Pattern formation in morphogenesis

We first treat the Gierer-Meinhardt equations by linear stability analysis to determine the critical parameter, at which the homogeneous distributions of activator and inhibitor concentrations become unstable. We find two types of instabilities: one leading to spatial pattern formation and another one leading to temporal oscillations. We consider the case where two instabilities are present. Using the method of generalized Ginzburg-Landau equations introduced earlier we then analyze the nonlinear equations. As we are mainly interested in spatial pattern formation on a sphere we consider the problem under an appropriate constraint. Combining the two occurring solutions we find patterns well-known in biology, such as a gradient system and temporal oscillations.     

Temporal encoding in a nervous system

We examined the extent to which temporal encoding may be implemented by single neurons in the cercal sensory system of the house cricket Acheta domesticus. We found that these neurons exhibit a greater-than-expected coding capacity, due in part to an increased precision in brief patterns of action potentials. We developed linear and non-linear models for decoding the activity of these neurons. We found that the stimuli associated with short-interval patterns of spikes (ISIs of 8 ms or less) could be predicted better by second-order models as compared to linear models. Finally, we characterized the difference between these linear and second-order models in a low-dimensional subspace, and showed that modification of the linear models along only a few dimensions improved their predictive power to parity with the second order models. Together these results show that single neurons are capable of using temporal patterns of spikes as fundamental symbols in their neural code, and that they communicate specific stimulus distributions to subsequent neural structures.     

Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis

Summary We first perform a linear stability analysis of the Gierer-Meinhardt model to determine the critical parameters where the homogeneous distribution of activator and inhibitor concentrations becomes unstable. There are two kinds of instabilities, namely, one leading to spatial patterns and another one leading to temporal oscillations. Focussing our attention on spatial pattern formation we solve the corresponding nonlinear equations by means of our previously introduced method of generalized Ginzburg-Landau equations. We explicitly consider the two-dimensional case and find both rolls and hexagon-like structures. The impact of different boundary conditions on the resulting patterns is also discussed. The occurrence of the new patterns has all the features of nonequilibrium phase transitions.     

Simple temporal models for ecological systems with complex spatial patterns

Spatial patterns are ubiquitous in nature. Because these patterns modify the temporal dynamics and stability properties of population densities at a range of spatial scales, their effects must be incorporated in temporal ecological models that do not represent space explicitly. We demonstrate a connection between a simple parameterization of spatial effects and the geometry of clusters in an individual‐based predator–prey model that is both nonlinear and stochastic. Specifically we show that clusters exhibit a power‐law scaling of perimeter to area with an exponent close to unity. In systems with a high degree of patchiness, similar power‐law scalings can provide a basis for applying simple temporal models that assume well‐mixed conditions.     

Pattern analysis in successional communities – An approach for studying shifts in ecological interactions

Abstract. Many ecological studies have addressed issues of vegetation spatial patterns in attempts to understand the processes generating them. We investigated changes in ecological processes during succession via the analysis of shrubs’ spatial patterns in a system of linear sand dunes, an arid ecosystem located in the Negev Desert in Israel during three consecutive years. We hypothesized that spatial patterns change from clustered to regular as succession progresses due to changes in the relative importance of facilitation and competition in this environment. In this ecosystem communities of early successional stages are frequently disturbed by high rates of sand movement, whereas in later successional stages sand stability is high. We mapped in the field individual shrubs on high‐resolution aerial photographs, and converted the digital images to a GIS data set. Using Ripley's K‐function we analysed spatial patterns at three levels: the single‐species level, among species and at the individual level, in three communities characterizing different successional stages. In the early successional communities we found clustered spatial patterns, in comparison with stable habitats where spatial patterns tended to be regular. We argue that these shifts in spatial patterns are indicative of the assumption that in this sand‐dune system ecological interactions change from facilitation to competition as succession progresses. Further, we argue that these interactions operate in different spatial scales at the different successional stages, and that the study of these processes should be conducted at the spatial scales specific to each community.     

