Brucellosis, botflies, and brainworms: the impact of edge habitats on pathogen transmission and species extinction

Ecological interactions between species that prefer different habitat types but come into contact in edge regions at the interfaces between habitat types are modeled via reaction-diffusion systems. The primary sort of interaction described by the models is competition mediated by pathogen transmission. The models are somewhat novel because the spatial domains for the variables describing the population densities of the interacting species overlap but do not coincide. Conditions implying coexistence of the two species or the extinction of one species are derived. The conditions involve the principal eigenvalues of elliptic operators arising from linearizations of the model system around equilibria with only one species present. The conditions for persistence or extinction are made explicit in terms of the parameters of the system and the geometry of the underlying spatial domains via estimates of the principal eigenvalues. The implications of the models with respect to conservation and refuge design are discussed. Received: 10 June 1999 / Revised version: 7 July 2000 / Published online: 20 December 2000     

Spatial Segregation of the Sexes and Nutrients affect Reproductive Success in a Dioecious Wind-pollinated Grass

In many dioecious plant species in which spatial distributions of males and females have been examined, the sexes are spatially segregated – usually along an environmental gradient. Unless pollen is uniformly distributed in a population, spatial segregation of the sexes should reduce the average mating success of individuals. In three Californian populations of Distichlis spicata – a wind-pollinated grass species that exhibits spatial segregation of the sexes – I examined patterns of pollen movement and the effects of pollen load and nutrient availability on seed set to determine whether spatial segregation of the sexes actually reduces mating success for both males and females. In two of the populations, pollen dispersal was restricted, and pollen augmentation consistently, significantly increased seed set. However, in the third population – which had the lowest seed set – I found that although there were some indications of pollen limitation, pollen dispersal was not restricted, and seed production was limited primarily by nutrient availability. These results imply that in some populations of D. spicata nutrient limitation on the production of seeds by females may be sufficiently strong that spatial segregation of the sexes causes a fairly low cost to reproductive success compared with a more random distribution of the sexes. However, in other populations, pollen does limit mating success, and the spatial segregation of males and females in these populations is reducing the fecundity of both males and females.     

Population synchrony and stability in environmentally forced metacommunities

A general prediction from simple metapopulation models is that spatially synchronized forcing can spatially synchronize population dynamics and destabilize metapopulations. In contrast, spatially asynchronous forcing is predicted to decrease population synchrony and promote temporal stability and population persistence, especially in the presence of dispersal. Only recently have studies begun to experimentally address these predictions. Moreover, few studies have experimentally examined how such processes operate in the context of competition communities. Stabilizing processes may continue to operate when placed within a metacommunity context with multiple competing consumers but only at low to intermediate levels of dispersal. High dispersal rates can reverse these predictions and lead to destabilization. We tested this under controlled conditions using an experimental aquatic system composed of three competing species of zooplankton. Metacommunities experienced different levels of dispersal and environmental forcing in the form of spatially synchronous or asynchronous pH perturbations. We found support that dispersal can have contrasting effects on population stability depending on the degree to which population dynamics were synchronized in space. Dispersal under synchronous forcing or no forcing had either neutral of positive effects on spatial population synchrony of all three zooplankton species. In these treatments, dispersal reduced population stability at the local and metapopulation levels for two of three species. In contrast, asynchronously varying environments reduced population synchrony relative to unforced systems, regardless of dispersal level. In these treatments, dispersal enhanced temporal stability and persistence of populations not by reducing population synchrony but by enhancing population minima and spatial averaging of abundances. High dispersal rates under asynchronous forcing reduced the abundance of one species, consistent with increasing regional competition and general metacommunity theory. However, no effects on its stability or persistence were observed. Our work highlights the context‐dependent effects of dispersal on population dynamics in varying environments.     

