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

Figure 5

The trajectory graphs and phase portrait of system (7) without delay for (𝛼,𝛽)∈𝐷₂ with 𝛼=0.2 and 𝛽=1.2. (a)-(b) 𝛾=0.02<𝛾₀. The positive equilibrium 𝐸^∗ is unstable with a large-amplitude stable periodic orbit. The initial value is (0.1578,0.18935). (c)-(d) 𝛾=0.04>𝛾₀. The positive equilibrium 𝐸^∗ is stable, and a small-amplitude unstable periodic solution and a large-amplitude stable periodic solution coexist. The initial values are (0.1578,0.21532) and (0.1578,0.21533). (e)-(f) 𝛾=0.06>𝛾₀. The positive equilibrium 𝐸^∗ is stable and there are no Hopf bifurcating periodic orbits. The initial value is (0.5,0.3).

(a)

(b)

(c)

(d)

(e)

(f)