Predator-prey dynamics and movement in fractal environments

Previous research suggests that local interactions and limited animal mobility can affect population dynamics. However, the spatial structure of the environment can further limit the mobility of animals. For example, an animal confined to a river valley or to a particular plant cannot move with equal ease in all directions. We show that spatial architecture could influence the population dynamics of predator-prey systems using individual-based computer simulations parameterized with allometric relationships from the literature. Spatial forms (representing geographical features or plant architecture) of differing fractal dimension were generated, and simulated predators and prey were introduced into these computer environments. We claim that the alteration in interaction rates and population dynamics found in these simulations can be explained as a consequence of the anomalously slow rates of movement associated with fractal spaces and the diffusion-limited nature of predator-prey interactions. As a result, functional responses and numerical responses are substantially reduced in fractal environments, and the overall stability of the system is determined by the interaction between individual mobility and spatial architecture.     

Effects of local interaction range and mobility on the spatio-temporal dynamics of competing animals in uniform habitats

Reduction of oscillations in population size is of fundamental importance to both theoretical and applied ecology. Spatial variability in population rates among different habitat regions is known to be an important mechanism that inhibits oscillations in population size. In the current study we used an individual-based model to simulate a single population of animals whose individual members are sensitive to competition only within their vicinity (i.e., within their competition neighborhood, CN). Our model extends previous studies by exploring how local interactions reduce population oscillations in competitive systems of animals, rather than in systems of plants. Our simulations explored the effects of animal mobility and interaction range separately on population dynamics. In our model, a decrease in CN dimensions tended to reduce population oscillations at all tested animal movement speeds. Yet, movement speed affected animal distribution patterns; an increase in movement speed led to more random distributions. We also found that mean population size was affected more by CN dimensions at lower mobility levels than when it was high.     

How localized consumption stabilizes predator-prey systems with finite frequency of mixing

Predator-prey theory began with aspatial models that assumed organisms interacted as if they were "well-mixed" particles that obey the laws of mass action, but it has become clear that both the spatial and individual nature of many organisms can change how the dynamics of such systems function. Here I examine how localized consumption of prey by predators changes the dynamics of predator-prey systems; I use an individual-based simulation of the Rosenzweig-MacArthur model in implicit space and its mean-field approximation. In combination with limited movement, localized consumption makes the predator-prey dynamics more stable than the comparable "well-mixed" Rosenzweig-MacArthur model. Using a spatial correlation, one can directly compare a simplified version of the individual-based model with the Rosenzweig-MacArthur model. While this comparison allows the changes in the dynamics to be captured by the "well-mixed" Rosenzweig-MacArthur model, the parameters of the functional response are now dependent on the movement parameters, and so the functional response must be estimated statistically from the dynamics of the individual-based model. Yet this implies that aspatial models may work in a scale-specific fashion for spatial systems. Unlike many recent spatial models, the localized consumption and limited movement in the model presented here cannot produce coherent spatial patterns and do not depend on a patchy structure, as found in metapopulation models. Instead, the individual nature of the interactions creates a diffusion-limited reaction, which appears closer to a form of ephemeral refuge.     

Physiologically structured models – from versatile technique to ecological theory