A mathematical model of activity–dependent,anatomical segregation induced by competition for neurotrophic support

Mathematical or computational models of activity–dependent neural competition typically impose competition in anatomically fixed networks by the use of synaptic normalisation, for which there is very little experimental support. Recent experimental evidence, however, strongly implicates neurotrophic factors in neural plasticity and competition, in addition to their well–known potent effects on neurite outgrowth and synaptogenesis. We therefore present a simple, mathematical model of anatomical segregation induced by activity–dependent competition for a limited supply of a neurotrophic factor provided by target cells to afferents. We extract the behaviour of the model in various regimes, in which the neurotrophic factor is either in critical supply or in abundant supply, by a combination of analytical and numerical methods, and study the effects of correlations in afferent inputs on competition. We apply the model to three different systems: ocular dominance column formation; elimination of polyneuronal innervation at the vertebrate neuromuscular junction; trigeminal brain stem whisker–related structure formation. Several classes of related predictions emerge, including the prediction that kittens reared with strabismus should require a higher concentration of neurotrophic factor infusion into their primary visual cortex than normally reared cats in order to induce the anatomical desegregation of ocular dominance columns. We also speculate on the mechanisms of support of inhibitory rather than excitatory neurons, and suggest the existence of a separate, Cl–mediated activity–dependent pathway for their neurotrophic support. Received: 17 May 1996 / Accepted in revised form: 27 July 1996     

Sustainability without coexistence state in Durrett–Levin hawk–dove model

Models that explain the sustainability of an exploiter–victim ecosystem admit, generally, a coexistence state of both species in the well-mixed limit. Even if this state is unstable, the extinction-prone system may acquire stability on spatial domains where different patches oscillate incoherently around the coexistence state. New experiments, however, suggest that a spatially segregated system may be stable even in the absence of such a coexistence state. Here we revisit the hawk–dove (case 3) model of Durrett and Levin, which has been shown to support persistent population for system of interacting particles. It turns out that this model does not admit a (stable or unstable) coexistence state on a single habitat. We analyze the peculiar mechanism that leads to persistence in this case and the role of demographic stochasticity with and without self-interaction, using numerical simulations and exact solutions in the infinite diffusion limit.     

Metapopulation dynamics with quasi-local competition

Stepping-stone models for the ecological dynamics of metapopulations are often used to address general questions about the effects of spatial structure on the nature and complexity of population fluctuations. Such models describe an ensemble of local and spatially isolated habitat patches that are connected through dispersal. Reproduction and hence the dynamics in a given local population depend on the density of that local population, and a fraction of every local population disperses to neighboring patches. In such models, interesting dynamic phenomena, e.g. the persistence of locally unstable predator-prey interactions, are only observed if the local dynamics in an isolated patch exhibit non-equilibrium behavior. Therefore, the scope of these models is limited. Here we extend these models by making the biologically plausible assumption that reproductive success in a given local habitat not only depends on the density of the local population living in that habitat, but also on the densities of neighboring local populations. This would occur if competition for resources occurs between neighboring populations, e.g. due to foraging in neighboring habitats. With this assumption of quasi-local competition the dynamics of the model change completely. The main difference is that even if the dynamics of the local populations have a stable equilibrium in isolation, the spatially uniform equilibrium in which all local populations are at their carrying capacity becomes unstable if the strength of quasi-local competition reaches a critical level, which can be calculated analytically. In this case the metapopulation reaches a new stable state, which is, however, not spatially uniform anymore and instead results in an irregular spatial pattern of local population abundance. For large metapopulations, a huge number of different, spatially non-uniform equilibrium states coexist as attractors of the metapopulation dynamics, so that the final state of the system depends critically on the initial conditions. The existence of a large number of attractors has important consequences when environmental noise is introduced into the model. Then the metapopulation performs a random walk in the space of all attractors. This leads to large and complicated population fluctuations whose power spectrum obeys a red-shifted power law. Our theory reiterates the potential importance of spatial structure for ecological processes and proposes new mechanisms for the emergence of non-uniform spatial patterns of abundance and for the persistence of complicated temporal population fluctuations.     

