Two-parameter bifurcations in LPA model (Q2409175)

The authors study a structured model of difference equations which describes the dynamics of a flour beetle (Tribolium) population which consists of 4 stages: eggs, larvae, pupae and adults, assuming that a certain level of cannibalism occurs between stages. To simplify the computations and due to the biological specifics of the system under consideration, the analysis is performed on the lower-dimensional LPA (larvae-pupae-adults) subsystem \[ \begin{aligned} L(t+1)&=bA(t)e^{-c_{EL}L(t)-c_{EA}A(t)}\\ P(t+1)&=(1-\mu_L)L(t)\\ A(t+1)&=P(t)e^{-c_{PA}A(t)}+(1-\mu _A)A(t). \end{aligned} \] In the above, \(b\) represents a natality rate, while \(\mu_L\) and \(\mu _A\) denote mortality rates and \(c_{EL}\), \(c_{EA}\), \(c_{PA}\) describe cannibalism rates. The paper focuses on two-parameters bifurcations which occur in the above model, \(c_{EL}\) and \(b\) being considered as parameters. The analysis is done primarily from a numerical viewpoint, although each numerical finding is complemented by a thorough and careful interpretation of the associated bifurcation diagram, interspersed with details regarding the appropriate theoretical background. In this regard, the paper is largely self-contained, an entire section being dedicated to summarizing the generic local two-parameter bifurcations of fixed points. The authors concentrate on the case in which \(R_0\), the basic reproductive number, is larger than 1 and determine the existence of two types of hysteresis: fixed point-loop and loop-loop. The connection between Chenciner bifurcations and strong 1:2 resonance is presented together with the possible consequences of both upon the dynamics of the Tribolium population (complicated dynamics, with possible coexistence of different types of invariant sets). Fixed points are observed to be easily destabilized, leading to weak or strong destabilization. The latter, caused by the passing of the LPC (the fold bifurcation of the invariant loop) possibly lead to the extinction of species.