Bifurcations of a Ratio-Dependent Holling-Tanner System with Refuge and Constant Harvesting

Figure 5

(a) The cusp of codimension 2 breaks into a stable focus 𝐸₁ and a hyperbolic saddle 𝐸₂ when 𝜇₁=−0.0098, 𝜇₂=−0.02. The change of stability of the focus yields an unstable limit cycle. (b) The unstable limit cycle is broken when 𝜇₁≈−0.0081768, 𝜇₂=−0.02, reachs the manifold of the saddle 𝐸₂, and leads to a homoclinic loop occur.

(a)

(b)