Persistence in fluctuating environments for interacting structured populations

Individuals within any species exhibit differences in size, developmental state, or spatial location. These differences coupled with environmental fluctuations in demographic rates can have subtle effects on population persistence and species coexistence. To understand these effects, we provide a general theory for coexistence of structured, interacting species living in a stochastic environment. The theory is applicable to nonlinear, multi species matrix models with stochastically varying parameters. The theory relies on long-term growth rates of species corresponding to the dominant Lyapunov exponents of random matrix products. Our coexistence criterion requires that a convex combination of these long-term growth rates is positive with probability one whenever one or more species are at low density. When this condition holds, the community is stochastically persistent: the fraction of time that a species density goes below \(\delta >0\) approaches zero as \(\delta \) approaches zero. Applications to predator-prey interactions in an autocorrelated environment, a stochastic LPA model, and spatial lottery models are provided. These applications demonstrate that positive autocorrelations in temporal fluctuations can disrupt predator-prey coexistence, fluctuations in log-fecundity can facilitate persistence in structured populations, and long-lived, relatively sedentary competing populations are likely to coexist in spatially and temporally heterogenous environments.     

Host plant quality,spatial heterogeneity,and the stability of mite predator–prey dynamics

Population dynamics models suggest that both the over-all level of resource productivity and spatial variability in productivity can play important roles in community dynamics. Higher productivity environments are predicted to destabilize consumer–resource dynamics. Conversely, greater heterogeneity in resource productivity is expected to contribute to stability. Yet the importance of these two factors for the dynamics of arthropod communities has been largely overlooked. I manipulated nutrient availability for strawberry plants in a multi-patch experiment, and measured effects of overall plant quality and heterogeneity in plant quality on the stability of interactions between the phytophagous mite Tetranychus urticae and its predator Phytoseiulus persimilis. Plant size, leaf N content and T. urticae population growth increased monotonically with increasing soil nitrogen availability. This gradient in plant quality affected two correlates of mite population stability, population variability over time (i.e., coefficient of variation) and population persistence (i.e., proportion of plant patches colonized). However, the highest level of plant quality did not produce the least stable dynamics, which is inconsistent with the “paradox of enrichment”. Heterogeneity in plant productivity had modest effects on stability, with the only significant difference being less variable T. urticae densities in the heterogeneous compared to the corresponding homogeneous treatment. These results are generally congruent with metapopulation theory and other models for spatially segregated populations, which predict that stability should be governed largely by relative movement rates of predators and prey—rather than patch quality.     

A decision-making Fokker–Planck model in computational neuroscience

In computational neuroscience, decision-making may be explained analyzing models based on the evolution of the average firing rates of two interacting neuron populations, e.g., in bistable visual perception problems. These models typically lead to a multi-stable scenario for the concerned dynamical systems. Nevertheless, noise is an important feature of the model taking into account both the finite-size effects and the decision’s robustness. These stochastic dynamical systems can be analyzed by studying carefully their associated Fokker–Planck partial differential equation. In particular, in the Fokker–Planck setting, we analytically discuss the asymptotic behavior for large times towards a unique probability distribution, and we propose a numerical scheme capturing this convergence. These simulations are used to validate deterministic moment methods recently applied to the stochastic differential system. Further, proving the existence, positivity and uniqueness of the probability density solution for the stationary equation, as well as for the time evolving problem, we show that this stabilization does happen. Finally, we discuss the convergence of the solution for large times to the stationary state. Our approach leads to a more detailed analytical and numerical study of decision-making models applied in computational neuroscience.     

Persistence in the chemostat

Sufficient conditions are obtained for persistence in chemostat models for interactions of a limiting nutrient (or substrate) and two populations, and for two limiting complementary substrates and a single population. The results of Freedman and Waltman are extended for the three interacting predator-prey populations.     