A ubiquitous feature of natural communities is the variation in size that can be observed between organisms, a variation that to a substantial degree is intraspecific. Size variation within species by necessity implies that ecological interactions vary both in intensity and type over the life cycle of an individual. Physiologically structured population models (PSPMs) constitute a modelling approach especially designed to analyse these size‐dependent interactions as they explicitly link individual level processes such as consumption and growth to population dynamics. We discuss two cases where PSPMs have been used to analyse the dynamics of size‐structured populations. In the first case, a model of a size‐structured consumer population feeding on a non‐structured prey was successful in predicting both qualitative (mechanisms) and quantitative (individual growth, survival, cycle amplitude) aspects of the population dynamics of a planktivorous fish population. We conclude that single generation cycles as a result of intercohort competition is a general outcome of size‐structured consumer–resource interactions. In the second case, involving both cohort competition and cannibalism, we show that PSPMs may predict double asymptotic growth trajectories with individuals ending up as giants. These growth trajectories, which have also been observed in field data, could not be predicted from individual level information, but are emergent properties of the population feedback on individual processes. In contrast to the size‐structured consumer–resource model, the dynamics in this case cannot be reduced to simpler lumped stage‐based models, but can only be analysed within the domain of PSPMs. Parameter values used in PSPMs adhere to the individual level and are derived independently from the system at focus, whereas model predictions involve both population level processes and individual level processes under conditions of population feedback. This leads to an increased ability to test model predictions but also to a larger set of variables that is predicted at both the individual and population level. The results turn out to be relatively robust to specific model assumptions and thus render a higher degree of generality than purely individual‐based models. At the same time, PSPMs offer a much higher degree of realism, precision and testing ability than lumped stage‐based or non‐structured models. The results of our analyses so far suggest that also in more complex species configurations only a limited set of mechanisms determines the dynamics of PSPMs. We therefore conclude that there is a high potential for developing an individual‐based, size‐dependent community theory using PSPMs.     

Time-ordered networks reveal limitations to information flow in ant colonies

Background

An important function of many complex networks is to inhibit or promote the transmission of disease, resources, or information between individuals. However, little is known about how the temporal dynamics of individual-level interactions affect these networks and constrain their function. Ant colonies are a model comparative system for understanding general principles linking individual-level interactions to network-level functions because interactions among individuals enable integration of multiple sources of information to collectively make decisions, and allocate tasks and resources.

Methodology/Findings

Here we show how the temporal and spatial dynamics of such individual interactions provide upper bounds to rates of colony-level information flow in the ant Temnothorax rugatulus. We develop a general framework for analyzing dynamic networks and a mathematical model that predicts how information flow scales with individual mobility and group size.

Conclusions/Significance

Using thousands of time-stamped interactions between uniquely marked ants in four colonies of a range of sizes, we demonstrate that observed maximum rates of information flow are always slower than predicted, and are constrained by regulation of individual mobility and contact rate. By accounting for the ordering and timing of interactions, we can resolve important difficulties with network sampling frequency and duration, enabling a broader understanding of interaction network functioning across systems and scales.     

On using integral projection models to generate demographically driven predictions of species' distributions: development and validation using sparse data

Knowledge of species' geographic distributions is critical for understanding and forecasting population dynamics, responses to environmental change, biodiversity patterns, and conservation planning. While many suggestive correlative occurrence models have been used to these ends, progress lies in understanding the underlying population biology that generates patterns of range dynamics. Here, we show how to use a limited quantity of demographic data to produce demographic distribution models (DDMs) using integral projection models for size‐structured populations. By modeling survival, growth, and fecundity using regression, integral projection models can interpolate across missing size data and environmental conditions to compensate for limited data. To accommodate the uncertainty associated with limited data and model assumptions, we use Bayesian models to propagate uncertainty through all stages of model development to predictions. DDMs have a number of strengths: 1) DDMs allow a mechanistic understanding of spatial occurrence patterns; 2) DDMs can predict spatial and temporal variation in local population dynamics; 3) DDMs can facilitate extrapolation under altered environmental conditions because one can evaluate the consequences for individual vital rates. To illustrate these features, we construct DDMs for an overstory perennial shrub in the Proteaceae family in the Cape Floristic Region of South Africa. We find that the species' population growth rate is limited most strongly by adult survival throughout the range and by individual growth in higher rainfall regions. While the models predict higher population growth rates in the core of the range under projected climates for 2050, they also suggest that the species faces a threat along arid range margins from the interaction of more frequent fire and drying climate. The results (and uncertainties) are helpful for prioritizing additional sampling of particular demographic parameters along these gradients to iteratively refine projections. In the appendices, we provide fully functional R code to perform all analyses.     

