Multiple time–scale dynamics of stage structured populations and derivative–free optimization
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The 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 population dynamic models can be estimated by minimizing a function of the solution of the ordinary differential equations and available data. Efficient and accurate methods for the solution of differential equations usually evaluate conditional statements. In this case, the objective function may be noisy, instead of continuously differentiable. Furthermore, an algorithm which is used to evaluate the objective function may unexpectedly fail to return a (plausible) value. Then, the optimization problem includes constraints which are only implicitly stated and hidden from the problem formulation. We demonstrate that derivative-free optimization methods find sufficiently accurate solutions for the challenging optimization problems. In the third paper, we compare the performances of several derivative-free methods for a set of optimization problems. We find that a derivative-free trust-region method is most robust to the choice of the initial iterate, but is in general outperformed by direct search methods. Additional numerical simulations in the thesis reveal that direct search methods which approximate a gradient or Hessian find the most accurate solutions. We observe that the optimization problems considered in this thesis are more challenging than a set of noisy benchmark problems. The thesis includes scientific contributions in addition to the results from the three papers.
Består avPaper A: Schaarschmidt, U., Steihaug, T., Subbey, S. A parametrized stock-recruitment relationship derived from a slow-fast population dynamic model. Math. Comput. Simul. 145 (2018), 171–185. The article is available in the main thesis. The article is also available at: https://doi.org/10.1016/j.matcom.2017.10.008
Paper B: Schaarschmidt, U., Subbey, S., Nash, R.D.M., and Frank, A.S.J. Emergent properties of a multi-stage population dynamic model. The article is not available in BORA.
Paper C: Schaarschmidt, U., Steihaug, T., Subbey, S. Derivative-free optimization for population dynamic models. In Model. Comput. & Optim. in Inf. Syst. & Manage. Sci. (2015), Le Thi, H.A., Pham Dinh, T., and Nguyen, N.T., Eds., vol. 359 of Advances in Intelligent Systems and Computing, Springer, Cham, 391–402. The article is not available in BORA due to publisher restrictions. The published version is available at: https://doi.org/10.1007/978-3-319-18161-5_33