The role of behavioral dynamics in determining the patch distributions of interacting species

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.

Food web functional responses

This article reviews the nature of functional responses that have commonly been used to represent feeding relationships in the ecological literature. It compares these with the range of functional response forms that are likely to characterize species in natural communities. The latter set of responses involves many more variables. The article reviews the history of functional response models, and examines previous work that has allowed the functional response of a predator to a single type of prey to depend on additional variables beyond the abundance of that prey type. While a number of more complex functional responses have been discussed over the years, many variables affecting feeding rates are still typically omitted from models of food webs. The influences on functional responses from trophic levels above that of the predator or below that of the prey are particularly likely to be ignored, although models and data have suggested that they can have large effects on the functional response. The influences of adaptive behavior and of the time-scale of response measurement are also too often ignored. Some of the known and unknown consequences of these omissions are discussed.

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.

A stochastic eco-epidemiological system with patchy structure and transport-related infection

In this paper, a stochastic eco-epidemiological system with patchy structure and transport-related infection is proposed and the stochastic dynamical behaviors are investigated. Firstly, by constructing suitable Lyapunov functions, it is revealed that there is a unique globally positive solution starting from the positive initial value. Secondly, it is proved that the presented system is stochastically ultimately bounded and the average in time of the second moment of solution is bounded. Thirdly, we prove that the large enough stochastic perturbations may lead the predator population and the diseases in the predator to be extinct while it is persistent in the deterministic system. Finally, some numerical simulations are given to test our theoretical results.

Specialization directs habitat selection responses to a top predator in semiaquatic but not aquatic taxa

Habitat selectivity has become an increasingly acknowledged mechanism shaping the structure of freshwater communities; however, most studies have focused on the effect of predators and competitors, neglecting habitat complexity and specialization. In this study, we examined the habitat selection of semiaquatic (amphibians: Bufonidae; odonates: Libellulidae) and aquatic organisms (true bugs: Notonectidae; diving beetles: Dytiscidae). From each family, we selected one habitat generalist species able to coexist with fish (Bufo bufo, Sympetrum sanguineum, Notonecta glauca, Dytiscus marginalis) and one species specialized in fishless habitats (Bufotes viridis, Sympetrum danae, Notonecta obliqua, Acilius sulcatus). In a mesocosm experiment, we quantified habitat selection decisions in response to the non-consumptive presence of fish (Carassius auratus) and vegetation structure mimicking different successional stages of aquatic habitats (no macrophytes; submerged and floating macrophytes; submerged, floating, and littoral-emergent macrophytes). No congruence between habitat specialists and generalists was observed, but a similar response to fish and vegetation structure defined both semiaquatic and aquatic organisms. While semiaquatic generalists did not distinguish between fish and fishless pools, specialists avoided fish-occupied pools and had a preferred vegetation structure. In aquatic taxa, predator presence affected habitat selection only in combination with vegetation structure, and all species preferred fishless pools with floating and submerged macrophytes. Fish presence triggered avoidance only in the generalist bug N. glauca. Our results highlight the significance of habitat selectivity for structuring freshwater ecosystems and illustrate how habitat selection responses to a top predator are dictated by specialization and life history.

A Non-local Cross-Diffusion Model of Population Dynamics I: Emergent Spatial and Spatiotemporal Patterns

We extend a spatially non-local cross-diffusion model of aggregation between multiple species with directed motion toward resource gradients to include many species and more general kinds of dispersal. We first consider diffusive instabilities, determining that for directed motion along fecundity gradients, the model permits the Turing instability leading to colony formation and persistence provided there are three or more interacting species. We also prove that such patterning is not possible in the model under the Turing mechanism for two species under directed motion along fecundity gradients, confirming earlier findings in the literature. However, when the directed motion is not along fecundity gradients, for instance, if foraging or migration is sub-optimal relative to fecundity gradients, we find that very different colony structures can emerge. This generalization also permits colony formation for two interacting species. In the advection-dominated case, aggregation patterns are more broad and global in nature, due to the inherent non-local nature of the advection which permits directed motion over greater distances, whereas in the diffusion-dominated case, more highly localized patterns and colonies develop, owing to the localized nature of random diffusion. We also consider the interplay between Turing patterning and spatial heterogeneity in resources. We find that for small spatial variations, there will be a combination of Turing patterns and patterning due to spatial forcing from the resources, whereas for large resource variations, spatial or spatiotemporal patterning can be modified greatly from what is predicted on homogeneous domains. For each of these emergent behaviors, we outline the theoretical mechanism leading to colony formation and then provide numerical simulations to illustrate the results. We also discuss implications this model has for studies of directed motion in different ecological settings.

