Scaling up predator-prey dynamics using spatial moment equations

Within Reach? Habitat Availability as a Function of Individual Mobility and Spatial Structuring

Organisms need access to particular habitats for their survival and reproduction. However, even if all necessary habitats are available within the broader environment, they may not all be easily reachable from the position of a single individual. Many species distribution models consider populations in environmental (or niche) space, hence overlooking this fundamental aspect of geographical accessibility. Here, we develop a formal way of thinking about habitat availability in environmental spaces by describing how limitations in accessibility can cause animals to experience a more limited or simply different mixture of habitats than those more broadly available. We develop an analytical framework for characterizing constrained habitat availability based on the statistical properties of movement and environmental autocorrelation. Using simulation experiments, we show that our general statistical representation of constrained availability is a good approximation of habitat availability for particular realizations of landscape-organism interactions. We present two applications of our approach, one to the statistical analysis of habitat preference (using step-selection functions to analyze harbor seal telemetry data) and a second that derives theoretical insights about population viability from knowledge of the underlying environment. Analytical expressions for habitat availability, such as those we develop here, can yield gains in analytical speed, biological realism, and conceptual generality by allowing us to formulate models that are habitat sensitive without needing to be spatially explicit.

How range residency and long-range perception change encounter rates

Encounter rates link movement strategies to intra- and inter-specific interactions, and therefore translate individual movement behavior into higher-level ecological processes. Indeed, a large body of interacting population theory rests on the law of mass action, which can be derived from assumptions of Brownian motion in an enclosed container with exclusively local perception. These assumptions imply completely uniform space use, individual home ranges equivalent to the population range, and encounter dependent on movement paths actually crossing. Mounting empirical evidence, however, suggests that animals use space non-uniformly, occupy home ranges substantially smaller than the population range, and are often capable of nonlocal perception. Here, we explore how these empirically supported behaviors change pairwise encounter rates. Specifically, we derive novel analytical expressions for encounter rates under Ornstein-Uhlenbeck motion, which features non-uniform space use and allows individual home ranges to differ from the population range. We compare OU-based encounter predictions to those of Reflected Brownian Motion, from which the law of mass action can be derived. For both models, we further explore how the interplay between the scale of perception and home-range size affects encounter rates. We find that neglecting realistic movement and perceptual behaviors can lead to systematic, non-negligible biases in encounter-rate predictions.

Micro-scale patchiness enhances trophic transfer efficiency and potential plankton biodiversity

Rather than spatial means of biomass, observed overlap in the intermittent spatial distributions of aquatic predators and prey is known to be more important for determining the flow of nutrients and energy up the food chain. A few previous studies have separately suggested that such intermittency enhances phytoplankton growth and trophic transfer to sustain zooplankton and ultimately fisheries. Recent observations have revealed that phytoplankton distributions display consistently high degrees of mm scale patchiness, increasing along a gradient from estuarine to open ocean waters. Using a generalized framework of plankton ecosystem models with different trophic configurations, each accounting for this intermittency, we show that it consistently enhances trophic transfer efficiency (TE), i.e. the transfer of energy up the food chain, and expands the model stability domain. Our results provide a new explanation for observation-based estimates of unexpectedly high TE in the vast oligotrophic ocean and suggest that by enhancing the viable trait space, micro-scale variability may potentially sustain plankton biodiversity.

Spatial structure arising from chase-escape interactions with crowding

Movement of individuals, mediated by localised interactions, plays a key role in numerous processes including cell biology and ecology. In this work, we investigate an individual-based model accounting for various intraspecies and interspecies interactions in a community consisting of two distinct species. In this framework we consider one species to be chasers and the other species to be escapees, and we focus on chase-escape dynamics where the chasers are biased to move towards the escapees, and the escapees are biased to move away from the chasers. This framework allows us to explore how individual-level directional interactions scale up to influence spatial structure at the macroscale. To focus exclusively on the role of motility and directional bias in determining spatial structure, we consider conservative communities where the number of individuals in each species remains constant. To provide additional information about the individual-based model, we also present a mathematically tractable deterministic approximation based on describing the evolution of the spatial moments. We explore how different features of interactions including interaction strength, spatial extent of interaction, and relative density of species influence the formation of the macroscale spatial patterns.

