No abstract

Deterministic population growth models can exhibit a large variety of flows, ranging from algebraic, exponential to hyper-exponential (with finite time explosion). They describe the growth for the size (or mass) of some population as time goes by. Variants of such models are introduced allowing logarithmic, exp-algebraic or even doubly exponential growth. The possibility of immigration is also raised. An important feature of such growth models is to decide whether the ground state 0 is reflecting or absorbing and also whether state ∞ is accessible or inaccessible.We then study a semi-stochastic catastrophe version of such models (also known as Piecewise-Deterministic-Markov Processes, in short, PDMP). Here, at some jump times, possibly governed by state-dependent rates, the size of the population shrinks by a random amount of its current size, an event possibly leading to instantaneous local (or total) extinction. A special separable shrinkage transition kernel is investigated in more detail, including the case of total disasters. Between the jump times, the new process grows, following the deterministic dynamics started at the newly reached state after each jump. We discuss the conditions under which such processes are either transient or recurrent (positive or null), the scale function playing a key role in this respect, together with the speed measure cancelling the Kolmogorov forward operator. The scale function is also used to compute, when relevant, the law of the height of excursions. The question of the finiteness of the time to extinction is investigated together (when finite), with the evaluation of the mean time to extinction, either local or global. Some information on the embedded chain to the PDMP is also required when dealing with the classification of states 0 and ∞ that we exhibit.

Catastrophe Markov chain population models have received a lot of attention in the recent past. We herewith consider two special cases of such models involving total disasters, both in discrete and in continuous-time. Depending on the parameters range, the two models can show up a recurrence/transience transition and, in the critical case, a positive/null recurrence transition. The collapse transition probabilities are chosen in such a way that the models are exactly solvable and, in case of positive recurrence, intimately related to the extended Sibuya and Pareto-Zipf distributions whose divisibility and self-decomposability properties are shown relevant. The study includes: existence and shape of the invariant measure, time-reversal, return time to the origin, contact probability at the origin, extinction probability, height and length of the excursions, a renewal approach to the fraction of time spent in the catastrophic state, scale function, first time to collapse and first-passage times, divisibility properties.