Individual-to-Resource Landscape Interaction Strength Can Explain Different Collective Feeding Behaviours

Taking in sufficient quantities of nutrients is vital for all living beings and in doing so, individuals interact with the local resource environment. Here, we focus explicitly on the interactions between feeding individuals and the resource landscape. In particular, we are interested in the emergent movement dynamics resulting from these interactions. We present an individual-based simulation model for the movement of populations in a resource landscape that allows us to vary the strength of the interactions mentioned above. The key assumption and novelty of our model is that individuals can cause the release of additional nutrients, as well as consuming them. Our model produces clear predictions. For example, we expect more tortuous individual movement paths and higher levels of aggregation in populations occupying homogeneous environments where individual movement makes more nutrients available. We also show how observed movement dynamics could change when local nutrient sources are depleted or when the population density increases. Our predictions are testable and qualitatively reproduce the different feeding behaviours observed in filter-feeding ducks, for example. We suggest that considering two-way interactions between feeding individuals and resource landscapes could help to explain fine-scale movement dynamics.     

Post‐reproductive life span and demographic stability

Recent field studies suggest that it is common in nature for animals to outlive their reproductive viability. Post‐reproductive life span has been observed in a broad range of vertebrate and invertebrate species. But post‐reproductive life span poses a paradox for traditional theories of life history evolution. The only commonly‐cited explanation is the ‘grandmother hypothesis’, which is limited to higher, social mammals. We propose that post‐reproductive life span evolves to stabilize population dynamics, avoiding local extinctions. Predator–prey and other ecosystem interactions tend to produce volatility that can create population crashes and local extinctions. Total fertility rates that exceed the ecosystem's recovery rate contribute to population overshoot, followed by collapse. These local extinctions may constitute a potent group selection mechanism, driving evolution toward controlled rates of population growth, even when there is a significant individual cost. In this paper, we consider the question: what life history characteristics support demographic homeostasis at the least cost to individual fitness? In individual‐based evolutionary simulations, we find that reduction in fertility is sufficient to avoid population instabilities leading to extinction, but that life histories that include senescence can accomplish the same thing at a lower cost to individual fitness. Furthermore, life histories that include the potential for a post‐reproductive period are yet more efficient at stabilizing population dynamics, while minimizing the impact on individual fitness.     

Evolution of reproductive effort in a metapopulation with local extinctions and ecological succession

Abstract Using a metapopulation model, we study how local extinctions, limited population life span, and local demographic disequilibrium affect the evolution of the reproductive effort in a species with overlapping generations but no senescence. We show that in a metapopulation with saturation of all sites and an infinite deme maximal life span (no succession), local extinctions simply constitute an additional source of extrinsic mortality. When either the hypothesis of an infinite deme maximal life span or the saturation hypothesis is relaxed, nontrivial predictions arise. in particular, we find interactions between the evolutionarily stable reproductive effort strategy and the demographic dynamics in the metapopulation. We predict that larger reproductive effort may be selected for in habitats of poorer productivity, contrary to what would be predicted in a single population. Also, we predict that higher dispersal rates should favor selection for lower reproductive efforts. However, metapopulation parameters that favor high dispersal rates also favor larger reproductive efforts. Conflicting selection pressures in the metapopulation also allow maintaining evolutionarily stable polymorphism between a low and high reproductive effort for particular trade-offs between survival and fecundity.     

Competition and evolutionary stability of plants in a spatially structured habitat

We modelled the population dynamics of two types of plants with limited dispersal living in a lattice structured habitat. Each site of the square lattice model was either occupied by an individual or vacant. Each individual reproduced to its neighbors. We derived a criterion for the invasion of a rare type into a population composed of a resident type based on a pair-approximation method, in which the dynamics of both average densities and the nearest neighbor correlations were considered. Based on this invasibility criterion, we showed that, when there is a tradeoff between birth and death rates, the evolutionarily stable type is the one that has the highest ratio of birth rate to mortality. If these types are different species, they form segregated spatial patterns in the lattice model in which intraspecific competitive interactions occur more frequently than interspecific interactions. However, stable coexistence is not possible in the lattice model contrary to results from completely mixed population models. This clearly shows that the casual conclusion, based on traditional well mixed population models, that different species can coexist if intraspecific competition is stronger than interspecific competition, does not hold for spatially structured population models.     

