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

Figure 9

The trajectory graphs and phase portrait of system (6) for (𝛼,𝛽)∈𝐷₃ with 𝛼=0.01,  𝛽=0.24, and 𝛾=0.5. (a)-(b) When 𝜏=0, the positive equilibria 𝐸^∗₁ and 𝐸^∗₃ are both asymptotically stable. The initial values are (0.2,0.05) and (0.6,0.05). (c)-(d) When 𝜏=0.37<min{𝜏₀₁,𝜏₀₃}, the positive equilibria 𝐸^∗₁ and 𝐸^∗₃ are both asymptotically stable and there exists a subcritical Hopf bifurcation near 𝐸^∗₁. Here, the initial values is (0.042,0.013). (e)-(f) When 𝜏=2.5>max{𝜏₀₁,𝜏₀₃}. All of the three equilibria become unstable and there exists a large-amplitude stable periodic orbit enclosing the three equilibria. Here, the initial value is (0.6,0.15).

(a)

(b)

(c)

(d)

(e)

(f)