Cellular automaton models for competition in patchy environments: Facilitation, inhibition, and tolerance

We have developed cellular automaton models for two species competing in a patchy environment. We have modeled three common types of competition: facilitation (in which the winning species can colonize only after the losing species has arrived) inhibition (in which either species is able to prevent the other from colonizing) and tolerance (in which the species most tolerant of reduced resource levels wins). The state of a patch is defined by the presence or absence of each species. State transition probabilities are determined by rates of disturbance, competitive exclusion, and colonization. Colonization is restricted to neighboring patches. In all three models, disturbance permits regional persistence of species that are excluded by competition locally. Persistence, and hence diversity, is maximized at intermediate disturbance frequencies. If disturbance and dispersal rates are sufficiently high, the inferior competitor need not have a dispersal advantage to persist. Using a new method for measuring the spatial patterns of nominal data, we show that none of these competition models generates patchiness at equilibrium. In the inhibition model, however, transient patchiness decays very slowly. We compare the cellular automaton models to the corresponding mean-field patch-occupancy models, in which colonization is not restricted to neighboring patches and depends on spatially averaged species frequencies. The patch-occupancy model does an excellent job of predicting the equilibrium frequencies of the species and the conditions required for coexistence, but not of predicting transient behavior.     

Competition and species coexistence in a metapopulation model: Can fast asymmetric migration reverse the outcome of competition in a homogeneous environment?

We investigate whether asymmetric fast migration can modify the predictions of classical competition theory and, in particular revert species dominance. We consider a model of two species competing for an implicit resource on a habitat divided into two patches. Both patches are connected through constant migration rates and in each patch local dynamics are driven by a Lotka-Volterra competition system.Local competition is asymmetric with the same superior competitor in both patches. Migration is asymmetric, species dependent and fast in comparison to local competitive interactions. The species and patches are taken to be otherwise similar: in both patches we assume the same carrying capacities for both species, and the same growth rates and pair-wise competition coefficients for each species.We show that global dynamics can be described by a classical Lotka-Volterra competition model. We found that by modifying the ratio of intraspecific migration rates for both species all possible combinations of global species relative dominance can be achieved. We find specific conditions for which the local superior competitor is globally excluded. This is to our knowledge the first study showing that fast asymmetric migration can lead to inferior competitor dominance in a homogeneous environment. We conclude that disparity of temporal scales between migration and local dynamics may have important consequences for the maintenance of biodiversity in spatially structured populations.     

Consequences of the Allee Effect and Intraspecific Competition on Population Persistence under Adverse Environmental Conditions

The impact of intraspecific interactions on ecological stability and population persistence in terms of steady state(s) existence is considered theoretically based on a general competition model. We compare persistence of a structured population consisting of a few interacting (competitive) subpopulations, or groups, to persistence of the corresponding unstructured population. For a general case, we show that if the intra-group competition is stronger than the inter-group competition, then the structured population is less prone to extinction, i.e. it can persist in a parameter range where the unstructured population goes extinct. For a more specific case of a population with hierarchical competition, we show that relative viability of structured and unstructured populations depend on the type of density dependence in the population growth. Namely, while in the case of logistic growth, structured and unstructured populations exhibit equivalent persistence; in the case of Allee dynamics, the persistence of a hierarchically structured population is shown to be higher. We then apply these results to the case of behaviourally structured populations and demonstrate that an extreme form of individual aggression can be beneficial at the population level and enhance population persistence.     

