Dining in danger: Resolving adaptive fish behavior increases realism of modeled ecosystem dynamics

Animals occupying higher trophic levels can have disproportionately large influence on ecosystem structure and functioning, owning to intricate behavioral responses to their environment, but the effects of behavioral adaptations on aquatic ecosystem dynamics are underrepresented, especially in model studies. Here, we explore how adaptive behavior of fish can affect the dynamics of aquatics ecosystems. We frame fish behavior in the context of the central trade-off between feeding and predation, calculating the optimal level of feeding determined by ambient food availability and predation risk. To explore whole-ecosystem consequences of fish behavior, we embed our behavioral model within the Water Ecosystems Tool (WET), a contemporary end-to-end aquatic ecosystem model. The principle of optimality provides a robust and mechanistic framework for representing animal behavior that is relevant for complex models, and can provide a stabilizing effect on model dynamics. The model predicts an emergent functional response similar to Holling type III, but with richer dynamics and a more rigorous theoretical foundation. We show how adaptive fish behavior works to stabilize food web dynamics compared to a control model with no optimal behavior, and how changing the strength of the underlying trade-off has profound effects on trophic control and food web structure. Furthermore, we demonstrate how including fish behavior allows for an overall more realistic response of the model system to environmental perturbation in the form of nutrient enhancement. We discuss the structuring effects of behavioral adaptations in real ecosystems, and how approaches like this one may benefit aquatic ecological modeling. Our study further highlights how a mechanistic approach based on concepts from theoretical ecology can be successfully implemented in complex operational models resulting in improved dynamics and descriptive power.

Copuling population dynamics and diel migration patterns

The diel vertical migration is one of the main drivers of population dynamics in the ocean. Population dynamical models of the ocean typically do not incorporate the behavioral aspects of the migration. We demonstrate a model with coupled population dynamics and behavior with the diel vertical migration emerging. We study the population dynamics and behavioral dynamics of a predator-prey system. We impose a cost of motion for both consumers and prey, and model each individual as following an Itô stochastic differential equation. We study the fixed-points of the ecosystem. Our modeling shows that as we increase the basal resource load, the strength of the diel vertical migration increases, as well as maximal velocity. In addition, a bimodal pattern emerges both for predators and consumers. The increase in the magnitude of the diel vertical migration causes a change in the allocation of copepod resources.

Population games with instantaneous behavior and the Rosenzweig-MacArthur model

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.

Solving multispecies population games in continuous space and time

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.

Coevolution of the reckless prey and the patient predator

The war of attrition in game theory is a model of a stand-off situation between two opponents where the winner is determined by its persistence. We model a stand-off between a predator and a prey when the prey is hiding and the predator is waiting for the prey to come out from its refuge, or when the two are locked in a situation of mutual threat of injury or even death. The stand-off is resolved when the predator gives up or when the prey tries to escape. Instead of using the asymmetric war of attrition, we embed the stand-off as an integral part of the predator-prey model of Rosenzweig and MacArthur derived from first principles. We apply this model to study the coevolution of the giving-up rates of the prey and the predator, using the adaptive dynamics approach. We find that the long term evolutionary process leads to three qualitatively different scenarios: the predator gives up immediately, while the prey never gives up; the predator never gives up, while the prey adopts any giving-up rate greater than or equal to a given positive threshold value; the predator goes extinct. We observe that some results are the same as for the asymmetric war of attrition, but others are quite different.

Running away or running to? Do prey make decisions solely based on the landscape of fear or do they also include stimuli from a landscape of safety?

Predator-prey interactions are a key part of ecosystem function, and non-consumptive effects fall under the landscape of fear theory. Under the landscape of fear, the antipredator responses of prey are based on the spatial and temporal distribution of predatory cues in the environment. However, the aversive stimuli (fear) are not the only stimuli prey can utilize when making behavioral decisions. Prey might also be using attractive stimuli that represent safety to guide decision making. Using a novel, orthogonal design, we were able to spatially separate aversive and attractive stimuli to determine whether prey are utilizing safety cues to navigate their environment. Crayfish Faxonius rusticus were placed in the center of a behavioral arena. Aversive stimuli of either predatory bass Micropterus salmoides cues or conspecific alarm cues increased along the x-axis of the behavioral arena. Safety cues (shelters) increased along the y-axis by decreasing the number of shelter openings in this direction. Crayfish were allowed two phases to explore the arena: one without the fearful stimuli and one with the stimuli. Linear mixed models were conducted to determine whether movement behaviors and habitat utilization were affected by the phase of the trial and the type of aversive stimuli. Crayfish responded more strongly to alarm cues than to fear cues, with only alarm cues significantly impacting habitat utilization. When responding to alarm cues, crayfish used safety cues as well as fear cues to relocate themselves within the arena. Based on these results, we argue that crayfish utilize a landscape of safety in conjunction with a landscape of fear when navigating their environment.

