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We present a theoretical framework, based on differential mean field games, for expressing diel vertical migration in the ocean as a game with a continuum of players. In such a game, each agent partially controls its own state by adjusting its vertical velocity but the vertical position in a water column is also subject to random fluctuations. A representative player has to make decisions based on aggregated information about the states of the other players. For this vertical differential game, we derive a mean field system of partial differential equations for finding a Nash equilibrium for the whole population. It turns out that finding Nash equilibria in the game is equivalent to solving a PDE-constrained optimization problem. We detail this equivalence when the expected fitness of the representative player can be approximated with a constant and solve both formulations numerically. We illustrate the results on simple numerical examples and construct several test cases to compare the two analytical approaches., (© 2023. The Author(s).)

How to determine the spatial distribution and population dynamics of animals are some of the key questions in ecology. These two have been coupled before, but there is no general method for determining spatial distributions based on instantanous behavior coupled with population dynamics. We propose modeling interacting populations with instantaneous habitat choice through mean-field games. By using the framework of variational inequalities, we are able to determine existence and uniqueness for habitat distributions of interacting populations, in both continuous and discrete habitats. With some additional restrictions, we are also able to show existence and uniqueness of fixed-points of the population dynamics along with spatial distributions. We illustrate our theoretical results by studying a Rosenzweig-MacArthur model where predators and consumers inhabit a continuous habitat. This study is conducted both theoretically and numerically. Analyzing the emergent dynamics is possible as viewing the system from the vantage point of variational inequalities allows for applying efficient numerical methods. The generality of our theoretical approach opens up for studying complex ecosystems, e.g. the impact of enrichment on spatial distributions in marine ecosystems., (© 2022. The Author(s).)

Game theory has emerged as an important tool to understand interacting populations in the last 50 years. Game theory has been applied to study population dynamics with optimal behavior in simple ecosystem models, but existing methods are generally not applicable to complex systems. In order to use game-theory for population dynamics in heterogeneous habitats, habitats are usually split into patches and game-theoretic methods are used to find optimal patch distributions at every instant. However, populations in the real world interact in continuous space, and the assumption of decisions based on perfect information is a large simplification. Here, we develop a method to study population dynamics for interacting populations, distributed optimally in continuous space. A continuous setting allows us to model bounded rationality, and its impact on population dynamics. This is made possible by our numerical advances in solving multiplayer games in continuous space. Our approach hinges on reformulating the instantaneous game, applying an advanced discretization method and modern optimization software to solve it. We apply the method to an idealized case involving the population dynamics and vertical distribution of forage fish preying on copepods. Incorporating continuous space and time, we can model the seasonal variation in the migration, separating the effects of light and population numbers. We arrive at qualitative agreement with empirical findings. Including bounded rationality gives rise to spatial distributions corresponding to reality, while the population dynamics for bounded rationality and complete rationality are equivalent. Our approach is general, and can easily be used for complex ecosystems., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2022 The Authors. Published by Elsevier Inc. All rights reserved.)

Toxic phytoplankton blooms have increased in many waterbodies worldwide with well-known negative impacts on human health, fisheries and ecosystems. However, why and how phytoplankton evolved toxin production is still a puzzling question, given that the producer that pays the costs often shares the benefit with other competing algae and thus provides toxins as a 'public good' (e.g. damaging a common competitor or predator). Furthermore, blooming phytoplankton species often show a high intraspecific variation in toxicity and we lack an understanding of what drives the dynamics of coexisting toxic and non-toxic genotypes. Here, by using an individual-based two-dimensional model, we show that small-scale patchiness of phytoplankton strains caused by demography can explain toxin evolution in phytoplankton with low motility and the maintenance of genetic diversity within their blooms. This patchiness vanishes for phytoplankton with high diffusive motility, suggesting different evolutionary pathways for different phytoplankton groups. In conclusion, our study reveals that small-scale spatial heterogeneity, generated by cell division and counteracted by diffusive cell motility and turbulence, can crucially affect toxin evolution and eco-evolutionary dynamics in toxic phytoplankton species. This contributes to a better understanding of conditions favouring toxin production and the evolution of public goods in asexually reproducing organisms in general.

Size-spectrum models are a recent class of models describing the dynamics of a whole community based on a description of individual organisms. The models are motivated by marine ecosystems where they cover the size range from multicellular plankton to the largest fish. We propose to extend the size-spectrum model with spatial components. The spatial dynamics is governed by a random motion and a directed movement in the direction of increased fitness, which we call 'fitness-taxis'. We use the model to explore whether spatial irregularities of marine communities can occur due to the internal dynamics of predator-prey interactions and spatial movements. This corresponds to a pattern-formation analysis generalized to an entire ecosystem but is not limited to one prey and one predator population. The analyses take the form of Fourier analysis and numerical experiments. Results show that diffusion always stabilizes the equilibrium but fitness-taxis destabilizes it, leading to non-stationary spatially inhomogeneous population densities, which are travelling in size. However, there is a strong asymmetry between fitness-induced destabilizing effects and diffusion-induced stabilizing effects with the latter dominating over the former. These findings reveal that fitness taxis acts as a possible mechanism behind pattern formations in ecosystems with high diversity of organism sizes, which can drive the emergence of spatial heterogeneity even in a spatially homogeneous environment., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2021 Elsevier Ltd. All rights reserved.)