Extrinsically and intrinsically generated spatial patterns of algal abundance in an experimental stream

Spatial heterogeneity in species abundance arises from both extrinsic (largely abiotic) and intrinsic (largely biotic) processes. The relative importance of these two types of processes can vary across ecological systems and across temporal and spatial scales. Numerous empirical studies have explored spatial patterns resulting from extrinsic and intrinsic processes, however the interaction of these two types of processes can result in complex patterns that are difficult to test. We used a unique model system consisting of periphytic algae grown on agar in an experimental stream to manipulate an extrinsic and an intrinsic process. We manipulated an extrinsic process by varying the spatial arrangement of nutrients creating both heterogeneous and homogeneous environments for the algae. We manipulated an intrinsic process by introducing a snail herbivore to the system. The resulting spatial algal patterns showed that both types of processes were important in producing spatial abundance patterns and that the patterns occurred at two distinct spatial scales in our system. At the scale of the imposed nutrient heterogeneity, algae “tracked” the differences in nutrient supply rates. The snail herbivores both reduced and promoted spatial patterns in algal abundance at different spatial scales reflecting their species-specific foraging behavior. An ability to detect differences in algal abundance allowed the snails to reduce the power of patterns at the scale of the imposed nutrient heterogeneity; however below a spatial scale of approximately 30 mm the snails could no longer detect differences in algal abundance and so foraged randomly. At this spatial scale the spatial heterogeneity in algal abundance increased and the resulting algal patterns were relatively spatially fixed through time. We suggest that this relative constancy may arise in part from a detected weak Allee effect in algal growth rates.     

Quantification of the spatial aspect of chaotic dynamics in biological and chemical systems

The need to study spatio-temporal chaos in a spatially extended dynamical system which exhibits not only irregular, initial-value sensitive temporal behavior but also the formation of irregular spatial patterns, has increasingly been recognized in biological science. While the temporal aspect of chaotic dynamics is usually characterized by the dominant Lyapunov exponent, the spatial aspect can be quantified by the correlation length. In this paper, using the diffusion-reaction model of population dynamics and considering the conditions of the system stability with respect to small heterogeneous perturbations, we derive an analytical formula for an ‘intrinsic length’ which appears to be in a very good agreement with the value of the correlation length of the system. Using this formula and numerical simulations, we analyze the dependence of the correlation length on the system parameters. We show that our findings may lead to a new understanding of some well-known experimental and field data as well as affect the choice of an adequate model of chaotic dynamics in biological and chemical systems.     

Biodiversity–productivity relationship in ponds: Community and metacommunity patterns along time and environmental gradients

Primary production correlates with diversity in various ways. These patterns may result from the interaction of various mechanisms related to the environmental context and the spatial and temporal scale of analysis. However, empirical evidence on diversity‐productivity patterns typically considers single temporal and spatial scales, and does not include the effect of environmental variables. In a metacommunity of macrophytes in ephemeral ponds, we analysed the diversity‐productivity relationship patterns in the field, the importance of the environmental variables of pond size and heterogeneity on such relationship, and the variation of these patterns at local (community level) and landscape scales (metacommunity level) across 52 ponds on twelve occasions, over five years (2005–2009). Combining all sampling dates, there were 377 ponds and 1954 sample‐unit observations. Vegetation biomass was used as a proxy for productivity, and biodiversity was represented by species richness, evenness, and their interaction. Environmental variables comprised pond area, depth and internal heterogeneity. Productivity and species richness were not directly related at the metacommunity level, and were positively related at the community level. Taking environmental variables into account revealed positive species richness‐productivity relationships at the metacommunity level and positive quadratic relationships at the community level. Productivity showed both positive and negative linear and nonlinear relationships with the size and heterogeneity of ponds. We found a weak relationship between productivity and evenness. The identity of variables associated with productivity changed between spatial scales and through time. The pattern of relationships between productivity and diversity depends on spatial scale and environmental context, and changes idiosyncratically through time within the same ecosystem. Thus, the diversity‐productivity relationship is not only a property of the study system, but also a consequence of environmental variations and the temporal and spatial scale of analysis.     