Lotka's Game in Predator–Prey Theory: Linking Populations to Individuals

This paper further examines an individual-based model of a spatially distributed predator–prey population that demonstrates strong spatial structuring in contrast with predictions from its representative analytic formulation. Examination of a small, localized population reveals that extinctions due to demographic stochasticity dominate the dynamics. Local extinction dynamics produce wave pulses and the interactions of these wave pulses constitute global dynamics. The results motivate a population-level cell-based model with each cell representing a local population and parameterized by local extinction probabilities, rather than individual-based interaction rates. A detailed comparison of spatiotemporal plots from the two modelling frameworks shows that the population-level model captures the broad range of dynamics exhibited by the individual-based model. The agreement between these two complementary theoretical frameworks, one formulated at the level of individuals, the other at the level of populations, provides a mechanistic understanding of the dynamics.     

How population loss through habitat boundaries determines the dynamics of a predator–prey system

The increased persistence of predator–prey systems when interactions are distributed through the space has been acknowledged by both empirical and theoretical studies. One salient feature of predator–prey interactions in heterogeneous space, for example, is the existence of cycles with reduced amplitude when compared with a homogeneous landscape. Although the role of spatial interactions in shaping the dynamics of predator–prey systems has been extensively studied, still very few works have focused on the effects of habitat loss and fragmentation on these systems. In this work, we study the population dynamics of a predator–prey system in a single finite habitat with flux at the boundaries. Species movement and growth are described through a reaction–diffusion model with Rosenzweig–MacArthur type local interactions. Conforming with the existing literature, we find that the reduction of habitat size, or increasing of species movement rates equivalently, has the potential to decrease the amplitude of oscillations and even bring the system to a steady coexistence equilibrium above a threshold. We observe, however, situations in which this trend is reversed. This occurs when species movement rates and response at patch boundaries interact to induce non-trivial patterns of species distributions. These distributions are characterized by anti-correlation between predator and prey, creating then spatial refugia for prey. Our results highlight the role of population loss through habitat boundaries in determining the dynamics of predator–prey interactions.     

Dealing with Uncertainty in Spatially Explicit Population Models

It has been argued that spatially explicit population models (SEPMs) cannot provide reliable guidance for conservation biology because of the difficulty of obtaining direct estimates for their demographic and dispersal parameters and because of error propagation. We argue that appropriate model calibration procedures can access additional sources of information, compensating the lack of direct parameter estimates. Our objective is to show how model calibration using population-level data can facilitate the construction of SEPMs that produce reliable predictions for conservation even when direct parameter estimates are inadequate. We constructed a spatially explicit and individual-based population model for the dynamics of brown bears (Ursus arctos) after a reintroduction program in Austria. To calibrate the model we developed a procedure that compared the simulated population dynamics with distinct features of the known population dynamics (=patterns). This procedure detected model parameterizations that did not reproduce the known dynamics. Global sensitivity analysis of the uncalibrated model revealed high uncertainty in most model predictions due to large parameter uncertainties (coefficients of variation CV 0.8). However, the calibrated model yielded predictions with considerably reduced uncertainty (CV 0.2). A pattern or a combination of various patterns that embed information on the entire model dynamics can reduce the uncertainty in model predictions, and the application of different patterns with high information content yields the same model predictions. In contrast, a pattern that does not embed information on the entire population dynamics (e.g., bear observations taken from sub-areas of the study area) does not reduce uncertainty in model predictions. Because population-level data for defining (multiple) patterns are often available, our approach could be applied widely.     

