Effects of Heterogeneity on Spread and Persistence in Rivers

The question how aquatic populations persist in rivers when individuals are constantly lost due to downstream drift has been termed the “drift paradox.” Recent modeling approaches have revealed diffusion-mediated persistence as a solution. We study logistically growing populations with and without a benthic stage and consider spatially varying growth rates. We use idealized hydrodynamic equations to link river cross-sectional area to flow speed and assume heterogeneity in the form of alternating patches, i.e., piecewise constant conditions. We derive implicit formulae for the persistence boundary and for the dispersion relation of the wave speed. We explicitly discuss the influence of flow speed, cross-sectional area and benthic stage on both persistence and upstream invasion speed.     

Seasonal influences on population spread and persistence in streams: spreading speeds

The drift paradox asks how stream-dwelling organisms can persist, without being washed out, when they are continuously subject to the unidirectional stream flow. To date, mathematical analyses of the stream paradox have investigated the interplay of growth, drift and flow needed for species persistence under the assumption that the stream environment is temporally constant. However, in reality, streams are subject to major seasonal variations in environmental factors that govern population growth and dispersal. We consider the influence of such seasonal variations on the drift paradox, using a time-periodic integrodifferential equation model. We establish upstream and downstream spreading speeds under the assumption of periodically fluctuating environments, and also show the existence of periodic traveling waves. The sign of the upstream spreading speed then determines persistence. Fluctuating environments are characterized by seasonal correlations between the flow, transfer rates, diffusion and settling rates, and we investigate the effect of such correlations on the population spread and persistence. We also show how results in this paper can formally connect to those for autonomous integrodifferential equations, through the appropriate weighted averaging methods. Finally, for a specific dispersal function, we show that the upstream spreading speed is nonnegative if and only if the critical domain size exists in this temporally fluctuating environment.     

Demography and dispersal: invasion speeds and sensitivity analysis in periodic and stochastic environments

Invasion speeds can be calculated from matrix integrodifference equation models that incorporate stage-specific demography and dispersal. These models also permit the calculation of the sensitivity and elasticity of invasion speed to changes in demographic and dispersal parameters. Such calculations have been used to understand the factors determining invasion speed and to explore possible tactics to manage invasive species. In this paper, we extend these calculations to temporally varying environments. We present formulas for the invasion speed and its sensitivity and elasticity in both periodic and stochastic environments. Periodic models can describe seasonal variation within a year, or can be used to study the frequency of occurrence of events (e.g., floods, fires) on interannual time scales. Stochastic models can incorporate variances, covariances, and temporal autocorrelation of parameters. We show that the invasion speed is calculated from a growth rate which is in turn calculated from a periodic or stochastic product of moment-generating function matrices. We present a new formulation of sensitivity analysis, using matrix calculus, that applies equally to constant, periodic, and stochastic environments.     

Meandering Rivers: How Important is Lateral Variability for Species Persistence?

Models for population dynamics in rivers and streams have highlighted the importance of spatial and temporal variations for population persistence. We present a novel model that considers the longitudinal variation as introduced by the sinuosity of a meandering river where a main channel is laterally extended to point bars in bends. These regions offer different habitat conditions for aquatic populations and therefore may enhance population persistence. Our model is a nonstandard reaction–advection–diffusion model where the domain of definition consists of the real line (representing the main channel) with periodically added intervals (representing the point bars). We give an existence and uniqueness proof for solutions of the equations. We then study population persistence as the (in-) stability of the trivial solution and population spread as the minimal wave speed of traveling periodic waves. We conduct a sensitivity analysis to highlight the importance of each parameter on the model outcome. We find that sinuosity can enhance species persistence.     

Persistence, spread and the drift paradox

We derive conditions for persistence and spread of a population where individuals are either immobile or dispersing by advection and diffusion through a one-dimensional medium with a unidirectional flow. Reproduction occurs only in the stationary phase. Examples of such systems are found in rivers and streams, marine currents, and areas with prevalent wind direction. In streams, a long-standing question, dubbed 'the drift paradox', asks why aquatic insects faced with downstream drift are able to persist in upper stream reaches. For our two-phase model, persistence of the population is guaranteed if, at low population densities, the local growth rate of the stationary component of the population exceeds the rate of entry of individuals into the drift. Otherwise the persistence condition involves all the model parameters, and persistence requires a critical (minimum) domain size. We calculate the rate at which invasion fronts propagate up- and downstream, and show that persistence and ability to spread are closely connected: if the population cannot advance upstream against the flow, it also cannot persist on any finite spatial domain. By studying two limiting cases of our model, we show that residence in the immobile state always enhances population persistence. We use our findings to evaluate a number of mechanisms previously proposed in the ecological literature as resolutions of the drift paradox.     