Fitness trade-offs of group formation and movement by Thomson's gazelles in the Serengeti ecosystem

Collective behaviours contributing to patterns of group formation and coordinated movement are common across many ecosystems and taxa. Their ubiquity is presumably due to altering interactions between individuals and their predators, resources and physical environment in ways that enhance individual fitness. On the other hand, fitness costs are also often associated with group formation. Modifications to these interactions have the potential to dramatically impact population-level processes, such as trophic interactions or patterns of space use in relation to abiotic environmental variation. In a wide variety of empirical systems and models, collective behaviour has been shown to enhance access to ephemeral patches of resources, reduce the risk of predation and reduce vulnerability to environmental fluctuation. Evolution of collective behaviour should accordingly depend on the advantages of collective behaviour weighed against the costs experienced at the individual level. As an illustrative case study, we consider the potential trade-offs on Malthusian fitness associated with patterns of group formation and movement by migratory Thomson's gazelles in the Serengeti ecosystem.

This article is part of the theme issue ‘Collective movement ecology’.

Spatial Moment Description of Birth-Death-Movement Processes Incorporating the Effects of Crowding and Obstacles

Birth-death-movement processes, modulated by interactions between individuals, are fundamental to many cell biology processes. A key feature of the movement of cells within in vivo environments is the interactions between motile cells and stationary obstacles. Here we propose a multi-species model of individual-level motility, proliferation and death. This model is a spatial birth-death-movement stochastic process, a class of individual-based model (IBM) that is amenable to mathematical analysis. We present the IBM in a general multi-species framework and then focus on the case of a population of motile, proliferative agents in an environment populated by stationary, non-proliferative obstacles. To analyse the IBM, we derive a system of spatial moment equations governing the evolution of the density of agents and the density of pairs of agents. This approach avoids making the usual mean-field assumption so that our models can be used to study the formation of spatial structure, such as clustering and aggregation, and to understand how spatial structure influences population-level outcomes. Overall the spatial moment model provides a reasonably accurate prediction of the system dynamics, including important effects such as how varying the properties of the obstacles leads to different spatial patterns in the population of agents.

Modelling and parameter inference of predator-prey dynamics in heterogeneous environments using the direct integral approach

Most bacterial habitats are topographically complex in the micro scale. Important examples include the gastrointestinal and tracheal tracts, and the soil. Although there are myriad theoretical studies that explore the role of spatial structures on antagonistic interactions (predation, competition) among animals, there are many fewer experimental studies that have explored, validated and quantified their predictions. In this study, we experimentally monitored the temporal dynamic of the predatory bacterium Bdellovibrio bacteriovorus, and its prey, the bacterium Burkholderia stabilis in a structured habitat consisting of sand under various regimes of wetness. We constructed a dynamic model, and estimated its parameters by further developing the direct integral method, a novel estimation procedure that exploits the separability of the states and parameters in the model. We also verified that one of our parameter estimates was consistent with its known, directly measured value from the literature. The ability of the model to fit the data combined with realistic parameter estimates indicate that bacterial predation in the sand can be described by a relatively simple model, and stress the importance of prey refuge on predation dynamics in heterogeneous environments.

