Stability and Bifurcation Analysis of a Delayed Leslie-Gower Predator-Prey System with Nonmonotonic Functional Response

Figure 4

The trajectory graphs and phase portrait of system (6) for (𝛼,𝛽)∈𝐷₁₁ with 𝛼=0.2,  𝛽=2, and 𝛾=0.2. (a)-(b) When 0≤𝜏<𝜏₀, the equilibrium 𝐸^∗ is stable. (c)-(d) Simulations of subcritical Hopf bifurcation. Starting from the initial value (𝑥(0),𝑦(0))=(0.0615,0.2897), the trajectory slowly gets away from a small-amplitude unstable periodic solution and finally converges to a large-amplitude stable periodic solution. Starting from the initial value (𝑥(0),𝑦(0))=(0.0615,0.2895), the trajectory converges to 𝐸^∗. (e)-(f) When 𝜏>𝜏₀, the equilibrium 𝐸^∗ is unstable with a large-amplitude stable periodic solution.

(a)

(b)

(c)

(d)

(e)

(f)