Patch Size as a Niche Dimension: Aquatic Insects Behaviorally Partition Enemy-Free Space across Gradients of Patch Size

Positive correlation of species richness with area is ubiquitous in nature, but the processes driving that relationship, as well as those constraining typical patterns, remain elusive. Patch size variation is pervasive in natural systems, and it is thus critical to understand how variation in patch size, as well as its potential interaction with factors like predation and isolation, affects community assembly. We crossed patch quality (fish presence/absence) with patch size to the examine effects of quality, size, and their interaction on colonization by aquatic insects. Overall, beetles favored small, fishless patches, but individual species sorted across patch size while hemipterans aggregated into large, fishless patches, producing sorting between Coleoptera and Hemiptera. Both patch size and predation risk generated significant variation in community structure and diversity. Patch size preferences for the 14 most abundant species and preeminence of species turnover in patterns of β-diversity reinforce patch size as a driver of regional species sorting via habitat selection. Species sorting at the immigration stage plays a critical role in community assembly. Identifying patch size as a component of perceived quality establishes patch size as a critical niche dimension and alters our view of its role in assembly dynamics and the maintenance of local and regional diversity.

Predation risk and patch size jointly determine perceived patch quality in ovipositing treefrogs, Hyla chrysoscelis

Two of the most important factors determining community structure and diversity within and among habitat patches are patch size and patch quality. Despite the importance of patch size in existing paradigms in island biogeography, metapopulation biology, landscape ecology, and metacommunity ecology, and growing conservation concerns with habitat fragmentation, there has been little investigation into how patch size interacts with patch quality. We crossed three levels of patch size (1.13 m² , 2.54 m² and 5.73 m² ) with two levels of patch quality (fish presence/absence, green sunfish [Lepomis cyanellus] and golden shiners [Notemigonus crysoleucus]) in six replicate experimental landscapes (3 × 2 × 6 = 36 patches). Both fish predators have been previously shown to elicit avoidance in ovipositing treefrogs. We examined how patch size and patch quality, as well as the interaction between size and quality, affected female oviposition preference and male calling site choice in a natural population of treefrogs (Hyla chrysoscelis). Females almost exclusively oviposited in the largest fishless patches, indicating that females use both risk, in the form of fish predators, and size itself, as components of patch quality. Females routinely use much smaller natural and experimental patches, suggesting that the responses to patch size are highly context dependent. Responses to fish were unaffected by patch size. Male responses largely mimicked those of females, but did not drive female oviposition. We suggest that patch size itself functions as another aspect of patch quality for H. chrysoscelis, and serves as another niche dimension across which species may behaviorally sort in natural systems. Because of strong, shared avoidance of fish (as well as other predators), among many colonizing taxa, patch size may be a critical factor in species sorting and processes of community assembly in freshwater habitats, allowing species to behaviorally segregate along gradients of patch size in fishless ponds. Conversely, lack of variation in patch size may concentrate colonization activity, leading to intensification of species interactions and/or increased use of lesser quality patches.

Giving predators a wide berth: quantifying behavioral predator shadows in colonizing aquatic beetles

The perceived quality of habitat patches in complex landscapes is highly context dependent. Characteristics of neighboring patches in such complex landscapes can influence perceived habitat quality, altering colonization dynamics and community structure. Spatial contagion of predation risk across patches has been observed over smaller spatial scales in aquatic systems. Naturally colonizing aquatic beetles were used to examine the spatial dynamics of risk contagion by quantifying the size of predator shadows around fish patches across spatial scales potentially involving numerous patches in natural landscapes. These consisted of fish free, replicate experimental mesocosm arrays radiating from larger central mesocosms containing fish, and allowed examination of the effect of distance to fish on beetle abundance, rarified species richness, and variation in species responses. Overall, beetles avoided pools closer to fish, but species varied in colonization pattern, resulting in species-specific predator shadows and potential behavioral species sorting. The spatial and phylogenetic extent of contagion and other context-dependent effects has implications for the role of complex behavior in the dynamics of communities and metacommunities.