How the concavity of reproduction/survival trade-offs impacts the evolution of life history strategies

Previous works using different mathematical techniques, however, show that the concavity of the trade-off relationship can alter the expected life history strategies. Thus we developed a model and found that the concavity of the reproduction-survival curve can still have a large impact on life history strategies in an ecological model with Darwinian evolution.

Co-adaptive behavior of interacting populations in a habitat selection game significantly impacts ecosystem functions

Individuals of different interacting populations often adjust to prevailing conditions by changing their behavior simultaneously, with consequences for trophic relationships throughout the system. While we now have a good theoretical understanding of how individuals adjust their behavior, the population dynamical consequences of co-adaptive behaviors are rarely described. Further, mechanistic descriptions of ecosystem functions are based on population models that seldom take behavior into account. Here, we present a model that combines the population dynamics and adaptive behavior of organisms of two populations simultaneously. We explore how the Nash equilibrium of a system - i.e. the optimal behavior of its constituent organisms - can shape population dynamics, and conversely how population dynamics impact the Nash equilibrium of the system. We illustrate this for the case of diel vertical migration (DVM), the daily movement of marine organisms between food-depleted but safe dark depths and more risky nutrition-rich surface waters. DVM represents the archetypal example of populations choosing between a foraging arena (the upper sunlit ocean) and a refuge (the dark depths). We show that population sizes at equilibrium are significantly different if organisms can adapt their behavior, and that optimal DVM behaviors within the community vary significantly if population dynamics are considered. As a consequence, ecosystem function estimates such as trophic transfer efficiency and vertical carbon export differ greatly when fitness seeking behavior is included. Ignoring the role of behavior in multi-trophic population modeling can potentially lead to inaccurate predictions of population biomasses and ecosystem functions.

Evolution and Adaptation of Anti-predation Response of Prey in a Two-Patchy Environment

When perceiving a risk from predators, a prey may respond by reducing its reproduction and decreasing or increasing (depending on the species) its mobility. We formulate a patch model to investigate the aforementioned fear effect which is indirect, in contrast to the predation as a direct effect, of the predator on the prey population. We consider not only cost but also benefit of anti-predation response of the prey, and explore their trade-offs together as well as the impact of the fear effect mediated dispersals of the prey. In the case of constant response level, if there is no dispersal and for some given response functions, the model indicates the existence of an evolutionary stable strategy which is also a convergence stable strategy for the response level; and if there is dispersal, the analysis of the model shows that it will enhance the co-persistence of the prey on both patches. Considering the trait as another variable, we continue to study the evolution of anti-predation strategy for the model with dispersal, which leads to a three-dimensional system of ordinary differential equations. We perform some numerical simulations, which demonstrate global convergence to a positive equilibrium with the response level evolving towards a positive constant level, implying the existence of an optimal anti-predation response level.

Plant coexistence mediated by adaptive foraging preferences of exploiters or mutualists

Coexistence of plants depends on their competition for common resources and indirect interactions mediated by shared exploiters or mutualists. These interactions are driven either by changes in animal abundance (density-mediated interactions, e.g., apparent competition), or by changes in animal preferences for plants (behaviorally-mediated interactions). This article studies effects of behaviorally-mediated interactions on two plant population dynamics and animal preference dynamics when animal densities are fixed. Animals can be either adaptive exploiters or adaptive mutualists (e.g., herbivores or pollinators) that maximize their fitness. Analysis of the model shows that adaptive animal preferences for plants can lead to multiple outcomes of plant coexistence with different levels of specialization or generalism for the mediator animal species. In particular, exploiter generalism promotes plant coexistence even when inter-specific competition is too strong to make plant coexistence possible without exploiters, and mutualist specialization promotes plant coexistence at alternative stable states when plant inter-specific competition is weak. Introducing a new concept of generalized isoclines allows us to fully analyze the model with respect to the strength of competitive interactions between plants (weak or strong), and the type of interaction between plants and animals (exploitation or mutualism).

