Foliage profiles of individual trees determine competition,self-thinning,biomass and NPP of a Cryptomeria japonica forest stand: A simulation study based on a stand-scale process-based forest model

A simulation study was carried out to investigate simultaneously the effects of eco-physiological parameters on competitive asymmetry, self-thinning, stand biomass and NPP in a temperate forest using an atmosphere–vegetation dynamics interactive model (MINoSGI). In this study, we selected three eco-physiological relevant parameters as foliage profiles (i.e. vertical distribution of leaf area density) of individual trees (distribution pattern is described by the parameter η), biomass allocation pattern in individual tree growth (χ) and the maximum carboxylation velocity (V_max). The position of the maximal leaf area density shifts upward in the canopy with increasing η. For scenarios with η < 4 (foliage concentrated in the lowest canopy layer) or η > 12 (foliage concentrated in the uppermost canopy layer), a low degree of competitive asymmetry was produced. These scenarios resulted in the survival of subordinate trees due to a brighter lower canopy environment when η < 4 or the generation of spatially separated foliage profiles between dominant and subordinate trees when η > 12. In contrast, competition between trees was most asymmetric when 4 ≤ η ≤ 12 (vertically widespread foliage profile in the canopy), especially when η = 8. In such cases, vertically widespread foliage of dominant trees lowered the opportunity of light acquisition for subordinate trees and reduced their carbon gain. The resulting reduction in carbon gain of subordinate trees yielded a higher degree of competitive asymmetry and ultimately higher mortality of subordinate trees. It was also shown that 4 ≤ η ≤ 12 generated higher self-thinning speed, smaller accumulated NPP, litter-fall and potential stand biomass as compared with the scenarios with η < 4 or η > 12. In contrast, our simulation revealed small effects of χ or V_max on the above-mentioned variables as compared with those of η. In particular, it is notable that greater V_max would not produce greater potential stand biomass and accumulated NPP although it has been thought that physiological parameters relevant to photosynthesis such as V_max influence dynamic changes in forest stand biomass and NPP (e.g. the greater the V_max, the greater the NPP). Overall, it is suggested that foliage profiles rather than biomass allocation or maximum carboxylation velocity greatly govern forest dynamics, stand biomass, NPP and litter-fall.     

Extension of the skin shear stress Li’s relationship to the flat bed

A proper estimation of the skin shear stress τ _s is necessary for a proper evaluation of sediment flux at the sediment–fluid interface. Several empirical formulas of the skin shear stress have been proposed in the literature for rippled bed as function of the factor form η/λ (η and λ are respectively the height and wavelength of the bedform). These formulas express that in the presence of bedform, τ _s is a partition of the total shear stress τ _b . In contrast, when the bottom is flat, τ _s is exactly equal to τ _b . Based on in situ measurements, Li (J Geophys Res 99:791–799, 1994) has proposed a new formula of τ _s depending on u _*/η (u _* is the friction velocity, ${u_{*}=\sqrt{\tau_{b}/\rho}}$ ), but not as a function of η/λ. This formulation appears physically more realistic, but does not cover the flat bottom range. The purpose of this short note is therefore the extension of the Li’s expression to this type of bottom.     

More asymmetric tree competition brings about more evapotranspiration and less runoff from the forest ecosystems: A simulation study