Predator–prey systems in streams and rivers

Many predator–prey systems are found in environments with a predominantly unidirectional flow such as streams and rivers. Alterations of natural flow regimes (e.g., due to human management or global warming) put biological populations at risk. The aim of this paper is to devise a simple method that links flow speeds (currents) with population retention (persistence) and wash-out (extinction). We consider systems of prey and specialist, as well as generalist, predators, for which we distinguish the following flow speed scenarios: (a) coexistence, (b) persistence of prey only or (c) predators only (provided they are generalists), and (d) extinction of both populations. The method is based on a reaction–advection–diffusion model and traveling wave speed approximations. We show that this approach matches well spread rates observed in numerical simulations. The results from this paper can provide a useful tool in the assessment of instream flow needs, estimating the flow speed necessary for preserving riverine populations.     

Seasonal Invasion Dynamics in a Spatially Heterogeneous River with Fluctuating Flows

A key problem in environmental flow assessment is the explicit linking of the flow regime with ecological dynamics. We present a hybrid modeling approach to couple hydrodynamic and biological processes, focusing on the combined impact of spatial heterogeneity and temporal variability on population dynamics. Studying periodically alternating pool-riffle rivers that are subjected to seasonally varying flows, we obtain an invasion ratchet mechanism. We analyze the ratchet process for a caricature model and a hybrid physical–biological model. The water depth and current are derived from a hydrodynamic equation for variable stream bed water flows and these quantities feed into a reaction-diffusion-advection model that governs population dynamics of a river species. We establish the existence of spreading speeds and the invasion ratchet phenomenon, using a mixture of mathematical approximations and numerical computations. Finally, we illustrate the invasion ratchet phenomenon in a spatially two-dimensional hydraulic simulation model of a meandering river structure. Our hybrid modeling approach strengthens the ecological component of stream hydraulics and allows us to gain a mechanistic understanding as to how flow patterns affect population survival.     

Low-head dams facilitate Round Goby <Emphasis Type="Italic">Neogobius melanostomus</Emphasis> invasion

Round Goby Neogobius melanostomus invasion of the Grand River (Ontario, Canada) presents an opportunity to assess the role of abiotic gradients in mediating the establishment and impact of nonnative benthic fishes in rivers. In this system, sequential low-head dams delineate uninvaded and invaded river reaches and create upstream gradients of increasing water velocity. We hypothesized that flow refugia created by impounded reservoirs above low-head dams enhance local Round Goby abundance. Round Goby influence on the native fish community was determined by variance partitioning, and we used generalized additive models to identify small-bodied benthic fish species most likely to be impacted by Round Goby invasion. Round Goby abundance declined as the degree of reservoir effect decreased upstream. The distributions of four species (including the endangered Eastern Sand Darter Ammocrypta pellucida) in invaded reaches were best explained by inclusion of both reservoir-associated abiotic variables and Round Goby abundance as model terms. To determine establishment potential of the uninvaded reach immediately upstream, four environmental habitat characteristics were used in discriminant function analysis (DFA) to predict three potential outcomes of introduction: non-invaded and either lower or higher Round Goby abundance (low and high invasion status, respectively) than the median number of Round Goby at invaded sites. Our DFA function correctly classified non-invaded and high-abundance invasion status sites > 85% of the time, with lower (73%) success in classifying low-abundance invasion status sites, and the spatial pattern of our results suggest that likelihood of establishment is greatest in impounded habitat.     

Pair-edge approximation for heterogeneous lattice population models

To increase the analytical tractability of lattice stochastic spatial population models, several approximations have been developed. The pair-edge approximation is a moment-closure method that is effective in predicting persistence criteria and invasion speeds on a homogeneous lattice. Here we evaluate the effectiveness of the pair-edge approximation on a spatially heterogeneous lattice in which some sites are unoccupiable, or "dead". This model has several possible interpretations, including a spatial SIS epidemic model, in which some sites are occupied by immobile host-species individuals while others are empty. We find that, as in the homogeneous model, the pair-edge approximation is significantly more accurate than the ordinary pair approximation in determining conditions for persistence. However, habitat heterogeneity decreases invasion speed more than is predicted by the pair-edge approximation, and the discrepancy increases with greater clustering of "dead" sites. The accuracy of the approximation validates the underlying heuristic picture of population spread and therefore provides qualitative insight into the dynamics of lattice models. Conversely, the situations where the approximation is less accurate reveals limitations of pair approximation in the presence of spatial heterogeneity.     

A two-phase epidemic driven by diffusion

In this paper, I present and analyse a model for the spatial dynamics of an epidemic following the point release of an infectious agent. Under conditions where the infectious agent disperses rapidly, relative to the dispersal rate of individuals, the resulting epidemic exhibits two distinct phases: a primary phase in which an epidemic wavefront propagates at constant speed and a secondary phase with a decelerating wavefront. The behavior of the primary phase is similar to standard results for diffusive epidemic models. The secondary phase may be attributed to the environmental persistence of the infectious agent near the release point. Analytic formulas are given for the invasion speeds and asymptotic infection levels. Qualitatively similar results appear to hold in an extended version of the model that incorporates virus shedding and dispersal of individuals.     

