Abstract
Understanding complex behavioural patterns of organisms observed in nature can be facilitated using mathematical modelling. The conventional paradigm in animal behavior modelling consists of maximisation of some evolutionary fitness function. However, the definition of fitness of an organism or population is generally subjective, and using different criteria can lead us to contradictory model predictions regarding optimal behaviour. Moreover, structuring of natural populations in terms of individual size or developmental stage creates an extra challenge for theoretical modelling. Here we revisit and formalise the definition of evolutionary fitness to describe long-term selection of strategies in deterministic self-replicating systems for generic modelling settings which involve an arbitrary function space of inherited strategies. Then we show how optimal behavioural strategies can be obtained for different developmental stages in a generic von-Foerster stage-structured population model with an arbitrary mortality term. We implement our theoretical framework to explore patterns of optimal diel vertical migration (DVM) of two dominant zooplankton species in the north-eastern Black Sea. We parameterise the model using 7 years of empirical data from 2007-2014 and show that the observed DVM can be explained as the result of a trade-off between depth-dependent metabolic costs for grazers, anoxia zones, available food, and visual predation.
Highlights
Understanding complex behavioural patterns of organisms observed in nature can be facilitated using mathematical modelling
It is intuitively clear that optimal behaviour and/or life traits of an organism should gradually alter with maturation and progression through different developmental stages: a successful behavioral strategy for juveniles may not be effective for adults, which often experience a different environment
We argue that mathematical modelling based on our revisited concept of fitness and backed up by long-term empirical observation can provide us with a better understanding of zooplankton DVM7,13,27,29,30
Summary
Understanding complex behavioural patterns of organisms observed in nature can be facilitated using mathematical modelling. We show how optimal behavioural strategies can be obtained for different developmental stages in a generic von-Foerster stage-structured population model with an arbitrary mortality term. The complex behavioral responses and sophisticated life traits of organisms which are observed in the natural world are often considered to be outcomes of long-term evolutionary processes, and their quantitative description via mathematical modelling is usually challenging. A new mathematically straightforward approach for identifying the evolutionary fitness has been proposed[16,17,18] This approach considers long-term dynamics of competing subpopulations which are described by different inherited units (e.g. behavioural strategies, life traits, genotypes). We show that the optimal patterns of behaviour which are observed in the model can be derived by applying the variational principle of natural selection Using this principle we obtain equations providing the optimal trajectories of regular vertical migration of zooplankton. We argue that mathematical modelling based on our revisited concept of fitness and backed up by long-term empirical observation can provide us with a better understanding of zooplankton DVM7,13,27,29,30
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