Dynamic heterogeneous spatio-temporal pattern formation in host-parasitoid systems with synchronised generations

In this paper we develop a general mathematical model describing the spatio-temporal dynamics of host-parasitoid systems with forced generational synchronisation, for example seasonally induced diapause. The model itself may be described as an individual-based stochastic model with the individual movement rules derived from an underlying continuum PDE model. This approach permits direct comparison between the discrete model and the continuum model. The model includes both within-generation and between-generation mechanisms for population regulation and focuses on the interactions between immobile juvenile hosts, adult hosts and adult parasitoids in a two-dimensional domain. These interactions are mediated, as they are in many such host-parasitoid systems, by the presence of a volatile semio-chemical (kairomone) emitted by the hosts or the hosts food plant. The model investigates the effects on population dynamics for different host versus parasitoid movement strategies as well as the transient dynamics leading to steady states. Despite some agreement between the individual and continuum models for certain motility parameter ranges, the model dynamics diverge when host and parasitoid motilities are unequal. The individual-based model maintains spatially heterogeneous oscillatory dynamics when the continuum model predicts a homogeneous steady state. We discuss the implications of these results for mechanistic models of phenotype evolution.P. Schofield gratefully acknowledges the financial support of the BBSRC and The Wellcome Trust.     

Two-patch dispersal-linked compensatory-overcompensatory spatially discrete population models

We study the role of asynchronous and synchronous dispersals on discrete-time two-patch dispersal-linked population models, where the pre-dispersal local patch dynamics are of mixed compensatory and overcompensatory types. Single-species dispersal-linked models behave as single-species single-patch models whenever all pre-dispersal local patch dynamics are compensatory and dispersal is synchronous. However, the dynamics of the corresponding two-patch population model connected by asynchronous dispersal depends on the dispersal rates. The species goes extinct on at least one patch when the asynchronous dispersal rates are high, while it persists when the rates are low. We use numerical simulations to show that in both synchronous and asynchronous mixed compensatory and overcompensatory systems, symmetric and asymmetric dispersals can control and impede the onset of cyclic population oscillations via period-doubling reversal bifurcations. Also, we show that in mixed systems both asynchronous and synchronous dispersals are capable of altering the pre-dispersal local patch dynamics from overcompensatory to compensatory dynamics. Dispersal-linked population models with ‘unstructured’ overcompensatory pre-dispersal local dynamics connected by synchronous dispersal can generate multiple attractors with fractal basin boundaries. However, mixed compensatory and overcompensatory systems appear to exhibit single attractors and not coexisting (multiple) attractors.     

Two-patch dispersal-linked compensatory-overcompensatory spatially discrete population models

We study the role of asynchronous and synchronous dispersals on discrete-time two-patch dispersal-linked population models, where the pre-dispersal local patch dynamics are of mixed compensatory and overcompensatory types. Single-species dispersal-linked models behave as single-species single-patch models whenever all pre-dispersal local patch dynamics are compensatory and dispersal is synchronous. However, the dynamics of the corresponding two-patch population model connected by asynchronous dispersal depends on the dispersal rates. The species goes extinct on at least one patch when the asynchronous dispersal rates are high, while it persists when the rates are low. We use numerical simulations to show that in both synchronous and asynchronous mixed compensatory and overcompensatory systems, symmetric and asymmetric dispersals can control and impede the onset of cyclic population oscillations via period-doubling reversal bifurcations. Also, we show that in mixed systems both asynchronous and synchronous dispersals are capable of altering the pre-dispersal local patch dynamics from overcompensatory to compensatory dynamics. Dispersal-linked population models with 'unstructured' overcompensatory pre-dispersal local dynamics connected by synchronous dispersal can generate multiple attractors with fractal basin boundaries. However, mixed compensatory and overcompensatory systems appear to exhibit single attractors and not coexisting (multiple) attractors.     