The present paper reports how stand size-structure dynamics due to competition between different-sized trees affect long-term forested water balance in Japanese cool-temperate planted stands (evergreen coniferous Cryptomeria japonica and deciduous coniferous Larix kaempferi stands) using a fully coupled multi-layered meteorological surface physics—terrestrial ecosystems model. The simulation captured the well-known annual variation in leaf area index (LAI) accurately with stand age in monocultured and even-aged stands; the occurrence of maximum LAI during the early growth stage and then a gradual decline followed by a steady state after the maximum LAI. The simulations also detected a high dependency of annual evapotranspiration (AETr) on LAI with stand age that is well known by prior observational researches. In the C. japonica (shade-tolerant late-successional species) stand, the relationship between annual net primary productivity of an individual tree (NPP_ind) and individual tree mass (w) changed from linear to a convex curve during self-thinning, indicating that the degree of asymmetric tree competition intensified with forest stand development. The higher degree of competitive asymmetry characterized by the convex-shaped NPPind-w relationship produced greater size inequality, i.e., the formation of trees stratified by height. Under such conditions, AETr and annual transpiration (ATr) were mainly regulated by larger trees. On the other hand, the NPPind-w relationships in the L. kaempferi (shade-intolerant early-successional species) stand were linear throughout the simulated period, indicating the lower degree of competitive asymmetry. Under such conditions, the growth of intermediate-sized trees was enhanced and these trees became a dominant source of AETr (and also ATr) during self-thinning. Furthermore, the sensitivity analysis of the effects of ecophysiological parameters such as foliage profile (i.e., vertical distribution of leaf area density) of an individual tree (distribution pattern is described by the parameter η), the maximum carboxylation velocity (V_cmax0) and biomass allocation pattern of individual plant growth (μ₁) on AETr, ATr and annual runoff (AR_off) showed that the temporal trends of AETr, ATr, AR_off and NPPind-w relationships were completely the same as those in the control simulations. However, the NPPind-w relationship during self-thinning indicated higher degrees of competitive asymmetry when η or V_cmax0 were greater than those in the control simulation and generated greater AETr and ATr and thus smaller AR_off. We found that more asymmetric tree competition brings about greater size inequality between different-sized trees and thus more evapotranspiration and less runoff in a forest stand. Overall, our simulation approach revealed that not only LAI dynamics but also plant competition, and thus size-structure dynamics, in a forest ecosystem are essential to long-term future projections of forested water balance.     

Application of a Random Forest algorithm to predict spatial distribution of the potential yield of Ruditapes philippinarum in the Venice lagoon, Italy

We present a modelling framework that combines machine learning techniques and Geographic Information Systems to support the management of an important aquaculture species, Manila clam (Ruditapes philippinarum). We use the Venice lagoon (Italy), the first site in Europe for the production of R. philippinarum, to illustrate the potential of this modelling approach. To investigate the relationship between the yield of R. philippinarum and a set of environmental factors, we used a Random Forest (RF) algorithm. The RF model was tuned with a large data set (n = 1698) and validated by an independent data set (n = 841). Overall, the model provided good predictions of site-specific yields and the analysis of marginal effect of predictors showed substantial agreement among the modelled responses and available ecological knowledge for R. philippinarum. The most influent environmental factors for yield estimation were percentage of sand in the sediment, salinity, and water depth. Our results agree with findings from other North Adriatic lagoons. The application of the fitted RF model to continuous maps of all the environmental variables allowed estimates of the potential yield for the whole basin. Such a spatial representation enabled site-specific estimates of yield in different farming areas within the lagoon. We present a possible management application of our model by estimating the potential yield under the current farming distribution and comparing it to a proposed re-organization of the farming areas. Our analysis suggests a reduction of total yield is likely to result from the proposed re-organization.     

Application of an evolutionary algorithm to the inverse parameter estimation of an individual-based model

Inverse parameter estimation of individual-based models (IBMs) is a research area which is still in its infancy, in a context where conventional statistical methods are not well suited to confront this type of models with data. In this paper, we propose an original evolutionary algorithm which is designed for the calibration of complex IBMs, i.e. characterized by high stochasticity, parameter uncertainty and numerous non-linear interactions between parameters and model output. Our algorithm corresponds to a variant of the population-based incremental learning (PBIL) genetic algorithm, with a specific “optimal individual” operator. The method is presented in detail and applied to the individual-based model OSMOSE. The performance of the algorithm is evaluated and estimated parameters are compared with an independent manual calibration. The results show that automated and convergent methods for inverse parameter estimation are a significant improvement to existing ad hoc methods for the calibration of IBMs.     

A reliability index for models

Since the prediction of environmental changes depends heavily on mathematical models, environmental planners have a definite need for a general test that can be easily applied to determine the reliability of a model or to choose between available models. In this paper a ‘reliability index’ for a model is defined. This index is a number k ? 1 which is determined from a set x₁,x₂,…,x_n of model predictions and a corresponding set y₁,y₂,…,y_n of observations. One interpretation of the index k is that the model predictions agree with observations, in some sense, within a factor of k. The index k is defined using an intuitive, geometric approach and is justified through agreement with another index developed using statistical techniques. Examples are given for a model which predicts the atmospheric movement of radon-222 originating from a concentrated source of radium-226, such as a uranium tailings pile.     