A spatial theory for emergent multiple predator-prey interactions in food webs

Predator–prey interaction is inherently spatial because animals move through landscapes to search for and consume food resources and to avoid being consumed by other species. The spatial nature of species interactions necessitates integrating spatial processes into food web theory and evaluating how predators combine to impact their prey. Here, we present a spatial modeling approach that examines emergent multiple predator effects on prey within landscapes. The modeling is inspired by the habitat domain concept derived from empirical synthesis of spatial movement and interactions studies. Because these principles are motivated by synthesis of short‐term experiments, it remains uncertain whether spatial contingency principles hold in dynamical systems. We address this uncertainty by formulating dynamical systems models, guided by core habitat domain principles, to examine long‐term multiple predator–prey spatial dynamics. To describe habitat domains, we use classical niche concepts describing resource utilization distributions, and assume species interactions emerge from the degree of overlap between species. The analytical results generally align with those from empirical synthesis and present a theoretical framework capable of demonstrating multiple predator effects that does not depend on the small spatial or temporal scales typical of mesocosm experiments, and help bridge between empirical experiments and long‐term dynamics in natural systems.

Micro-scale variability enhances trophic transfer and potentially sustains biodiversity in plankton ecosystems

We develop moment closure approximations to represent micro-scale spatial variability in the concentrations of nutrients (N), phytoplankton (P) and zooplankton (Z) in an NPZ model, which we apply to examine the impact of different levels of micro-scale variability on both ecosystem dynamics and trophic transfer. Accounting explicitly for both the mean-field and fluctuating components of each prognostic variable in the NPZ model yields different dynamics for the mean-field concentrations, as well as lower phytoplankton biomass and greater zooplankton biomass, compared to the conventional NPZ model without micro-scale variability. The biomass of zooplankton consistently increases with increasing total micro-scale variability, and a minimum threshold of such variability is required for the existence of stable steady state solutions in the NPZ closure model. Compared to the conventional NPZ model, the domain of parameter space over which stable solutions exist is larger than for the NPZ closure model, and this stable domain widens with increasing total variability. The latter result suggests that natural systems with greater micro-scale variability may have the potential to sustain greater biodiversity. We find that with the NPZ closure model: (1) the stability domains increases with micro-scale variability, (2) increase of the level of total micro-scale variability enhances trophic transfer, i.e. increases the biomass of zooplankton, and (3) the coefficient of variation (CV_P) of phytoplankton increases with micro-scale variability.

A spatial theory for characterizing predator-multiprey interactions in heterogeneous landscapes

Trophic interactions in multiprey systems can be largely determined by prey distributions. Yet, classic predator-prey models assume spatially homogeneous interactions between predators and prey. We developed a spatially informed theory that predicts how habitat heterogeneity alters the landscape-scale distribution of mortality risk of prey from predation, and hence the nature of predator interactions in multiprey systems. The theoretical model is a spatially explicit, multiprey functional response in which species-specific advection-diffusion models account for the response of individual prey to habitat edges. The model demonstrates that distinct responses of alternative prey species can alter the consequences of conspecific aggregation, from increasing safety to increasing predation risk. Observations of threatened boreal caribou, moose and grey wolf interacting over 378 181 km(2) of human-managed boreal forest support this principle. This empirically supported theory demonstrates how distinct responses of apparent competitors to landscape heterogeneity, including to human disturbances, can reverse density dependence in fitness correlates.

Indirect effects of primary prey population dynamics on alternative prey

We develop a theory of generalist predation showing how alternative prey species are affected by changes in both mean abundance and variability (coefficient of variation) of their predator's primary prey. The theory is motivated by the indirect effects of cyclic rodent populations on ground-breeding birds, and developed through progressive analytic simplifications of an empirically-based model. It applies nonetheless to many other systems where primary prey have fast life-histories and can become superabundant, thus facilitating impact on alternative prey species and generating highly asymmetric interactions. Our results suggest that predator effects on alternative prey should generally decrease with mean primary prey abundance, and increase with primary prey variability (low to high CV)-unless predators have strong aggregative responses, in which case these results can be reversed. Approximations of models including predator dynamics (general numerical response with possible delays) confirm these results but further suggest that negative temporal correlation between predator and primary prey is harmful to alternative prey. Finally, we find that measurements of predator numerical responses are crucial to predict-even qualitatively-the response of ecosystems to changes in the dynamics of outbreaking prey species.