Multisensory fusion and the stochastic structure of postural sway

 We analyze the stochastic structure of postural sway and demonstrate that this structure imposes important constraints on models of postural control. Linear stochastic models of various orders were fit to the center-of-mass trajectories of subjects during quiet stance in four sensory conditions: (i) light touch and vision, (ii) light touch, (iii) vision, and (iv) neither touch nor vision. For each subject and condition, the model of appropriate order was determined, and this model was characterized by the eigenvalues and coefficients of its autocovariance function. In most cases, postural-sway trajectories were similar to those produced by a third-order model with eigenvalues corresponding to a slow first-order decay plus a faster-decaying damped oscillation. The slow-decay fraction, which we define as the slow-decay autocovariance coefficient divided by the total variance, was usually near 1. We compare the stochastic structure of our data to two linear control-theory models: (i) a proportional–integral–derivative control model in which the postural system's state is assumed to be known, and (ii) an optimal-control model in which the system's state is estimated based on noisy multisensory information using a Kalman filter. Under certain assumptions, both models have eigenvalues consistent with our results. However, the slow-decay fraction predicted by both models is less than we observe. We show that our results are more consistent with a modification of the optimal-control model in which noise is added to the computations performed by the state estimator. This modified model has a slow-decay fraction near 1 in a parameter regime in which sensory information related to the body's velocity is more accurate than sensory information related to position and acceleration. These findings suggest that: (i) computation noise is responsible for much of the variance observed in postural sway, and (ii) the postural control system under the conditions tested resides in the regime of accurate velocity information. Received: 20 March 2001 / Accepted: 17 April 2002 Acknowledgements. We thank Tjeerd Dijkstra for bringing the slow-decay component of postural sway to our attention. Funding for this research was provided by National Institutes of Health grant R29 N35070–01A2, John J. Jeka, PI. Correspondence to: T. Kiemel (Tel.: +1-301-4056176, Fax: +1-301-3149358 e-mail: kiemel@glue.umd.edu)     

Perspective of practical biological control and population theories

We have not yet had sufficient theoretical explanation for successful biological control in which a key pest is controlled after an introduction of natural enemies. I compare here real features of successful biological control and theoretical host–parasitoid population models to reduce the gap between theory and practice. I first review the historical interaction between classical biological control projects and theoretical population models. Second, I consider the importance of host refuges in host–parasitoid population dynamics as concerns the mechanisms of low and stable host density. The importance of density–dependent parasitism through parasitoid reproduction in multivoltine host–parasitoid systems and supplemental generalist natural enemies are also discussed. Finally, I consider the difference in tactics for classical biological control and for augmentation of natural enemies in annual crop systems. Received: December 20, 1998 / Accepted: January 15, 1999     

Implications of nest-site limitation on density-dependent nest predation at variable spatial scales in a cavity-nesting bird

Nest‐site limitation may have different implications in the spatial distribution of breeding pairs depending on the availability of suitable habitat and the types of nest‐sites. Distribution of cavities suitable as nest sites may allow circumstantial aggregation or active choice of colonial nesting, which may have different implications on breeding performance through effects on breeding density, with variable costs and benefits depending on the consequences of intraspecific competition, social interactions and predation. We evaluated the effects of breeding density derived from nesting site limitation on breeding performance and predation at different spatial scales and considering multiple social, population and environmental limiting factors in the red‐billed chough Pyrrhocorax pyrrhocorax. The results indicate that variable breeding density may arise within the population depending on the availability and spatial distribution of nest‐sites. Nest‐site availability and distribution may also determine social breeding systems (isolated or aggregated) at variable densities, thus resembling differences found at different spatially distant populations under contrasting environmental conditions. Breeding performance was related to density‐dependent processes of population regulation, especially density‐dependent nest predation due to predator attraction to nest clusters. Results also indicate that predation pressure depend on density patterns at large scales. This suggest that predation may have important consequences on population dynamics of spatially structured populations depending on the strength of this kind of density dependence, which in turn may depend on habitat features affecting the prey but also the spatially variable guild of predators. Because habitat and nesting site availability may vary spatially depending on multiple human influences, understanding the strength and form in which breeding density and nest predation at different spatial scales may influence the size and persistence of populations can help to manage them more adequately.     

