. Moreover, there is a subgraph G″ of G with (1/2 + γ - o(1))|E| edges that does not contain a cycle of length at least cn.