Modelling optimal behavioural strategies in structured populations using a novel theoretical framework

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.

Mathematical modelling evolutionarily stable behavior of zooplankton with state constraints

“…2 ) + 1), where P 0 represents the maximal phytoplankton density [2], S x = tanh(σx+ C 2 )+1, G = cosh ξ x+ C…”

“…This approach was later developed in the works [6,27,28,32], where algorithms for deriving the fitness function depending on the type of the system of competing communities were proposed.This approach would be applied to construct the fitness function in the zooplankton community [2,[29][30][31]. In the simplest case, the fitness function consists of the energy gain recieved with food, energy expenditures for vertical movements and losses as a result of predation, unfavorable temperature conditions, and hydrogen sulfide concentration.…”

“…In Figs. 10, 11, the calculated trajectory of zooplankton movement is superimposed on the graphs of the revealed migratory vertical movement of zooplankton in the northeastern part of the Black Sea [2]. It can be seen that real and calculated trajectory almost coincide.…”

“…Information technology has long been used in biology (see [1] for a short review). However, they recently began to be used to identify evolutionarily stable hereditary characteristics of living organisms [2][3][4][5].…”

Exploring Evolutionary Fitness in Biological Systems Using Machine Learning Methods

“…It was previously shown, using both empirical data and mathematical modeling, that the timing and amplitude of DVM of herbivorous zooplankton are mainly determined by the following environmental factors [ 22 , 42 ]: the density of food (phytoplankton) at the depth x of the column, the distribution of visual predators consuming zooplankton (e.g., fish), and their daily activity other factors resulting to zooplankton mortality such as the unfavorable temperature or/and hydrogen sulfide concentration [ 10 , 16 ]. All of these factors can be considered as mathematical functions of the vertical coordinate x and the time of day t (the time of day is rescaled, so corresponds to 24 h).…”