Allee threshold and extinction threshold for spatially explicit metapopulation dynamics with Allee effects

In this paper, we investigate a spatially explicit metapopulation model with Allee effects. We refer to the patch occupancy model introduced by Levins (Bull Entomol Soc Am 15:237–240, 1969) as a spatially implicit metapopulation model, i.e., each local patch is either occupied or vacant and a vacant patch can be recolonized by a randomly chosen occupied patch from anywhere in the metapopulation. When we transform the model into a spatially explicit one by using a lattice model, the obtained model becomes theoretically equivalent to a “lattice logistic model” or a “basic contact process”. One of the most popular or standard metapopulation models with Allee effects, developed by Amarasekare (Am Nat 152:298–302, 1998), supposes that those effects are introduced formally by means of a logistic equation. However, it is easier to understand the ecological meaning of associating Allee effects with this model if we suppose that only the logistic colonization term directly suffers from Allee effects. The resulting model is also well defined, and therefore we can naturally examine it by Monte Carlo simulation and by doublet and triplet decoupling approximation. We then obtain the following specific features of one-dimensional lattice space: (1) the metapopulation as a whole does not have an Allee threshold for initial population size even when each local population follows the Allee effects; and (2) a metapopulation goes extinct when the extinction rate of a local population is lower than that in the spatially implicit model. The real ecological metapopulation lies between two extremes: completely mixing interactions between patches on the one hand and, on the other, nearest neighboring interactions with only two nearest neighbors. Thus, it is important to identify the metapopulation structure when we consider the problems of invasion species such as establishment or the speed of expansion.     

Reinfection induced disease in a spatial SIRI model

Spatial models are widely used in epidemiology to investigate persistence and extinction of disease as well as their spatial patterns. One of the most important issues in studying epidemic models is the role of infection on the persistence and extinction of the disease. In this paper, we investigate a spatial susceptible–infected–recovered–infected model using cellular automata. We show that, in the regime where disease disappears in the susceptible–infected–recovered–susceptible model, spiral and target waves will emerge in the two-dimensional space due to the reinfection. The obtained results may point out that reinfection has great influence on the epidemic spreading, which enriches the findings of spatiotemporal dynamics in epidemic models.     

Assessing the spatial variability of density dependence in waterfowl populations

Population models commonly assume that the demographic parameters are spatially invariant, but there is considerable evidence that population growth rate (r) and the strength of density dependence (β) can vary over a species' range. To address this issue we developed a spatially explicit Gompertz population model based on the spatially varying coefficients approach to assess the spatial variation in population drivers. The model was fit to spatially stratified time series population estimates of the mallard Anas platyrhynchos in western North America. We included precipitation during the previous year and spring maximum temperature in the current year as environmental factors in the density dependent population model. Because density dependent models can give biased estimates for time series of abundance data, we fit a naïve model without informative priors and a model where we constrained the mean and variance of r to biologically realistic values that were derived via a comparative demography approach. In the naïve model, r and β were not separately identifiable and their values were overestimated, leading to unrealistic population growth. The naïve model also implied spatial variation in population r and the return time to equilibrium [1?(– β)] across the survey area. In contrast, in the informative model, r and the return time to equilibrium did not vary markedly among populations and were generally equal across populations. The effects of the climatic factors were similar across models. Population growth rates in the Prairie‐pothole region were positively correlated with precipitation, while in Alaska rates were positively correlated with spring temperature. Although it has been argued in the past that adding ecological realism could help avoid the pitfalls associated with density dependent models, our results demonstrate that imposing constraints on the population parameters is still the best course of action.     

Parasitoid-mediated effects: apparent competition and the persistence of host–parasitoid assemblages

Indirect effects such as apparent competition (in which two hosts that do not compete for resources interact via a shared natural enemy) are increasingly being shown to be prevalent in the structure and function of ecological assemblages. Here, we review the empirical and theoretical evidence for these enemy-mediated effects in host–parasitoid assemblages. We first address questions about the design of experiments to test for apparent competition. Second, we consider factors likely to affect the coexistence of host species that share a parasitoid and are involved in apparent competition. We show that parasitoid aggregation, and the switching effect that this can generate when hosts occur in separate patches, not only promotes persistence but is also strongly stabilizing. The broader consequences of these effects are discussed. Received: November 6, 1998 / Accepted: January 13, 1999