Gravity currents in two-layer stratified media

An analytical, experimental and numerical study of boundary gravity currents propagating through a two-layer stratified ambient of finite vertical extent is presented. Gravity currents are supposed to originate from a lock-release apparatus; the (heavy) gravity current fluid is assumed to span the entire channel depth, H, at the initial instant. Our theoretical discussion considers slumping, supercritical gravity currents, i.e. those that generate an interfacial disturbance whose speed of propagation matches the front speed, and follows from the classical analysis of Benjamin (J Fluid Mech 31:209?C248, 1968). In contrast to previous investigations, we argue that the interfacial disturbance must be parameterized so that its amplitude can be straightforwardly determined from the ambient layer depths. Our parameterization is based on sensible physical arguments; its accuracy is confirmed by comparison against experimental and numerical data. More generally, measured front speeds show positive agreement with analogue model predictions, which remain strictly single-valued. From experimental and numerical observations of supercritical gravity currents, it is noted that this front speed is essentially independent of the interfacial thickness, ??, even in the limiting case where ?? = H so that the environment is comprised of a uniformly stratified ambient with no readily discernible upper or lower ambient layer. Conversely, when the gravity current is subcritical, there is a mild increase of front speed with ??. Our experiments also consider the horizontal distance, X, at which the front begins to decelerate. The variation of X with the interface thickness and the depths and densities of the ambient layers is discussed. For subcritical gravity currents, X may be as small as three lock lengths whereas with supercritical gravity currents, the gravity current may travel long distances at constant speed, particularly as the lower layer depth diminishes.     

Fish assemblage compositions after implementation of the IndVal method on the Narew River system

Indicator species index (IndVal) was used as a new method for an already published study, and allowed of a more convincing way of fish assemblages characterization in a large river system. Three sites clusters (AB, CD, EF) were distinguish using the self-organizing map (SOM, Artificial neural network algorithm) in the lowland Narew River system, which comprised the most characteristic species of the total of 36 present. AB included Pungitius pungitius, Barbatula barbatula, Gasterosteus aculeatus and Gobio gobio (natural and slightly modified sites from small rivers), EF included Leuciscus idus, Perca fluviatilis, Rutilus rutilus, Blicca bjoerkna, Esox lucius, Lota lota and Alburnus alburnus (sites from the main channel and lower courses of biggest tributaries, and CD without characteristic species (containing sites from small and large river ditches impacted by pollution, engineering and both). The IndVal method applied here gives precise and accurate information on fish species habitat preferences.     

Influences of canopy structure and physiological traits on flux partitioning between understory and overstory in an eastern Siberian boreal larch forest

Boreal forests play an important role in the global balance of energy and CO₂. Our previous study of elaborate eddy covariance observations in a Siberian boreal larch forest, conducted both above the forest canopy and at the forest floor, revealed a significant contribution of latent heat flux (LE) from the cowberry understory to the whole ecosystem LE. Thus, in the present study, we examined what factors control the partitioning of whole ecosystem LE and CO₂ flux into the understory and overstory vegetation, using detailed leaf-level physiology (for both understory and overstory vegetation) and soil respiration property measurements as well as a multilayer soil-vegetation-atmosphere transfer (SVAT) model. The modeling results showed that the larch overstory's leaf area index (LAI) and vertical profile of leaf photosynthetic capacity were major factors determining the flux partitioning in this boreal forest ecosystem. This is unlike other forest ecosystems that tend to have dense LAI. We concluded that control of the larch overstory's LAI had a relationship with both the coexistence of the larch with the cowberry understory and with the water resources available to the total forest ecosystem.     

Analysis of behavioral changes of zebrafish (Danio rerio) in response to formaldehyde using Self-organizing map and a hidden Markov model

Two computational methods were applied to classification of movement patterns of zebrafish (Danio rerio) to elucidate Markov processes in behavioral changes before and after treatment of formaldehyde (0.1 mg/L) in semi-natural conditions. The complex data of the movement tracks were initially classified by the Self-organizing map (SOM) to present different behavioral states of test individuals. Transition probabilities between behavioral states were further evaluated to fit Markov processes by using the hidden Markov model (HMM). Emission transition probability was also obtained from the observed variables (i.e., speed) for training with the HMM. Experimental transition and emission probability matrices were successfully estimated with the HMM for recognizing sequences of behavioral states with accuracy rates in acceptable ranges at central and boundary zones before (77.3-81.2%) and after (70.1-76.5%) treatment. A heuristic algorithm and a Markov model were efficiently combined to analyze movement patterns and could be a means of in situ behavioral monitoring tool.     

