On population growth with catastrophes

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.

On decay-surge population models

“…Writing D + (R) for all distributions having support in [0, ∞), we may define the distribution δ t Π t,x by Notice that the measure Π t,x (dy) has support [x t (x) , ∞] with an atom at x t (x) with mass P (T 1 (x) > t) . Moreover, an analogous argument as the one used in the proof of Proposition 4 in [8] shows that Π t,x is absolutely continuous on (x t (x), ∞).…”

“…Similarly as in [8], considering the family of test functions u (y) = e λ (y) := e −λy , λ ≥ 0, we get, putting Π t,x (y) =…”

“…In this Section, we finally exhibit a natural relationship between Decay/Surge population models, as studied here, to the ones of growth/collapse models as developed in [2], [8], [9] and [10]. Let X t be a decay-surge process as defined by the triple (α, β, k) and consider the new process X t = 1/X t .…”

“…We are thus led to consider the piecewise deterministic Markov process (PDMP) with state-space [0, ∞) obeying: (8)…”

“…When simulating the embedded chain, a natural question is to relate its invariant measure to the one of the underlying continuous time process. We recall the following result from [8].…”