Indirect Genetic Effects and the Dynamics of Social Interactions

BackgroundIndirect genetic effects (IGEs) occur when genes expressed in one individual alter the expression of traits in social partners. Previous studies focused on the evolutionary consequences and evolutionary dynamics of IGEs, using equilibrium solutions to predict phenotypes in subsequent generations. However, whether or not such steady states may be reached may depend on the dynamics of interactions themselves.ResultsIn our study, we focus on the dynamics of social interactions and indirect genetic effects and investigate how they modify phenotypes over time. Unlike previous IGE studies, we do not analyse evolutionary dynamics; rather we consider within-individual phenotypic changes, also referred to as phenotypic plasticity. We analyse iterative interactions, when individuals interact in a series of discontinuous events, and investigate the stability of steady state solutions and the dependence on model parameters, such as population size, strength, and the nature of interactions. We show that for interactions where a feedback loop occurs, the possible parameter space of interaction strength is fairly limited, affecting the evolutionary consequences of IGEs. We discuss the implications of our results for current IGE model predictions and their limitations.     

Population dynamical consequences of gregariousness in a size-structured consumer-resource interaction

Many animal species live in groups. Group living may increase exploitation competition within the group, and variation among groups in intra-group competition intensity could induce life-history variability among groups. Models of physiologically structured populations generally predict single generation cycles, driven by exploitation competition within and between generations. We expect that life-history variability and habitat heterogeneity induced by group living may affect such competition-driven population dynamics. In this study, we vary the gregariousness (the tendency to aggregate in groups) of a size-structured consumer population in a spatially explicit environment. The consumer has limited mobility, and moves according to a probabilistic movement process. We study the effects on the population dynamics, as mediated through the resource and the life-history of the consumer. We find that high gregariousness leads to large spatial resource variation, and highly variable individual life-history, resulting in highly stochastic population dynamics. At reduced gregariousness, life-history of consumers synchronizes, habitat heterogeneity is reduced, and single generation cycles appear. We expect this pattern to occur for any group living organism with limited mobility. Our results indicate that constraints set by population dynamical feedback may be an important aspect in understanding group living in nature.     

Bacterial colonization and extinction on marine aggregates: stochastic model of species presence and abundance

Organic aggregates provide a favorable habitat for aquatic microbes, are efficiently filtered by shellfish, and may play a major role in the dynamics of aquatic pathogens. Quantifying this role requires understanding how pathogen abundance in the water and aggregate size interact to determine the presence and abundance of pathogen cells on individual aggregates. We build upon current understanding of the dynamics of bacteria and bacterial grazers on aggregates to develop a model for the dynamics of a bacterial pathogen species. The model accounts for the importance of stochasticity and the balance between colonization and extinction. Simulation results suggest that while colonization increases linearly with background density and aggregate size, extinction rates are expected to be nonlinear on small aggregates in a low background density of the pathogen. Under these conditions, we predict lower probabilities of pathogen presence and reduced abundance on aggregates compared with predictions based solely on colonization. These results suggest that the importance of aggregates to the dynamics of aquatic bacterial pathogens may be dependent on the interaction between aggregate size and background pathogen density, and that these interactions are strongly influenced by ecological interactions and pathogen traits. The model provides testable predictions and can be a useful tool for exploring how species‐specific differences in pathogen traits may alter the effect of aggregates on disease transmission.     

Predators Reduce Extinction Risk in Noisy Metapopulations

Background

Spatial structure across fragmented landscapes can enhance regional population persistence by promoting local “rescue effects.” In small, vulnerable populations, where chance or random events between individuals may have disproportionately large effects on species interactions, such local processes are particularly important. However, existing theory often only describes the dynamics of metapopulations at regional scales, neglecting the role of multispecies population dynamics within habitat patches.

Findings

By coupling analysis across spatial scales we quantified the interaction between local scale population regulation, regional dispersal and noise processes in the dynamics of experimental host-parasitoid metapopulations. We find that increasing community complexity increases negative correlation between local population dynamics. A potential mechanism underpinning this finding was explored using a simple population dynamic model.

Conclusions

Our results suggest a paradox: parasitism, whilst clearly damaging to hosts at the individual level, reduces extinction risk at the population level.