Integrodifference equations in patchy landscapes

We analyze integrodifference equations (IDEs) in patchy landscapes. Movement is described by a dispersal kernel that arises from a random walk model with patch dependent diffusion, settling, and mortality rates, and it incorporates individual behavior at an interface between two patch types. Growth follows a simple Beverton–Holt growth or linear decay. We obtain explicit formulae for the critical domain-size problem, and we illustrate how different individual behavior at the boundary between two patch types affects this quantity. We also study persistence conditions on an infinite, periodic, patchy landscape. We observe that if the population can persist on the landscape, the spatial profile of the invasion evolves into a discontinuous traveling periodic wave that moves with constant speed. Assuming linear determinacy, we calculate the dispersion relation and illustrate how movement behavior affects invasion speed. Numerical simulations justify our approach by showing a close correspondence between the spread rate obtained from the dispersion relation and from numerical simulations.     

Are Anomalous Invasion Speeds Robust to Demographic Stochasticity?

Two important issues for conservation are the range expansion of species as a result of climate change and the invasion of exotic species. Being able to predict the rate at which species spread is key for successful management. In deterministic models, the invasion speed of a polymorphic population can be faster than that of any of the component phenotypes, and these “anomalous” invasion speeds persist even when the mutation rate between phenotypes is vanishingly small. Here we investigate whether the same phenomenon is observed in a model with demographic stochasticity. The model that we use is discrete in time and space and we carry out numerical simulations to determine the invasion speed of a population that has two morphs which differ in their dispersal abilities. We find that anomalous speeds are observed in the stochastic model, but only when the carrying capacity of the population is large or the mutation rate between morphs is high enough. These results suggest that only species with large population sizes, such as many insect species, may be able to invade faster if they are polymorphic than if there is only a single morph present in the population.     

Spreading speed, traveling waves, and minimal domain size in impulsive reaction-diffusion models

How growth, mortality, and dispersal in a species affect the species' spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively, they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the population at the end of a pulse as a possibly nonmonotone function of the density of the population at the beginning of the pulse. The dynamics in the dispersal stage is governed by a nonlinear reaction-diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species' spreading speeds, traveling wave speeds, as well as minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also give an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results.     

The contribution of newly established populations to the dynamics of range expansion in a one-dimensional fluvial-estuarine system: rainbow trout (Oncorhynchus mykiss) in Eastern Quebec

Aim Rainbow trout (Oncorhynchus mykiss, Walbaum 1792) is an exotic salmonid invading eastern Canada. First introduced for recreational fishing in Ontario, Quebec and the Maritime provinces, the species is now spreading in salmon rivers located in Eastern Quebec, where its stocking is strictly forbidden. Newly established populations have been found along the St Lawrence Estuary. To effectively mitigate the potential threat the invasion poses to native salmonids, we aimed to document the invasion’s origin and progress in the St Lawrence River and estuary. We first determined genetic origins among several potential wild and cultured source populations, found at the upstream and downstream extremities of the St Lawrence system. Thereafter, we studied the range expansion, predicting that the invasion process conforms to a one‐dimensional stepping‐stone dispersion model. Location Recently invaded salmon rivers that flow into the Estuary and Gulf of St Lawrence in Quebec, and watercourses supporting naturalized populations (Lake Ontario, Lake Memphremagog and Prince‐Edward‐Island rivers). Methods Rainbow trout from 10 potential source populations (wild and domestic strains) and 243 specimens captured in salmon rivers were genotyped at 10 microsatellite loci. Genetic origins of specimens and relationship between colonies were assessed using assignment analyses based on individuals and clusters. Results Invasion of rainbow trout in Eastern Quebec is directed downstream, driven by migrants from upstream naturalized populations, found in the Ganaraska River (Lake Ontario), and, to a lesser extent, in Lake Memphremagog. Populations from the Maritime provinces and domestic strains do not contribute to the colonisation process. A recently established population in Charlevoix (Eastern Quebec) supplies other downstream colonies. Main conclusions Rainbow trout is spreading from Lake Ontario downstream to Eastern Quebec using the St Lawrence River system as an invasion corridor. Range expansion in a downstream direction is driven by a more complex stepping‐stone dispersion model than predicted. Management options to protect native salmonids include reducing the effective size of the Charlevoix population, impeding reproduction in recently colonized rivers, halting the upstream migration of anadromous spawners, and curtailing stocking events inside the stocking area.     