Temporal variation in phytoplankton beta diversity patterns and metacommunity structures across subtropical reservoirs

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High-content behavioral analysis of Caenorhabditis elegans in precise spatiotemporal chemical environments

To quantitatively understand chemosensory behaviors, it is desirable to present many animals with repeatable, well-defined chemical stimuli. To that end, we describe a microfluidic system to analyze Caenorhabditis elegans behavior in defined temporal and spatial stimulus patterns. A 2 cm × 2 cm structured arena allowed C. elegans to perform crawling locomotion in a controlled liquid environment. We characterized behavioral responses to attractive odors with three stimulus patterns: temporal pulses, spatial stripes and a linear concentration gradient, all delivered in the fluid phase to eliminate variability associated with air-fluid transitions. Different stimulus configurations preferentially revealed turning dynamics in a biased random walk, directed orientation into an odor stripe and speed regulation by odor. We identified both expected and unexpected responses in wild-type worms and sensory mutants by quantifying dozens of behavioral parameters. The devices are inexpensive, easy to fabricate, reusable and suitable for delivering any liquid-borne stimulus.     

Multiscale responses of soil stability and invasive plants to removal of non-native grazers from an arid conservation reserve

Disturbances and ecosystem recovery from disturbance both involve numerous processes that operate on multiple spatial and temporal scales. Few studies have investigated how gradients of disturbance intensity and ecosystem responses are distributed across multiple spatial resolutions and also how this relationship changes through time during recovery. We investigated how cover of non-native species and soil-aggregate stability (a measure of vulnerability to erosion by water) in surface and subsurface soils varied spatially during grazing by burros and cattle and whether patterns in these variables changed after grazer removal from Mojave National Preserve, California, USA. We compared distance from water and number of ungulate defecations — metrics of longer-term and recent grazing intensity, respectively, — as predictors of our response variables. We used information-theoretic analyses to compare hierarchical linear models that accounted for important covariates and allowed for interannual variation in the disturbance–response relationship at local and landscape scales. Soil stability was greater under perennial vegetation than in bare interspaces, and surface soil stability decreased with increasing numbers of ungulate defecations. Stability of surface samples was more affected by time since removal of grazers than was stability of subsurface samples, and subsurface soil stability in bare spaces was not related to grazing intensity, time since removal, or any of our other predictors. In the high rainfall year (2003) after cattle had been removed for 1–2 years, cover of all non-native plants averaged nine times higher than in the low-rainfall year (2002). Given the heterogeneity in distribution of large-herbivore impacts that we observed at several resolutions, hierarchical analyses provided a more complete understanding of the spatial and temporal complexities of disturbance and recovery processes in arid ecosystems.     

Pattern formation in a spatial plant-wrack model with tide effect on the wrack

Spatial patterns are a subfield of spatial ecology, and these patterns modify the temporal dynamics and stability properties of population densities at a range of spatial scales. Localized ecological interactions can generate striking large-scale spatial patterns in ecosystems through spatial self-organization. Possible mechanisms include oscillating consumer–resource interactions, localized disturbance–recovery processes, and scale-dependent feedback. However, in this paper, our main aim is to study the effect of tide on the pattern formation of a spatial plant-wrack model. We discuss the changes of the wavelength, wave speed, and the conditions of the spatial pattern formation, according to the dispersion relation formula. Both the mathematical analysis and numerical simulations reveal that the tide has great influence on the spatial pattern. More specifically, typical traveling spatial patterns can be obtained. Our obtained results are consistent with the previous observation that wracks exhibit traveling patterns, which is useful to help us better understand the dynamics of the real ecosystems.     

Schedules of reinforcement as regulators of dispersion patterns

An eight-rat eight-station operant conditioning arena was used to study the spatial structure and temporal stability of foraging dispersion patterns. Food was obtained by bar pressing as the population was exposed to an ascending series of the fixed and variable aspects of ratio and interval schedules of reinforcement. Dispersion patterns, defined by the number of rats simultaneously foraging at each of the eight stations, and the temporal changes in these patterns, were the dependent variables. Both variables exhibited a unique relationship to each schedule type and value. The absence of such relationships when either food supply or response costs were examined suggests that these factors were not the determinants of spatio-temporal structure. An account is provided of how schedules may interact with behavioral foraging chains to explain dispersion patterns.     