Two-Species Migration and Clustering in Two-Dimensional Domains

We extend two-species models of individual aggregation or clustering to two-dimensional spatial domains, allowing for more realistic movement of the populations compared with one spatial dimension. We assume that the domain is bounded and that there is no flux into or out of the domain. The motion of the species is along fitness gradients which allow the species to seek out a resource. In the case of competition, species which exploit the resource alone will disperse while avoiding one another. In the case where one of the species is a predator or generalist predator which exploits the other species, that species will tend to move toward the prey species, while the prey will tend to avoid the predator. We focus on three primary types of interspecies interactions: competition, generalist predator-prey, and predator-prey. We discuss the existence and stability of uniform steady states. While transient behaviors including clustering and colony formation occur, our stability results and numerical evidence lead us to believe that the long-time behavior of these models is dominated by spatially homogeneous steady states when the spatial domain is convex. Motivated by this, we investigate heterogeneous resources and hazards and demonstrate how the advective dispersal of species in these environments leads to asymptotic steady states that retain spatial aggregation or clustering in regions of resource abundance and away from hazards or regions or resource scarcity.

Spatial disease dynamics of free-living pathogens under pathogen predation

The epidemiological dynamics of potentially free-living pathogens are often studied with respect to a specific pathogen species (e.g., cholera) and most studies concentrate only on host-pathogen interactions. Here we show that metacommunity-level interactions can alter conventional spatial disease dynamics. We introduce a pathogen eating consumer species and investigate a deterministic epidemiological model of two habitat patches, where both patches can be occupied by hosts, pathogens, and consumers of free-living pathogens. An isolated habitat patch shows periodic disease outbreaks in the host population, arising from cyclic consumer-pathogen dynamics. On the other hand, consumer dispersal between the patches generate asymmetric disease prevalence, such that the host population in one patch stays disease-free, while disease outbreaks occur in the other patch. Such asymmetry can also arise with host dispersal, where infected hosts carry pathogens to the other patch. This indirect movement of pathogens causes also a counter-intuitive effect: decreasing morbidity in a focal patch under increasing pathogen immigration. Our results underline that community-level interactions influence disease dynamics and consistent spatial asymmetry can arise also in spatially homogeneous systems.

Bad neighbors: how spatially disjunct habitat degradation can cause system-wide population collapse

Movement of individuals links the effects of local variation in habitat quality with growth and persistence of populations at the landscape scale. When the populations themselves are linked by interspecific interactions, such as predation, differential movement between habitats may lead to counterintuitive system-wide dynamics. Understanding the interaction between local drivers and dynamics of widely dispersed species is necessary to predict the impacts of habitat fragmentation and degradation, which may be transmitted across habitat boundaries by species' movements. Here we model predator-prey interactions across unaltered and degraded habitat areas, and we explore the additional effects of adaptive habitat choice by predators on the resilience of prey populations. We show how movement between habitats can produce the "bad neighbor effect," in which predators' response to localized habitat degradation causes system-wide loss of prey populations. This effect arises because adaptive foraging results in the concentration of predators in the more productive unaltered habitat, even when this habitat can not support the increased prey mortality. The mechanisms underlying this effect are especially sensitive to prey dispersal rate and adaptive predator behavior.

Local contagion and regional compression: habitat selection drives spatially explicit, multiscale dynamics of colonisation in experimental metacommunities

Habitat selection, including oviposition site choice, is an important driver of community assembly in freshwater systems. Factors determining patch quality are assessed by many colonising organisms and affect colonisation rates, spatial distribution and community structure. For many species, the presence/absence of predators is the most important factor affecting female oviposition decisions. However, individual habitat patches exist in complex landscapes linked by processes of dispersal and colonisation, and spatial distribution of factors such as predators has potential effects beyond individual patches. Perceived patch quality and resulting colonisation rates depend both on risk conditions within a given patch and on spatial context. Here we experimentally confirm the role of one context-dependent processes, spatial contagion, functioning at the local scale, and provide the first example of another context-dependent process, habitat compression, functioning at the regional scale. Both processes affect colonisation rates and patterns of spatial distribution in naturally colonised experimental metacommunities.

Patch quality and context, but not patch number, drive multi-scale colonization dynamics in experimental aquatic landscapes

Colonization and extinction are primary drivers of local population dynamics, community structure, and spatial patterns of biological diversity. Existing paradigms of island biogeography, metapopulation biology, and metacommunity ecology, as well as habitat management and conservation biology based on those paradigms, emphasize patch size, number, and isolation as primary characteristics influencing colonization and extinction. Habitat selection theory suggests that patch quality could rival size, number, and isolation in determining rates of colonization and resulting community structure. We used naturally colonized experimental landscapes to address four issues: (a) how do colonizing aquatic beetles respond to variation in patch number, (b) how do they respond to variation in patch quality, (c) does patch context affect colonization dynamics, and (d) at what spatial scales do beetles respond to habitat variation? Increasing patch number had no effect on per patch colonization rates, while patch quality and context were critical in determining colonization rates and resulting patterns of abundance and species richness at multiple spatial scales. We graphically illustrate how variation in immigration rates driven by perceived predation risk (habitat quality) can further modify dynamics of the equilibrium theory of island biogeography beyond predator-driven effects on extinction rates. Our data support the importance of patch quality and context as primary determinants of colonization rate, occupancy, abundance, and resulting patterns of species richness, and reinforce the idea that management of metapopulations for species preservation, and metacommunities for local and regional diversity, should incorporate habitat quality into the predictive equation.