Representation of motion onset and offset in an augmented Barlow-Levick model of motion detection

Kinetic occlusion produces discontinuities in the optic flow field, whose perception requires the detection of an unexpected onset or offset of otherwise predictably moving or stationary contrast patches. Many cells in primate visual cortex are directionally selective for moving contrasts, and recent reports suggest that this selectivity arises through the inhibition of contrast signals moving in the cells’ null direction, as in the rabbit retina. This nulling inhibition circuit (Barlow-Levick) is here extended to also detect motion onsets and offsets. The selectivity of extended circuit units, measured as a peak evidence accumulation response to motion onset/offset compared to the peak response to constant motion, is analyzed as a function of stimulus speed. Model onset cells are quiet during constant motion, but model offset cells activate during constant motion at slow speeds. Consequently, model offset cell speed tuning is biased towards higher speeds than onset cell tuning, similarly to the speed tuning of cells in the middle temporal area when exposed to speed ramps. Given a population of neurons with different preferred speeds, this asymmetry addresses a behavioral paradox—why human subjects in a simple reaction time task respond more slowly to motion offsets than onsets for low speeds, even though monkey neuron firing rates react more quickly to the offset of a preferred stimulus than to its onset.     

Spatial patterns and coexistence mechanisms in systems with unidirectional flow

River ecosystems are the prime example of environments where unidirectional flow influences the dispersal of individuals. Spatial patterns of community composition and species replacement emerge from complex interplays of hydrological, geochemical, biological, and ecological factors. Local processes affecting algal dynamics are well understood, but a mechanistic basis for large scale emerging patterns is lacking. To understand how these patterns could emerge in rivers, we analyze a reaction-advection-diffusion model for two competitors in heterogeneous environments. The model supports waves that invade upstream up to a well-defined "upstream invasion limit". We discuss how these waves are produced and present their key properties. We suggest that patterns of species replacement and coexistence along spatial axes reflect stalled waves, produced from diffusion, advection, and species interactions. Emergent spatial scales are plausible given parameter estimates for periphyton. Our results apply to other systems with unidirectional flow such as prevailing winds or climate-change scenarios.     

How does stochasticity in colonization accelerate the speed of invasion in a cellular automaton model?

We investigate the speed of invasion waves for a single species generated by stochastic short- and/or long-distance colonizations in a time-continuous cellular automaton (CA) model on a two-dimensional homogenous landscape. By simulating the CA models, we demonstrate that stochasticity can dramatically increase the speed of invasion compared to the corresponding deterministic CA model or the corresponding one-dimensional stochastic CA model. To explain this phenomenon, we first develop a mathematical model for the invasion involving only short-distance colonization (i.e., colonization only occurs from the eight adjacent cells), and present several approximation methods for solving the model. Our analyses show that the increased wave speed in the stochastic model is due to irregularity in the shape of the wavefront. Further extension of this model to include long-distance colonization demonstrates that stochasticity influences speeds to even greater extents in this case. Using dimension analysis, we deduced a semi-empirical formula for the speed as a function of three parameters intrinsic to short- and long-distance colonization, which agrees well with simulation results. Based on these results, we discuss how important stochasticity in colonization and spatial dimensionality are in the acceleration of invasion speed.     

Speeds of invasion in a model with strong or weak Allee effects

We study an invasion model based on a reaction-diffusion equation with an Allee effect. We use a special, piecewise-linear, population growth rate. This function allows us to obtain traveling wave solutions and to compute wave speeds for a full range of Allee effects, including weak Allee effects. Some investigators claim that linearization fails to give the correct speed of invasion if there is an Allee effect. We show that the minimum speed for a sufficiently weak Allee may, in fact, be the same as that derived by means of linearization.     

Rat muscle blood flows during high-speed locomotion

We previously studied blood flow distribution within and among rat muscles as a function of speed from walking (15 m/min) through galloping (75 m/min) on a motor-driven treadmill. The results showed that muscle blood flows continued to increase as a function of speed through 75 m/min. The purpose of the present study was to have rats run up to maximal treadmill speeds to determine if blood flows in the muscles reach a plateau as a function of running speed over the animals' normal range of locomotory speeds. Muscle blood flows were measured with radiolabeled microspheres at 1 min of running at 75, 90, and 105 m/min in male Sprague-Dawley rats. The data indicate that even at these relatively high treadmill speeds there was still no clear evidence of a plateau in blood flow in most of the hindlimb muscles. Flows in most muscles continued to increase as a function of speed. These observed patterns of blood flow vs. running speed may have resulted from the rigorous selection of rats that were capable of performing the high-intensity exercise and thus only be representative of a highly specific population of animals. On the other hand, the data could be interpreted to indicate that the cardiovascular potential during exercise is considerably higher in laboratory rats than has normally been assumed and that inadequate blood flow delivery to the muscles does not serve as a major limitation to their locomotory performance.