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In this paper, we consider a Hastings–Powell type model, which consists of a three-species food chain model with fear effect. For more realistic formulation, we incorporated two time delays into the model, one for prey density another for the gestation of the middle and top predator populations. By choosing time delays as the bifurcation parameters, the essential mathematical features and their dynamics are studied in terms of local stability and Hopf bifurcation analysis. Linearizing the system at the positive equilibrium point and analyzing the distribution of the roots of the associated characteristic equation, the conditions for the existence of Hopf bifurcation are obtained. It is shown that for the gradual increase of the magnitude of delay, the stability of equilibrium point changes and the system exhibits a Hopf bifurcation as the time delay passes through some critical values. Furthermore, an explicit formula for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic orbits are investigated by using the normal form method and center manifold theory. Finally, several numerical simulations are provided to verify the effectiveness of the derived theoretical results and to examine the analytical findings. [ABSTRACT FROM AUTHOR]

• The three-strategy game model with continuous distributed time delay is analyzed. • The two-community three-strategy games with two delays is discussed. • The existence of Hopf bifurcation in replicator equation is proved. • The rock-paper-scissors game is simulated as an example. In this paper, we combine evolutionary game with dynamics, and discuss two kinds of game models with time delay. First, based on continuous distributed (kernel function) stochastic delay, we give the asymptotic stability condition of general three-strategy game model. Second, we study the replicator dynamics of two-community three-strategy game model with two fixed time delays (i.e. τ 1 within the community and τ 2 between different communities). Based on Hopf bifurcation theory, by calculating the critical conditions for the existence of bifurcation, the influence of bifurcation on the stability of equilibrium points is analyzed. Finally, we take rock-paper-scissors game as an example to verify the correctness of the theoretical results, when the delay τ meets certain conditions or doesn't exceed the critical value, the stability of the system will not change. [ABSTRACT FROM AUTHOR]