A parametrized stock-recruitment relationship derived from a slow-fast population dynamic model

Abstract

AcknowledgementsMy deepest and sincere thank you goes to Trond Steihaug, for his support and guidance throughout my research work. I appreciate that you were never tired of giving me advice for any problem, often in the form of some metaphor.I would like to thank Sam Subbey for his support and for many fruitful discussions. The atmosphere at the Institute of Informatics was delightful, and I would like to thank the administration for letting me stay at my working place until submission. Many of my colleagues have made the last years a great experience. To mention but a few, Joanna Bauer, Arne Klein and Marika Ivanova, I enjoyed your company.Thank you to my friends for listening to me talking about my research and for making sure that I did not forget life outside of academia. Thank you for proofreading parts of my work.Sindre Saether has been very supportive. Danke, dass du mir den Rücken frei gehalten hast. Ohne dich wäre der Kühlschrank leer und die Kleidung in der Wäsche geblieben.To my family: Takk til Saether-familien for at jeg får lov til å vaere en del av familien. Marianne, danke dafür, dass du immer mein Bestes willst. Manche Leute haben den Grundstein für etwas gelegt, das damals noch mehr oder weniger unabsehbar in der Zukunft lag. Mein Dank gebührt auch dir, Hannelore, denn du warst wichtig für mich. Den wichtigsten Stein aber hast du gelegt, Bodo. Ich hätte gerne mit dir über mein Thema geredet und ich wünschte, du wüsstest, wie dankbar ich dir bin.ii Acknowledgements AbstractThe parent-progeny (adult fish-juvenile) relationship is central to understanding the dynamics of fish populations. Management and harvest decisions are based on the assumption of a stock-recruitment function that relates the number of adults to their progeny. Multi-stage population dynamic models provide a modelling framework for understanding this relationship, since they describe the dynamics of fish in several life stages (such as adults, eggs, larvae, and juveniles). Biological processes at various life stages usually evolve at distinct time scales.This thesis contains three papers, which address challenges in modelling and parameter estimation for multiple time-scale dynamics of stage structured populations. A major question is, whether a multi-stage population dynamic model supports the assumption of a stock-recruitment function.In the first paper, we address the parent-progeny relationship admitted by slow-fast systems of differential equations that model the dynamics of a fish population with two stages. We introduce a slow-fast population dynamic model which replicates several well-known stock-recruitment functions.Traditionally, the dynamics of fish populations are described by difference equations. In the second paper, discrete time models for several life stages are formulated. We demonstrate that a multi-stage model may not admit a stock-recruitment function. Sufficient conditions for the validity of two hypotheses about the existence and structure of a parent-progeny function are established.Parameters in populat...

Mathematical Preconditions for Existence of the Stock-Recruitment Function

“…This specificity enables a more accurate representation of the dynamics at, and between each stage. While it is possible to categorize the adult population by length (Callahan et al 2019) or size (weight) (Meng et al 2013), we align with previous research on multi-stage SR relationships (Schaarschmidt et al 2018; Touzeau and Gouzé 1998), and focus on an age-structured adult population model.…”

“…Assumption T4 can be justified from the fisheries literature, with egg and larva stage duration of a few months, maturation after a few years, and life spans of several years (Petitgas et al 2013). Furthermore, faster evolution of prerecruits in comparison to adults is an assumption underlying other multi-stage models for SR (Schaarschmidt et al 2018; Touzeau and Gouzé 1998).…”

“…For instance, when parameters such as fecundity, reproductive rates, or predation rates vary with age, the traditional SR function may fail to emerge (Touzeau and Gouzé 1998). In contrast, in multi-stage models that account for recruitment as a function of an age-structured parental population, the resulting function becomes multivariate, diverging from the conventional SR function, which typically relies on a single variable (Schaarschmidt et al 2018). Thus, the quest of the paper is to determine necessary and sufficient conditions for existence of a (I) closed-form SR function, and (II) multivariate SR function.…”

“…In contrast to studies that rely on inherently variable and uncertain data to investigate SR relationships (Myers and Barrowman 1996; Gilbert 1997), our methodology consistently provides a mathematical representation of the parent-progeny relationship. Our approach involving research on multi-stage SR relationships is aligned with literature (Quinn and Deriso 1999; Schaarschmidt et al 2018; Touzeau and Gouzé 1998), but differ in the following way. In contrast to (Schaarschmidt et al 2018; Touzeau and Gouzé 1998), we consider all life history stages (eggs, larvae, juveniles and adults), rather than a broad classification of life stages into pre-recruits and recruits.…”

“…Our approach involving research on multi-stage SR relationships is aligned with literature (Quinn and Deriso 1999; Schaarschmidt et al 2018; Touzeau and Gouzé 1998), but differ in the following way. In contrast to (Schaarschmidt et al 2018; Touzeau and Gouzé 1998), we consider all life history stages (eggs, larvae, juveniles and adults), rather than a broad classification of life stages into pre-recruits and recruits. We consider a more comprehensive life cycle process than in Quinn and Deriso (1999).…”