Effects of seawater pCO2 and temperature on shell growth, shell stability, condition and cellular stress of Western Baltic Sea Mytilus edulis (L.) and Arctica islandica (L.)

Acidification of the World’s oceans may directly impact reproduction, performance and shell formation of marine calcifying organisms. In addition, since shell production is costly and stress in general draws on an organism’s energy budget, shell growth and stability of bivalves should indirectly be affected by environmental stress. The aim of this study was to investigate whether a combination of warming and acidification leads to increased physiological stress (lipofuscin accumulation and mortality) and affects the performance [shell growth, shell breaking force, condition index (C_i)] of young Mytilus edulis and Arctica islandica from the Baltic Sea. We cultured the bivalves in a fully-crossed 2-factorial experimental setup (seawater (sw) pCO₂ levels “low”, “medium” and “high” for both species, temperature levels 7.5, 10, 16, 20 and 25 °C for M. edulis and 7.5, 10 and 16 °C for A. islandica) for 13 weeks in summer. Mytilus edulis and A. islandica appeared to tolerate wide ranges of sw temperature and pCO₂. Lipofuscin accumulation of M. edulis increased with temperature while the C_i decreased, but shell growth of the mussels only sharply decreased while its mortality increased between 20 and 25 °C. In A. islandica, lipofuscin accumulation increased with temperature, whereas the C_i, shell growth and shell breaking force decreased. The pCO₂ treatment had only marginal effects on the measured parameters of both bivalve species. Shell growth of both bivalve species was not impaired by under-saturation of the sea water with respect to aragonite and calcite. Furthermore, independently of water temperatures shell breaking force of both species and shell growth of A. islandica remained unaffected by the applied elevated sw pCO₂ for several months. Only at the highest temperature (25 °C), growth arrest of M. edulis was recorded at the high sw pCO₂ treatment and the C_i of M. edulis was slightly higher at the medium sw pCO₂ treatment than at the low and high sw pCO₂ treatments. The only effect of elevated sw pCO₂ on A. islandica was an increase in lipofuscin accumulation at the high sw pCO₂ treatment compared to the medium sw pCO₂ treatment. Our results show that, despite this robustness, growth of both M. edulis and A. islandica can be reduced if sw temperatures remain high for several weeks in summer. As large body size constitutes an escape from crab and sea star predation, this can make bivalves presumably more vulnerable to predation—with possible negative consequences on population growth. In M. edulis, but not in A. islandica, this effect is amplified by elevated sw pCO₂. We follow that combined effects of elevated sw pCO₂ and ocean warming might cause shifts in future Western Baltic Sea community structures and ecosystem services; however, only if predators or other interacting species do not suffer as strong from these stressors.     

Changes of microbial composition during wastewater reclamation and distribution systems revealed by high-throughput sequencing analyses

This study employed 454-pyrosequencing to investigate microbial and pathogenic communities in two wastewater reclamation and distribution systems. A total of 11972 effective 16S rRNA sequences were acquired from these two reclamation systems, and then designated to relevant taxonomic ranks by using RDP classifier. The Chao index and Shannon diversity index showed that the diversities of microbial communities decreased along wastewater reclamation processes. Proteobacteria was the most dominant phylum in reclaimed water after disinfection, which accounted for 83% and 88% in two systems, respectively. Human opportunistic pathogens, including Clostridium, Escherichia, Shigella, Pseudomonas and Mycobacterium, were selected and enriched by disinfection processes. The total chlorine and nutrients (TOC, NH₃-N and NO₃-N) significantly affected the microbial and pathogenic communities during reclaimed water storage and distribution processes. Our results indicated that the disinfectant-resistant pathogens should be controlled in reclaimed water, since the increases in relative abundances of pathogenic bacteria after disinfection implicate the potential public health associated with reclaimed water.     

