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On decay–surge population models

Part of: Markov processes

Published online by Cambridge University Press:  08 November 2022

Branda Goncalves ,
Thierry Huillet  and
Eva Löcherbach Open the ORCID record for Eva Löcherbach [Opens in a new window]
Branda Goncalves*
Affiliation:
CY Cergy Paris Université
Thierry Huillet*
Affiliation:
CY Cergy Paris Université
Eva Löcherbach*
Affiliation:
Université Paris 1 Panthéon-Sorbonne
*
*Postal address: LPTM, Laboratoire de Physique Théorique et Modélisation, CNRS UMR-8089, 2 avenue Adolphe-Chauvin, 95302 Cergy-Pontoise, France.
*Postal address: LPTM, Laboratoire de Physique Théorique et Modélisation, CNRS UMR-8089, 2 avenue Adolphe-Chauvin, 95302 Cergy-Pontoise, France.
****Postal address: SAMM, Statistique, Analyse et Modélisation Multidisciplinaire, EA 4543 et FR FP2M 2036 CNRS, 90 rue de Tolbiac, 75013 Paris, France. Email address: eva.locherbach@univ-paris1.fr
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Abstract

We consider continuous space–time decay–surge population models, which are semi-stochastic processes for which deterministically declining populations, bound to fade away, are reinvigorated at random times by bursts or surges of random sizes. In a particular separable framework (in a sense made precise below) we provide explicit formulae for the scale (or harmonic) function and the speed measure of the process. The behavior of the scale function at infinity allows us to formulate conditions under which such processes either explode or are transient at infinity, or Harris recurrent. A description of the structures of both the discrete-time embedded chain and extreme record chain of such continuous-time processes is supplied.

Keywords

Declining population modelssurge processespiecewise deterministic Markov processesHawkes processesscale functionHarris recurrence

MSC classification

Primary: 60J75: Jump processes
Secondary: 92D25: Population dynamics (general)
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 2 , June 2023 , pp. 444 - 472
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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