The relationship between variable host grouping and functional responses among parasitoids of Antispila nysaefoliella (Lepidoptera: Heliozelidae)

Our study investigated the importance of variability in the parasitoid community as a source of selection on host group size using a field population of the tupelo leafminer, Antispila nysaefoliella Clemens, which specializes on tupelo, Nyssa sylvatica Marsh. Larvae were collected from leaves with variable numbers of larvae and screened for parasitism using polymerase chain reaction of mitochondrial cytochrome oxidase I using markers designed specifically for amplifying parasitoid DNA while excluding host DNA. This method of selective PCR was effective for detecting the presence and identifying species of immature stages of three hymenopteran superfamilies: Chalcidoidea, Ichneumonoidea and Platygastroidea, which represented 83.4%, 16.0% and 0.6% of the total detectable parasitism, respectively. Our resulting sequences were then calibrated with sequences from identified adult parasitoids that had been either reared or field-captured. A cluster analysis revealed 10 distinct clades that showed differences in attack patterns with respect to host traits and season. Total parasitism followed an inverse density-dependent or density-independent pattern with respect to host density (number per leaf). However, when parasitoid taxa were considered separately, one clade, which could be a cryptic species of Pnigalio maculipes Crawford (Chalcidoidea: Eulophidae), was found to increase its per leaf attack rate with host density. Our results suggest that parasitoid community composition and differences among species in their attack strategies can play a large role in determining the adaptive advantage of host grouping.

Two-patch population models with adaptive dispersal: the effects of varying dispersal speeds

The population-dispersal dynamics for predator-prey interactions and two competing species in a two patch environment are studied. It is assumed that both species (i.e., either predators and their prey, or the two competing species) are mobile and their dispersal between patches is directed to the higher fitness patch. It is proved that such dispersal, irrespectively of its speed, cannot destabilize a locally stable predator-prey population equilibrium that corresponds to no movement at all. In the case of two competing species, dispersal can destabilize population equilibrium. Conditions are given when this cannot happen, including the case of identical patches.

Generalized movement strategies for constrained consumers: ignoring fitness can be adaptive

Abstract Movements made by real organisms--such as movements involved in dispersal, migration, and habitat selection--are expected to occasionally be suboptimal because of realistic constraints imposed by incomplete information, perceptual limitations, and stochasticity. Previous theory considering such constraints has shown that movements appropriately conditioned on habitat or resource characteristics can balance out suboptimal components of movement and thereby lead organisms to ideal free distributions and fitness maxima, whereas movements conditioned on fitness differentials cannot. These findings suggest a somewhat paradoxical hypothesis: even if organisms have information about their fitness, movement strategies that maximize fitness may be conditioned on something other than fitness per se. We test this hypothesis by investigating the evolutionary stability of generalized, conditional movement strategies that vary in their use of information on fitness versus information on habitat characteristics. We show that when costs of sensory machinery are included, natural selection should favor movement strategies that completely ignore fitness information. Finally, we synthesize previous work by showing how several previous important theoretical results for adaptive movement strategies are united under our one general model.

The roles of spatial heterogeneity and adaptive movement in stabilizing (or destabilizing) simple metacommunities

Adaptive consumer movement and between-patch heterogeneity have both been suggested to reduce population fluctuations in spatially subdivided systems. These conjectures are explored using models of two-patch consumer-resource systems with fitness dependent consumer movement and cyclic dynamics in at least one of the patches; neither conjecture applies generally to such systems. Under relatively low heterogeneity, highly accurate and rapid adaptive movement most often increases both the between-patch correlation of density and the variation in the total density of both species compared to a similar system having a low rate of random movement. However, such adaptive movement can decrease between-patch correlation and global population variability when (1) the consumer's movement is moderately sensitive to fitness differences and heterogeneity is relatively low, or (2) one of the patches would be stable in isolation, and the stable patch supports a sufficiently large consumer population. In both cases, the dynamics are typically either a stable equilibrium or a simple anti-phase cycle with low variation in total population size. Under adaptive movement, population variability is often lowest for intermediate levels of heterogeneity, but monotonic increases or decreases with increasing spatial heterogeneity are possible, depending on the fitness sensitivity of movement and how the characteristic that differs between patches affects within-patch stability and population size. High rates of random movement can lead to greater stability than adaptive movement when consumers are very efficient.