Tough as a rock-boring urchin: adult Echinometra sp. EE from the Red Sea show high resistance to ocean acidification over long-term exposures

Ocean acidification, a process caused by the continuous rise of atmospheric CO₂ levels, is expected to have a profound impact on marine invertebrates. Findings of the numerous studies conducted in this field indicate high variability in species responses to future ocean conditions. This study aimed at understanding the effects of long-term exposure to elevated pCO₂ conditions on the performance of adult Echinometra sp. EE from the Gulf of Aqaba (Red Sea). During an 11-month incubation under high pCO₂ (1,433 μatm, pH_NBS 7.7) and control (435 μatm, pH_NBS 8.1) conditions, we examined the urchins’ somatic and gonadal growth, gametogenesis and skeletal microstructure. Somatic and gonadal growths were exhibited with no significant differences between the treatments. In addition, all urchins in the experiment completed a full reproductive cycle, typical of natural populations, with no detectable impact of increased pCO₂ on the timing, duration or progression of the cycle. Furthermore, scanning electron microscopy imaging of urchin tests and spines revealed no signs of the usual observed effects of acidosis, such as skeletal dissolution, widened stereom pores or non-smoothed structures. Our results, which yielded no significant impact of the high pCO₂ treatment on any of the examined processes in the urchins studied, suggest high resistance of adult Echinometra sp. EE to near future ocean acidification conditions. With respect to other findings in this area, the outcome of this study provides an example of the complicated and diverse responses of echinoids to the predicted environmental changes.     

In situ thermal dynamics of shallow water corals is affected by tidal patterns and irradiance

We studied the diel variation of in situ coral temperature, irradiance and photosynthetic performance of hemispherical colonies of Porites lobata and branching colonies of Porites cylindrica during different bulk water temperature and tidal scenarios on the shallow reef flat of Heron Island, Great Barrier Reef, Australia. Our study presents in situ evidence that coral tissue surface temperatures can exceed that of the surrounding water under environmental conditions typically occurring during low tide in shallow reef or lagoon environments. Such heating may be a regular occurrence on shallow reef flats, triggered by the combined effects of high irradiance and low water flow characteristic of low Spring tides. At these times, solar heating of corals coincides with times of maximum water temperature and high irradiance, where the slow flow and consequent thick boundary layers impede heat exchange between corals and the surrounding water. Despite similar light-absorbing properties, the heating effect was more pronounced for the hemispherical P. lobata than for the branching P. cylindrica. This is consistent with previous laboratory experiments showing the evidence of interspecific variation in coral thermal environment and may result from morphologically influenced variation in convective heat transfer and/or thermal properties of the skeleton. Maximum coral surface warming did not coincide with maximum irradiance, but with maximum water temperature, well into the low-tide period with extremely low water flow in the partially drained reef flat, just prior to flushing by the rising tide. The timing of low tide thus influences the thermal exposure and photophysiological performance of corals, and the timing of tidally driven coral surface warming could potentially have different physiological impacts in the morning or in the afternoon.     

Elucidation of ecosystem attributes of an oligotrophic lake in Hokkaido,Japan, using Ecopath with Ecosim (EwE)

The fishing practices in the oligotrophic Lake Toya, Hokkaido, Japan, have profound implications in the ecosystem sustainability. The status of the sockeye salmon (Oncorhynchus nerka) population has become a serious concern among the lake managers and policy makers during the last decades. While the decline of the sockeye salmon population has been well documented in Lake Toya, there is considerable uncertainty with regards to the impact on the broader system dynamics. In this study, our objective is to address this knowledge gap by undertaking a synthesis of the Lake Toya food web using the mass-balance modeling software Ecopath with Ecosim (EwE). Our primary research question is to examine the repercussions of the declining sockeye salmon population on the trophic dynamics of the lake. Namely, we assess if there are any competing species that might have benefited from the decrease of sockeye salmon standing biomass and to what extent do these changes propagate through the Lake Toya food web? Our analysis pinpoints the critical role of the Japanese smelt (Hypomesus transpacificus nipponensis) in the system, which demonstrates a wide range of effects on several functional groups at both higher and lower trophic levels in Lake Toya. In particular, being a substantial portion of the masu salmon (Oncorhynchus masou) and adult sockeye salmon diets, the Japanese smelt has a positive impact on the top predators of the system. Amphipods, insects, and shrimp strongly benefit from the autochthonous and allochthonous organic matter in the system, while the tight coupling between phytoplankton and zooplankton seems to be particularly critical for the integrity of the Lake Toya food web. Whereas the values of the different ecosystem attributes (e.g., primary production/biomass, biomass/total throughput, system omnivory index, amount of recycled throughput, Finn's cycling index) provide evidence that Lake Toya is an immature system, we note that the internal redundancy and the system overhead estimates suggest that the lake possesses substantial reserves to overcome external perturbations. We also examined the effects of a variety of fishing policies on the biomass of masu salmon and adult sockeye salmon, which verify the belief that the adult sockeye population is quite fragile with high likelihood to collapse. Our analysis also predicts that sockeye will not rebound unless the fishing pressure exerted is substantially reduced (>50% of the reference levels used). Masu salmon seems to benefit under all the scenarios examined indicating that the intensity of the current fishing activities is significantly lower than its biomass accumulation rate in the system.     

