Bifurcations of invariant tori in predator-prey models with seasonal prey harvesting. (English) Zbl 1294.34050

Chen, Jing; Huang, Jicai; Ruan, Shigui; Wang, Jihua

SIAM J. Appl. Math. 73, No. 5, 1876-1905 (2013).

Subject of the paper is a predator-prey model whose non-monotonous functional response decays exponentially, and where prey harvesting takes place. First, the harvesting is assumed to have constant yield \(h\), and six possible equilibria in the closed positive quadrant are determined, three of them on the boundary. Depending on \(h\) and three additional parameters, the existence, types, and degeneracies of these equilibria are next investigated. From these results, and using a known statement on the generalized Liénard system, the authors then derive the different possibilities of bifurcation: saddle-node and (degenerate) Hopf bifurcation with the existence of up to two limit cycles. Moreover, a Bogdanov-Takens bifurcation is proved at one of the inner equilibria. In the last part of the paper, the authors add to the constant-yield harvesting h a time-periodic harvesting with small amplitude \(A\), and then prove the existence of an asymptotically stable periodic solution. Finally, under some non-resonant conditions, an asymptotically stable invariant torus is shown to bifurcate from a stable limit cycle that exists for \(A=0\). Numerical simulations illustrate the preceding results and also indicate the bifurcation of a stable invariant homoclinic torus from the homoclinic loop that exists when \(A=0\).

Reviewer: Josef Hainzl (Freiburg)

MSC:

34C60	Qualitative investigation and simulation of ordinary differential equation models
34C23	Bifurcation theory for ordinary differential equations
92D25	Population dynamics (general)
37G15	Bifurcations of limit cycles and periodic orbits in dynamical systems
37G20	Hyperbolic singular points with homoclinic trajectories in dynamical systems
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D20	Stability of solutions to ordinary differential equations

Keywords:

predator-prey model; seasonal harvesting; Bogdanov-Takens bifurcation; degenerate Hopf bifurcation; invariant torus; homoclinic torus