Local explosions and extinction of continuous state branching processes with logistic competition

We study by duality methods the extinction and explosion times of continuousstate branching processes with logistic competition (LCSBPs) and identify the local time at ∞ of the process when it is instantaneously reflected at ∞. The main idea is to introduce a certain "bidual" process V of the LCSBP Z. The latter is the Siegmund dual process of the process U , that was introduced in [Fou19] as the Laplace dual of Z. By using both dualities, we shall relate local explosions and the extinction of Z to local extinctions and the explosion of the process V. The process V being a one-dimensional diffusion on [0, ∞], many results on diffusions can be used and transfered to Z. A concise study of Siegmund duality for regular one-dimensional diffusions is also provided