| Abstract: |
| Oscillatory regimes typically arise from the presence of periodic orbits in dynamical systems. We present three new results for models of competing predators with Holling type II functional response.
First, we introduce a new mechanism that generates periodic orbits, which we call hybrid bifurcations: a classical bifurcation occurring at a bifurcation without parameters. We establish existence and stability criteria for the bifurcating periodic solutions and obtain stable periodic coexistence states far from extinction regimes, where all populations remain bounded away from zero.
Second, we prove global continuation and stability of families of periodic orbits with respect to a parameter using a computer-assisted framework. As a consequence, we solve a conjecture of Butler and Waltman (1981) concerning one-parameter families of periodic orbits connecting two extinction regimes.
Finally, we analyze a candidate Poincar\`e return map for these systems. Using computer-assisted techniques, we prove the existence of periodic orbits together with associated period-doubling bifurcations, indicating a transition to a chaotic regime. |
|