Formation of tidal starting-jet vortices through idealized barotropic inlets with finite length

This paper presents a surface particle image velocimetry study to investigate the dynamics of shallow starting-jet dipoles formed by tidal flow through inlets and their interaction with vorticity formed at the inlet channel lateral boundaries. Vortical structure in the flow field is identified using a local swirl strength criterion evaluated from the two-dimensional flow field. The starting jet dipole vortices and vortices formed as the lateral boundary layers are expelled during flow reversal are characterized by their trajectory, size, and circulation. Using these quantities, a model is developed to predict the size and strength of the expelled lateral boundary layer vortices based on the inlet velocity, channel length, and width of the lateral boundary layer. The expelled boundary layer vortices are found to disrupt the formation of the primary tidal jet dipole through two mechanisms. First, because the boundary layer vortices themselves form a dipole with each half of the starting-jet dipole, the starting-jet vortices are pulled apart and advected away from the inlet mouth early in the tidal cycle, resulting in a reduction in the spin-up time and the amount of vorticity input during starting-jet vortex formation. Second, the advection of each dipole away from the inlet disconnects each starting-jet vortex from the starting jet; hence, the vortices are not fed by fluid in the jet or energized by shear in the jet boundary layers. These influences of the lateral boundary layer on the starting-jet vortices’ formation and propagation are found to be a function of the channel length L, maximum velocity U, and tidal period T, resulting in a predictive value to characterize their trajectory, strength, and evolution.     

Sensitivity of modelled gross primary productivity to topographic effects on surface radiation: A case study in the Cotter River Catchment, Australia

Gross primary productivity (GPP) is a critical response variable for many environmental problems, including terrestrial carbon accounting and the calculation of catchment water balances. Various approaches for modelling GPP have been developed and applied at continental and landscapes scales, but little attention has been given to the sensitivity of GPP to the spatial scale of its driving variables. A key driving variable is surface radiation (R_s) which is influenced by both meso-scale factors (latitude, time of year, cloudiness) and the topographic variables of slope, aspect and horizon shading. We compared the sensitivity of modelled GPP to two different sources of surface radiation (R_s): (1) the ANUCLIM method which only captures the meso-scaled factors; and (2) the SRAD method which incorporates the topographic effects GPP was calculated using the radiation use efficiency (RUE) model (Roderick et al., 2001) to discern general patterns of vegetation productivity at a sub-catchment (i.e. sub-water shed) scale. The radiation use efficiency approach uses the normalized difference vegetation index (NDVI) derived from satellite data (MODIS TERRA), along with estimates of solar radiation at the top of the atmosphere (R_o) and canopy (R_s). In this approach, R_o and R_s capture the influence of diffuse irradiance in canopy photosynthesis and vegetation productivity respectively. This research showed that R_s calculated using the SRAD program provides important discrimination of GPP regimes at a sub-catchment scale, as the result of minimum and maximum daily radiation varying between shaded and exposed surfaces. However, mean daily radiation at a whole-of-catchment scale did not differ between the two sources as the differences in the minimum and maximum daily values tend to cancel each other out. Applications of GPP models therefore need to consider whether topographic factors are important and select the appropriate source of R_s values. GPP models should also reflect understanding of radiation use efficiency. However, further research is required especially with respect to the influence of water stress on plant response.     

Estimating animal densities and home range in regions with irregular boundaries and holes: A lattice-based alternative to the kernel density estimator

Density estimates based on point processes are often restrained to regions with irregular boundaries or holes. We propose a density estimator, the lattice-based density estimator, which produces reasonable density estimates under these circumstances. The estimation process starts with overlaying the region with nodes, linking these together in a lattice and then computing the density of random walks of length k on the lattice. We use an approximation to the unbiased crossvalidation criterion to find the optimal walk length k. The technique is illustrated using walleye (Sander vitreus) radiotelemetry relocations in Lake Monroe, Indiana. We also use simulation to compare the technique to the traditional kernel density estimate in the situation where there are no significant boundary effects.