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Figure 6 | Advances in Difference Equations

Figure 6

From: Stability and Hopf bifurcation for a stage-structured predator–prey model incorporating refuge for prey and additional food for predator

Figure 6

Time series of \(u_{1}(t)\), \(u_{2}(t)\), \(v(t)\) and phase portrait of the model (1.5) with \(\tau= 0.5 > \tau_{0} = 0.1509514710143546\). An orbit from the initial condition \((u_{1}(0),u_{2}(0),v(0)) = (7, 1.2, 125)\) located in a sufficiently small neighborhood of the equilibrium \(E_{2}(5.303571428572600, 1.650000000000400, 133.712142857161200)\) converges to a periodic solution. The simulation results indicate that the equilibrium \(E_{2}\) is unstable and periodic solution is stable. The periodic attractor bifurcates from the equilibrium and surrounds the equilibrium \(E_{2}\) as the delay τ crosses the critical value \(\tau_{0} = 0.1509514710143546\)

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