Diel vertical migration (DVM), the daily movement of organisms through oceanic water columns, is mainly driven by spatio-temporal variations in the light affecting the intensity of predator-prey interactions. Migration patterns of an organism are intrinsically linked to the distribution of its conspecifics, its prey and its predators, each with their own fitness-seeking imperatives. We present a mechanistic, trait-based model of DVM for the different components of a pelagic community. Specifically, we consider size, sensory mode and feeding mode as key traits, representing a community of copepods that prey on each other and are, in turn, preyed upon by fish. Using game-theoretic principles, we explore the optimal distribution of the main groups of a planktonic pelagic food web simultaneously. Within one single framework, our model reproduces a whole suite of observed patterns, such as size-dependent DVM patterns of copepods and reverse migrations. These patterns can only be reproduced when different trophic levels are considered at the same time. This study facilitates a quantitative understanding of the drivers of DVM, and is an important step towards mechanistically underpinned predictions of DVM patterns and biologically mediated carbon export.

Aquatic positional telemetry offers vast opportunities to study in vivo behaviour of wild animals, but there is room for improvement in the data quality provided by current procedures for estimating positions. Here we present a novel positioning method called YAPS (Yet Another Positioning Solver), involving Maximum Likelihood analysis of a state-space model applied directly to time of arrival (TOA) data in combination with a movement model. YAPS avoids the sequential positioning-filtering-approach applied in alternative tools by using all available data in a single model, and offers better accuracy and error control. Feasibility and performance of YAPS was rigorously tested in a simulation study and by applying YAPS to data from an acoustic transmitter towed in a receiver array. Performance was compared to an alternative positioning model and proprietary software. The simulation study and field test revealed that YAPS performance was better and more consistent than alternatives. We conclude that YAPS outperformed the compared alternative methods, and that YAPS constitute a vast improvement to currently available positioning software in acoustic telemetry. Additionally, in contrast to vendor-supplied solutions, YAPS is transparent, flexible and can easily be adapted and extended for further improvements or to meet study specific requirements such as three-dimensional positioning.

Bigeye tuna are known for remarkable daytime vertical migrations between deep water, where food is abundant but the water is cold, and the surface, where water is warm but food is relatively scarce. Here we investigate if these dive patterns can be explained by dynamic optimal foraging theory, where the tuna maximizes its energy harvest rate. We assume that foraging efficiency increases with body temperature, so that the vertical migrations are thermoregulatory. The tuna's state is characterized by its mean body temperature and depth, and we solve the optimization problem numerically using dynamic programming. With little calibration of model parameters, our results are consistent with observed data on vertical movement: we find that small tuna should display constant-depth strategies while large tuna should display vertical migrations. The analysis supports the hypothesis that the tuna behaves such as to maximize its energy gains. The model therefore provides insight into the processes underlying observed behavioral patterns and allows generating predictions of foraging behavior in unobserved environments., (© 2016 by the Ecological Society of America.)

We consider organisms which use a renewal strategy such as run-tumble when moving in space, for example to perform chemotaxis in chemical gradients. We derive a diffusion approximation for the motion, applying a central limit theorem due to Anscombe for renewal-reward processes; this theorem has not previously been applied in this context. Our results extend previous work, which has established the mean drift but not the diffusivity. For a classical model of tumble rates applied to chemotaxis, we find that the resulting chemotactic drift saturates to the swimming velocity of the organism when the chemical gradients grow increasingly steep. The dispersal becomes anisotropic in steep gradients, with larger dispersal across the gradient than along the gradient. In contrast to one-dimensional settings, strong bias increases dispersal. We next include Brownian rotation in the model and find that, in limit of high chemotactic sensitivity, the chemotactic drift is 64% of the swimming velocity, independent of the magnitude of the Brownian rotation. We finally derive characteristic timescales of the motion that can be used to assess whether the diffusion limit is justified in a given situation. The proposed technique for obtaining diffusion approximations is conceptually and computationally simple, and applicable also when statistics of the motion is obtained empirically or through Monte Carlo simulation of the motion.

