More degeneracy but fewer bifurcations in a predator-prey system having fully null linear part (Q2144356)

The authors study the generalized Lotka-Volterra (GLV) model with a completely vanishing linear part. They point out that in [\textit{J. Guckenheimer}, Lect. Notes Math. 898, 99--142 (1981; Zbl 0482.58006); \textit{S. Ruan} et al., J. Differ. Equations 249, No. 6, 1410--1435 (2010; Zbl 1205.34038)], the authors studied the vector fields with fully null degeneracy, for which the Poincaré normal form theory is not available. In particular, in [Zbl 1205.34038] the following Lotka-Volterra model has been considered: \[ \dot{x}=x\left(\alpha x+(\alpha-1)y-\alpha x^2-\alpha xy \right),\] \[\dot{y}=y\left((\kappa-\beta)x-\beta y\right),\] in the first quadrant \(\mathbb{R}_+^2:=\left\{(x,y):x\geqslant 0,y\geqslant0\right\}\) with positive parameter \(\alpha,\beta\) and \(\kappa\). Under the generic condition \((\alpha-1)\left\{2(\alpha\kappa-\kappa+\beta)+\beta(\kappa-\alpha-\beta)\right\}\neq 0\), the above system can be reduced into the GLV normal form \[\dot{x}=x\left(\alpha x+(\alpha-1)y+\tilde{a}_{20} x^2+O(|(x,y)|^3) \right),\] \[\dot{y}=y\left((\kappa-\beta)x-\beta y+O(|(x,y)|^3) \right).\] On the other hand, the nondegeneracy condition \[ (\alpha-1)(\kappa-\beta)(\kappa-\alpha\kappa-\beta)\left\{(\kappa-\alpha-\beta)^2+(1-\alpha-\beta)^2\right\}\tilde{a}_{20}\neq 0, \] implies that the above GLV normal form has degeneracy of codimension 2 at the origin within the GLV class. In the present paper the authors discuss the critical case \(\kappa-\beta=0\). It turns our that the normal form reads \[\dot{x}=x\left(\alpha x+(\alpha-1)y-\alpha x^2-\alpha xy \right),\qquad \dot{y}=y\left(-\beta y\right).\tag{\(*\)}\] The authors prove that the quadratic term of the original system cannot be eliminated in the GLV normal form, i.e., the quadratic term \((\kappa - \beta)xy\) has an essential effect to degeneracy. Thus, under the critical condition \(\kappa - \beta = 0\) and new nondegeneracy condition \(\alpha -1 \neq 0\), the authors obtain the following results: (a) System (\(*\)) has degeneracy of codimension 3 in the GLV class and the authors prove a versal unfolding of the system in the class; (b) For the unfolding system of (\(*\)) with three unfolding parameters, there are at most three equilibria that bifurcate from the origin, one of those equilibria lies in the interior of \(\mathbb{R}_+^2\) and the others lie on the boundary. Neither a closed orbit nor a heteroclinic loop exists; (c) There is no Hopf bifurcation and heteroclinic bifurcation in system (\(*\)), but two transcritical bifurcations at different equilibria may occur simultaneously, which is impossible in the codimension-2 case.