Inferring parameters of prey switching in a 1 predator-2 prey plankton system with a linear preference tradeoff

We construct two ordinary-differential-equation models of a predator feeding adaptively on two prey types, and we evaluate the models' ability to fit data on freshwater plankton. We model the predator's switch from one prey to the other in two different ways: (i) smooth switching using a hyperbolic tangent function; and (ii) by incorporating a parameter that changes abruptly across the switching boundary as a system variable that is coupled to the population dynamics. We conduct linear stability analyses, use approximate Bayesian computation (ABC) combined with a population Monte Carlo (PMC) method to fit model parameters, and compare model results quantitatively to data for ciliate predators and their two algal prey groups collected from Lake Constance on the German-Swiss-Austrian border. We show that the two models fit the data well when the smooth transition is steep, supporting the simplifying assumption of a discontinuous prey-switching behavior for this scenario. We thus conclude that prey switching is a possible mechanistic explanation for the observed ciliate-algae dynamics in Lake Constance in spring, but that these data cannot distinguish between the details of prey switching that are encoded in these different models.

L-shaped prey isocline in the Gause predator-prey experiments with a prey refuge

Predator and prey isoclines are estimated from data on yeast-protist population dynamics (Gause et al., 1936). Regression analysis shows that the prey isocline is best fitted by an L-shaped function that has a vertical and a horizontal part. The predator isocline is vertical. This shape of isoclines corresponds with the Lotka-Volterra and the Rosenzweig-MacArthur predator-prey models that assume a prey refuge. These results further support the idea that a prey refuge changes the prey isocline of predator-prey models from a horizontal to an L-shaped curve. Such a shape of the prey isocline effectively bounds amplitude of predator-prey oscillations, thus promotes species coexistence.

Costs of predator-induced phenotypic plasticity: a graphical model for predicting the contribution of nonconsumptive and consumptive effects of predators on prey

Defensive modifications in prey traits that reduce predation risk can also have negative effects on prey fitness. Such nonconsumptive effects (NCEs) of predators are common, often quite strong, and can even dominate the net effect of predators. We develop an intuitive graphical model to identify and explore the conditions promoting strong NCEs. The model illustrates two conditions necessary and sufficient for large NCEs: (1) trait change has a large cost, and (2) the benefit of reduced predation outweighs the costs, such as reduced growth rate. A corollary condition is that potential predation in the absence of trait change must be large. In fact, the sum total of the consumptive effects (CEs) and NCEs may be any value bounded by the magnitude of the predation rate in the absence of the trait change. The model further illustrates how, depending on the effect of increased trait change on resulting costs and benefits, any combination of strong and weak NCEs and CEs is possible. The model can also be used to examine how changes in environmental factors (e.g., refuge safety) or variation among predator-prey systems (e.g., different benefits of a prey trait change) affect NCEs. Results indicate that simple rules of thumb may not apply; factors that increase the cost of trait change or that increase the degree to which an animal changes a trait, can actually cause smaller (rather than larger) NCEs. We provide examples of how this graphical model can provide important insights for empirical studies from two natural systems. Implementation of this approach will improve our understanding of how and when NCEs are expected to dominate the total effect of predators. Further, application of the models will likely promote a better linkage between experimental and theoretical studies of NCEs, and foster synthesis across systems.

SIRT1 activates MAO-A in the brain to mediate anxiety and exploratory drive

SIRT1 is a NAD(+)-dependent deacetylase that governs a number of genetic programs to cope with changes in the nutritional status of cells and organisms. Behavioral responses to food abundance are important for the survival of higher animals. Here we used mice with increased or decreased brain SIRT1 to show that this sirtuin regulates anxiety and exploratory drive by activating transcription of the gene encoding the monoamine oxidase A (MAO-A) to reduce serotonin levels in the brain. Indeed, treating animals with MAO-A inhibitors or selective serotonin reuptake inhibitors (SSRIs) normalized anxiety differences between wild-type and mutant animals. SIRT1 deacetylates the brain-specific helix-loop-helix transcription factor NHLH2 on lysine 49 to increase its activation of the MAO-A promoter. Both common and rare variations in the SIRT1 gene were shown to be associated with risk of anxiety in human population samples. Together these data indicate that SIRT1 mediates levels of anxiety, and this regulation may be adaptive in a changing environment of food availability.

