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.

Figure 1

(A,B) Typical pattern of DVM of two dominant zooplankton herbivores Calanus euxinus and Pseudocalanus elongatus observed in the north-eastern Black Sea in summer. The samples were collected on 21/06/2011. For each developmental stage the centers of vertical abundance distribution is shown, see the main text and SM2 for detail. (C,D) Annual variation of the depths of DVM for migrating females (denoted by filled/open squares for night/day depths) and non-migrating stages CI-CIII (denoted by semi-filled circles).

Figure 2

(A) Seasonal variation of the depths of the upper and lower unfavorable zones for C. euxinus and P. elongatus observed in the north-eastern Black Sea. The upper zone is dictated by temperatures higher than 12 °C, whereas the boundary of the lower zone is given by ${\sigma }_{\theta }$ curve achieving 15.7 (see SM3). Dependence of the upper (B) and the lower (C) depths of DVM (females) on the depths of the unfavorable zones corresponding to panel (A). Triangles and circles denote, respectively, C. euxinus and P. elongatus. The solid and the dashed lines fit, respectively, the data for P. elongatus and C. euxinus. (D) The average (across all observations) vertical profile of chlorophyll a. The curve fitting to the data is discussed in text.

Figure 3

(A) The abiotic and biotic components across the water column which constitute the environment for zooplankton grazers in summer ($h_d=140$m, $h_u=20$m). The scaling in the vertical direction for each curve can be obtained considering the maximal values provided in Table 1. (B) Typical pattern of optimal DVM in the model for C. euxinus in the environment shown in panel (A). The other parameters are given in Table 1. (C) Optimal DVM of C. euxinus constructed for the hypothetical modelling scenario with a depth-independent basal metabolic cost. In (B,C) blue and red lines denote, respectively, females (stage IV) and juveniles (CVI-CV).

Figure 4

Connection between the upper and lower unfavorable zones (h_d and h_u) in the water column and the upper and the lower depths ($H_{A0},H_{A1}$) of females in the model obtained for C. euxinus. The impact of variation of h_u (for a constant $h_d=140$m) on H_A0 is shown via empty squares (curve 1). The impact of h_d ($h_u=20$m) on H_A1 is denoted by circles 2 (constructed for spatially variable metabolic costs) or by triangles 2′ (constructed for spatially constant metabolic costs). The other parameters are taken from Table 1.

Figure 5

(A) Influence of predation pressure on the optimal DVM of females (CVI) of C. euxinus in the model. Trajectories (1–5) correspond to ${\gamma }_A=0.44;0.60;0.80;1.0;1.2$ 1/day, respectively. (B) Influence of the available food on the optimal DVM of females (CVI) of C. euxinus in the model. Trajectories (1–5) correspond to $P_0=36;30;25;20;16$ μg C/l, respectively. Here $h_d=140$m, $h_u=20$m, the other parameters are as in Table 1.