Space race functional responses

We derive functional responses under the assumption that predators and prey are engaged in a space race in which prey avoid patches with many predators and predators avoid patches with few or no prey. The resulting functional response models have a simple structure and include functions describing how the emigration of prey and predators depend on interspecific densities. As such, they provide a link between dispersal behaviours and community dynamics. The derived functional response is general but is here modelled in accordance with empirically documented emigration responses. We find that the prey emigration response to predators has stabilizing effects similar to that of the DeAngelis-Beddington functional response, and that the predator emigration response to prey has destabilizing effects similar to that of the Holling type II response. A stability criterion describing the net effect of the two emigration responses on a Lotka-Volterra predator-prey system is presented. The winner of the space race (i.e. whether predators or prey are favoured) is determined by the relationship between the slopes of the species' emigration responses. It is predicted that predators win the space race in poor habitats, where predator and prey densities are low, and that prey are more successful in richer habitats.

Spatial point processes and moment dynamics in the life sciences: a parsimonious derivation and some extensions

Mathematical models of dynamical systems in the life sciences typically assume that biological systems are spatially well mixed (the mean-field assumption). Even spatially explicit differential equation models typically make a local mean-field assumption. In effect, the assumption is that diffusive movement is strong enough to destroy spatial structure or that interactions between individuals are sufficiently long-range that the effects of spatial structure are weak. However, many important biophysical processes, such as chemical reactions of biomolecules within cells, disease transmission among humans, and dispersal of plants, have characteristic spatial scales that can generate strong spatial structure at the scale of individuals, with important effects on the behaviour of biological systems. This calls for mathematical methods that incorporate spatial structure. Here, we focus on one method, spatial-moment dynamics, which is based on the idea that important information about a spatial point process is held in its low-order spatial moments. The method goes beyond the dynamics of the first moment, i.e. the mean density or concentration of agents in space, in which no information about spatial structure is retained. By including the dynamics of at least the second moment, the method retains some information about spatial structure. Whereas mean-field models effectively use a closure assumption for the second moment, spatial-moment models use a closure assumption for the third (or a higher-order) moment. The aim of the paper was to provide a parsimonious and intuitive derivation of spatial-moment dynamic equations that is accessible to non-specialists. The derivation builds naturally from the first moment to the second, and we show how it can be extended to higher-order moments. Rather than tying the model to a specific biological example, we formulate a general model of movement, birth, and death of multiple types of interacting agents. This model can be applied to problems from a range of disciplines, some of which we discuss. The derivation is performed in a spatially non-homogeneous setting, to facilitate future investigations of biological scenarios, such as invasions, in which the spatial patterns are non-stationary over space.

Incorporating spatial correlations into multispecies mean-field models

In biology, we frequently observe different species existing within the same environment. For example, there are many cell types in a tumour, or different animal species may occupy a given habitat. In modeling interactions between such species, we often make use of the mean-field approximation, whereby spatial correlations between the locations of individuals are neglected. Whilst this approximation holds in certain situations, this is not always the case, and care must be taken to ensure the mean-field approximation is only used in appropriate settings. In circumstances where the mean-field approximation is unsuitable, we need to include information on the spatial distributions of individuals, which is not a simple task. In this paper, we provide a method that overcomes many of the failures of the mean-field approximation for an on-lattice volume-excluding birth-death-movement process with multiple species. We explicitly take into account spatial information on the distribution of individuals by including partial differential equation descriptions of lattice site occupancy correlations. We demonstrate how to derive these equations for the multispecies case and show results specific to a two-species problem. We compare averaged discrete results to both the mean-field approximation and our improved method, which incorporates spatial correlations. We note that the mean-field approximation fails dramatically in some cases, predicting very different behavior from that seen upon averaging multiple realizations of the discrete system. In contrast, our improved method provides excellent agreement with the averaged discrete behavior in all cases, thus providing a more reliable modeling framework. Furthermore, our method is tractable as the resulting partial differential equations can be solved efficiently using standard numerical techniques.