An initial modeling approach was applied to analyze how a single, nonmotile, free-living, heterotrophic bacterial cell may optimize the deployment of its extracellular enzymes. Free-living cells live in a dilute and complex substrate field, and to gain enough substrate, their extracellular enzymes must be utilized efficiently. The model revealed that surface-attached and free enzymes generate unique enzyme and substrate fields, and each deployment strategy has distinctive advantages. For a solitary cell, surface-attached enzymes are suggested to be the most cost-efficient strategy. This strategy entails potential substrates being reduced to very low concentrations. Free enzymes, on the other hand, generate a radically different substrate field, which suggests significant benefits for the strategy if free cells engage in social foraging or experience high substrate concentrations. Swimming has a slight positive effect for the attached-enzyme strategy, while the effect is negative for the free-enzyme strategy. The results of this study suggest that specific dissolved organic compounds in the ocean likely persist below a threshold concentration impervious to biological utilization. This could help explain the persistence and apparent refractory state of oceanic dissolved organic matter (DOM). Microbial extracellular enzyme strategies, therefore, have important implications for larger-scale processes, such as shaping the role of DOM in ocean carbon sequestration., (Copyright © 2015, American Society for Microbiology. All Rights Reserved.)

The optimal allocation of resources to repair vs. reproduction in an organism may depend on the magnitude and pattern of the external mortality it is experiencing, which, in turn, may depend on its feeding and mate-finding behavior. Thus, the fundamental activities of an organism, i.e., to feed, to survive, and to reproduce, are interrelated through trade-offs. Here, we use small planktonic copepods to examine how adult longevity and ageing patterns in a protected laboratory environment relate to the feeding mode (active searching vs. passive ambush feeding), mate-finding behavior, and spawning mode of the species. We show that average adult longevity varies between species by an order of magnitude and is independent of body size. Ambush feeders that carry their eggs have longer average life spans and experience higher mortality later in life relative to active feeders that broadcast their eggs. Males generally have shorter life spans and experience higher mortality earlier in life than females, and this difference may be accentuated in species where dangerous mate-finding is male biased. We finally show a trade-off between longevity and fecundity, with ambush feeders producing eggs at a rate five to 10 times lower than the active feeders, consistent with predictions from optimal resource allocation theory.

Spatiotemporal patterns, indicating the spatiotemporal variability of individual abundance, are a pronounced scenario in ecological interactions. Most of the existing models for spatiotemporal patterns treat species as homogeneous groups of individuals with average characteristics by ignoring intraspecific physiological variations at the individual level. Here we explore the impacts of size variation within species resulting from individual ontogeny, on the emergence of spatiotemporal patterns in a fully size-structured population model. We found that size dependency of animal's diffusivity greatly promotes the formation of spatiotemporal patterns, by creating regular spatiotemporal patterns out of temporal chaos. We also found that size-dependent diffusion can substitute large-amplitude base harmonics with spatiotemporal patterns with lower amplitude oscillations but with enriched harmonics. Finally, we found that the single-generation cycle is more likely to drive spatiotemporal patterns compared to predator-prey cycles, meaning that the mechanism of Hopf bifurcation might be more common than hitherto appreciated since the former cycle is more widespread than the latter in case of interacting populations. Due to the ubiquity of individual ontogeny in natural ecosystems we conclude that diffusion variability within populations is a significant driving force for the emergence of spatiotemporal patterns. Our results offer a perspective on self-organized phenomena, and pave a way to understand such phenomena in systems organized as complex ecological networks.

The comb jelly Mertensia ovum, widely distributed in Arctic regions, has recently been discovered in the northern Baltic Sea. We show that M. ovum also exists in the central Baltic but that the population consists solely of small-sized larvae (less than 1.6 mm). Despite the absence of adults, eggs were abundant. Experiments revealed that the larvae were reproductively active. Egg production and anticipated mortality rates suggest a self-sustaining population. This is the first account of a ctenophore population entirely recruiting through larval reproduction (paedogenesis). We hypothesize that early reproduction is favoured over growth to compensate for high predation pressure.

In this paper we consider the advective/diffusive transport of a solute near a hovering zooplankter. We approximate the fluid flow with that of a Stokeslet, corresponding to the plankter exerting a point force on the water, and assume that the plankter acts as a point source for the transported solute, located at the same point as the force. We find an analytical expression in closed form for the steady-state concentration of the solute. We also discuss the situation where the plankter performs Brownian motion. Finally we apply the results to the courtship of the marine copepod Pseudocalanus elongatus, where the male performs a mating dance below the hovering female. For this situation, our model supports the hypothesis that the mating dance is guided by the plume of a signalling pheromone.

This paper derives optimal life histories for fishes or other animals in relation to the size spectrum of the ecological community in which they are both predators and prey. Assuming log-linear size-spectra and well known scaling laws for feeding and mortality, we first construct the energetics of the individual. From these we find, using dynamic programming, the optimal allocation of energy between growth and reproduction as well as the trade-off between offspring size and numbers. Optimal strategies were found to be strongly dependent on size spectrum slope. For steep size spectra (numbers declining rapidly with size), determinate growth was optimal and allocation to somatic growth increased rapidly with increasing slope. However, restricting reproduction to a fixed mating season changed optimal allocations to give indeterminate growth approximating a von Bertalanffy trajectory. The optimal offspring size was as small as possible given other restrictions such as newborn starvation mortality. For shallow size spectra, finite optimal maturity size required a decline in fitness for large size or age. All the results are compared with observed size spectra of fish communities to show their consistency and relevance.