To promote the reasonable allocation of tasks and improve the occurrence of cooperative dilemma, we consider the interference of the third strategy based on the two-strategy division of labor game. When the cost difference of performing tasks exceeds the cooperation benefits, the strategies (players performing task A, players performing task B, and destroyers) coexist. The system also experiences bistability and occurs a transcritical bifurcation at \(E^*\) . Information often cannot be transmitted in a timely manner, so the effects of the delays in obtaining fitness information for task performers and destroyers is investigated, respectively. We perform the existence of local Hopf bifurcation occurring at \(E^*\) and the stability of \(E^*\) in the delayed system. We determine the direction, stability, and period of these periodic solutions bifurcating from steady state by the central manifold theorem and the normal form rule. The results illustrate that a small information delay does not affect the final evolutionary outcomes of the three strategies. The information delay exceeds the critical value, the frequencies of the three strategies oscillate. It is worth noting that a sufficiently large information delay leads to a mutual transformation between the existence of a single strategy and the coexistence of two strategies. Based on the above results, we can establish effective mechanisms to promote the completion of all tasks and maximize the benefits of the group.