On the Gause predator-prey model with a refuge: a fresh look at the history

This article re-analyses a prey-predator model with a refuge introduced by one of the founders of population ecology Gause and his co-workers to explain discrepancies between their observations and predictions of the Lotka-Volterra prey-predator model. They replaced the linear functional response used by Lotka and Volterra by a saturating functional response with a discontinuity at a critical prey density. At concentrations below this critical density prey were effectively in a refuge while at a higher densities they were available to predators. Thus, their functional response was of the Holling type III. They analyzed this model and predicted existence of a limit cycle in predator-prey dynamics. In this article I show that their model is ill posed, because trajectories are not well defined. Using the Filippov method, I define and analyze solutions of the Gause model. I show that depending on parameter values, there are three possibilities: (1) trajectories converge to a limit cycle, as predicted by Gause, (2) trajectories converge to an equilibrium, or (3) the prey population escapes predator control and grows to infinity.

Comparing the qualitatively different effects rapidly evolving and rapidly induced defences have on predator-prey interactions

Interspecific interactions depend not only on the population densities of the interacting species, but on their phenotypes as well. Phenotypic variation can be plastic or heritable and both mechanisms can drive phenotypic change at rates comparable to or faster than those of ecological dynamics (e.g. changes in population abundances or spatial distributions). In this study, we compare the effects rapidly induced and rapidly evolving defences have on community dynamics by considering the fast phenotypic change limit using fast-slow systems theory. Our approach allows us to study phenotypically plastic and evolving systems with one overarching theory, thus capturing the effects rapidly induced defences have on ecological dynamics and how those effects differ from the effects of evolving defences. Our results show that rapidly induced defences tend to stabilize community dynamics and that some behaviours observed in rapidly evolving systems cannot be produced by phenotypic plasticity.

Adaptive changes in prey vulnerability shape the response of predator populations to mortality

Simple models are used to explore how adaptive changes in prey vulnerability alter the population response of their predator to increased mortality. If the mortality is an imposed harvest, the change in prey vulnerability also influences the relationship between harvest effort and yield of the predator. The models assume that different prey phenotypes share a single resource, but have different vulnerabilities to the predator. Decreased vulnerability is assumed to decrease resource consumption rate. Adaptive change may occur by phenotypic changes in the traits of a single species or by shifts in the abundances of a pair of coexisting species or morphs. The response of the predator population is influenced by the shape of the predator's functional response, the shape of resource density dependence, and the shape of the tradeoff between vulnerability and food intake in the prey. Given a linear predator functional response, adaptive prey defense tends to produce a decelerating decline in predator population size with increased mortality. Prey defense may also greatly increase the range of mortality rates that allow predator persistence. If the predator has a type-2 response with a significant handling time, adaptive prey defense may have a greater variety of effects on the predator's response to mortality, sometimes producing alternative attractors, population cycles, or increased mean predator density. Situations in which there is disruptive selection on prey defense often imply a bimodal change in yield as a function of harvesting effort, with a minimum at intermediate effort. These results argue against using single-species models of density dependent growth to manage predatory species, and illustrate the importance of incorporating anti-predator behavior into models in applied population ecology.

Lotka-Volterra systems in environments with randomly disordered temporal periodicity

A generalized Lotka-Volterra model for a pair of interacting populations of predators and prey is studied. The model accounts for the prey's interspecies competition and therefore is asymptotically stable, whereas its oscillatory behavior is induced by temporal variations in environmental conditions simulated by those in the prey's reproduction rate. Two models of the variations are considered, each of them combining randomness with "hidden" periodicity. The stationary joint probability density function (PDF) of the number of predators and prey is calculated numerically by the path integration (PI) method based on the use of characteristic functions and the fast Fourier transform. The numerical results match those for the asymptotic case of white-noise variations for which an analytical solution is available. Several examples are studied, with calculations of important characteristics of oscillations, for example the expected rate of up-crossings given the level of the predator number. The calculated PDFs may be of predominantly random (unimodal) or predominantly periodic nature (bimodal). Thus, the PI method has been demonstrated to be a powerful tool for studies of the dynamics of predator-prey pairs. The method captures the random oscillations as observed in nature, taking into account potential periodicity in the environmental conditions.