Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting. (English) Zbl 0917.34029

Xiao, Dongmei; Ruan, Shigui

Ruan, Shigui (ed.) et al., Differential equations with applications to biology. Proceedings of the international conference, Halifax, Canada, July 25–29, 1997. Providence, RI: American Mathematical Society. Fields Inst. Commun. 21, 493-506 (1999).

Consider the differential system \[ dx/dt=rx\left( 1-{x\over k}\right) -{yx\over a+x},\qquad dy/dt=y \left(-d+{x \over a+x}\right)-h,\tag{*} \] describing a predator-prey system with constant rate harvesting. \(k,d,r,a\) and \(h\) are positive parameters. The paper contains a bifurcation analysis of (*). The existence of a Bogdanov-Takens bifurcation is established under some conditions.
For the entire collection see [Zbl 0903.00038].

Reviewer: Klaus R.Schneider (Berlin)

MSC:

34C23	Bifurcation theory for ordinary differential equations
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37G99	Local and nonlocal bifurcation theory for dynamical systems

Keywords:

bifurcation; Bogdanov-Takens singularity