Adaptation and habitat selection in the eco-evolutionary process

The struggle for existence occurs through the vital rates of population growth. This basic fact demonstrates the tight connection between ecology and evolution that defines the emerging field of eco-evolutionary dynamics. An effective synthesis of the interdependencies between ecology and evolution is grounded in six principles. The mechanics of evolution specifies the origin and rules governing traits and evolutionary strategies. Traits and evolutionary strategies achieve their selective value through their functional relationships with fitness. Function depends on the underlying structure of variation and the temporal, spatial and organizational scales of evolution. An understanding of how changes in traits and strategies occur requires conjoining ecological and evolutionary dynamics. Adaptation merges these five pillars to achieve a comprehensive understanding of ecological and evolutionary change. I demonstrate the value of this world-view with reference to the theory and practice of habitat selection. The theory allows us to assess evolutionarily stable strategies and states of habitat selection, and to draw the adaptive landscapes for habitat-selecting species. The landscapes can then be used to forecast future evolution under a variety of climate change and other scenarios.

Tactical population movements and distributions for ideally motivated competitors

The spatial distributions of populations are a reflection of underlying rules for movement behavior in the context of the environment encountered by individuals. Here I study how ideal directed movement--in which individuals travel in the direction offering the most immediate perceived improvement to their personal fitness--dictates the spatial position of two populations occupying the same relative niche and engaged in competition via interference to an individual's ability to gather resources. Drawing on the analytic derivation of equilibria, numerical simulations, and graphical assessments, I provide conditions under which sympatry, parapatry, or regional exclusion is expected during different phases of the community's development. I also demonstrate that specific competitive asymmetries produce distinguishable distributions and invasion patterns and identify which populations are found centrally or peripherally. Dynamic and dispersal equilibria were examined for differences in the sensitivity to spatial variations in fitness, per capita mortality, metabolic efficiency, the strength of interspecific interference, resource collection speed, and the optimal location of each population along an environmental cline. These asymmetries were studied both in isolation and pairwise in fitness trade-off scenarios.

Ratio- and predator-dependent functional forms for predators optimally foraging in patches

Functional forms of predator-prey interactions are developed for predators optimally foraging on prey distributed in patches. The model uses mean free-path-length theory to develop functional forms for two idealized behaviors of prey in patches. For congregating prey that maintain a fixed density, for example, fish schools, the predation rate has a ratio-dependent form, and predator interference depends only on predator density. For sessile prey, which maintain a fixed patch size, a new predator-dependent form emerges in which predator interference depends on both prey and predator densities. The Beddington-DeAngelis equation is a special case of the sessile form. The model provides behavioral and biological criteria with which to select the functional form and ranges of coefficients appropriate for a particular food web. Finally, the model illustrates that behavior is an essential factor in predator-prey dynamics.

Adaptation of prey and predators between patches

Mathematical models are proposed to simulate migrations of prey and predators between patches. In the absence of predators, it is shown that the adaptation of prey leads to an ideal spatial distribution in the sense that the maximal capacity of each patch is achieved. With the introduction of co-adaptation of predators, it is proved that both prey and predators achieve ideal spatial distributions when the adaptations are weak. Further, it is shown that the adaptation of prey and predators increases the survival probability of predators from the extinction in both patches to the persistence in one patch. It is also demonstrated that there exists a pattern that prey and predators cooperate well through adaptations such that predators are permanent in every patch in the case that predators become extinct in each patch in the absence of adaptations. For strong adaptations, it is proved that the model admits periodic cycles and multiple stability transitions.

The structure of clines with fitness-dependent dispersal

Spatial models commonly assume that dispersal does not depend on environmental conditions or phenotype. For example, these assumptions underpin explanations for clines on the basis of a trade-off between dispersal and local adaptation. We reexamine clines when an individual's decisions over whether and where to disperse depend on its fitness. We compare fitness-dependent dispersal with cases where dispersal responds to juvenile survivorship only. Clines are steeper the more responsive dispersal is to environmental conditions for all dispersal behaviors that we consider. Clines eventually become stepped as the responsiveness of dispersal to environmental conditions is increased for half of the dispersal behaviors we consider, but smooth clines are maintained for the remaining cases. Smooth clines are maintained by the biased movement of individuals out of the hybrid zone when individuals move directionally in response to gradients in juvenile survivorship, which is a different mechanism to that maintaining smooth clines in classic cline theory.