Bet‐hedging as a complex interaction among developmental instability,environmental heterogeneity,dispersal, and life‐history strategy

One potential evolutionary response to environmental heterogeneity is the production of randomly variable offspring through developmental instability, a type of bet‐hedging. I used an individual‐based, genetically explicit model to examine the evolution of developmental instability. The model considered both temporal and spatial heterogeneity alone and in combination, the effect of migration pattern (stepping stone vs. island), and life‐history strategy. I confirmed that temporal heterogeneity alone requires a threshold amount of variation to select for a substantial amount of developmental instability. For spatial heterogeneity only, the response to selection on developmental instability depended on the life‐history strategy and the form and pattern of dispersal with the greatest response for island migration when selection occurred before dispersal. Both spatial and temporal variation alone select for similar amounts of instability, but in combination resulted in substantially more instability than either alone. Local adaptation traded off against bet‐hedging, but not in a simple linear fashion. I found higher‐order interactions between life‐history patterns, dispersal rates, dispersal patterns, and environmental heterogeneity that are not explainable by simple intuition. We need additional modeling efforts to understand these interactions and empirical tests that explicitly account for all of these factors.     

Predicting the occurrence of persistent hotspots in ecosystem variables

Ecological resources and services (e.g. organisms, nutrient cycling) are distributed heterogeneously across landscapes. While spatial variation has been studied extensively, the pattern of hotspots and coolspots persisting over time – called persistent spatial variation (PSV) – has not. Yet this pattern imparts key information to managers about whether resources will be found consistently in certain locations or vary unpredictably. Anticipating whether an ecosystem variable will display PSV is thus a valuable prospect. We tested the ability of attributes of variables (e.g. niche breadth, abundance, temporal scale) to predict the occurrence of PSV. Using a new measure of PSV based on the F‐value of analysis of variance, we were able to 1) decompose the pattern of persistent hotspots into spatial and temporal components – ‘spatial variation’ of site mean values and ‘stability’ of time series at each site – and 2) identify predictors of these patterns in temperate lakes and tropical coastal rock pools. We found PSV to be highly predictable (R2 = up to 0.80) from an estimate of stability taken at a single site, as well as from other factors related to stability. These factors included whether the variable was environmental (stable, slow) or was an aggregate of other variables (stabilized by statistical averaging). Species properties like niche position and abundance were modest predictors because they correlated with PSV components of site occupancy, spatial variation and stability. We conclude that PSV and the distribution of resources in space and time might be predicted from simple temporal indicators (e.g. stability at a single location) when data are scarce.     

Mismatches between birds' spatial and temporal dynamics reflect their delayed response to global changes

Global changes alter the dynamics of biodiversity, and are forecasted to continue or worsen in the decades to come. Modelling approaches used to anticipate these impacts are mainly based on the equivalence between spatial and temporal response to environmental forcings, generally called space-for-time substitution. However, several processes are known to generate deviations between spatial and temporal responses, potentially undermining the prediction based on space-for-time substitution. We here used high-resolution data from the french breeding bird survey to quantify and map the deviation between spatial and temporal patterns of bird abundances resulting from the dynamics of 124 species monitored in 2133 sites between 2001 and 2012. Using independent empirical data, we then tested specific predictions linked to the determinants (anthropogenic activities) and processes (lagged responses to environmental changes) potentially generating these deviations. We found that deviations between spatial and temporal patterns of abundances were particularly structured in space for bird communities. Following our predictions, these space–time deviations were positively correlated with the human influence on ecosystems, and linked with colonization–extinction ratios and community completeness, two markers of ongoing delayed responses to environmental changes. Our results suggest that the deviations between spatial and temporal patterns are related to recent anthropogenic environmental changes and disequilibrium responses to these changes. Investigating deviations between spatial and temporal patterns of biodiversity might open promising perspectives for a formal quantification of disequilibrium state of biodiversity at large spatial scale.     

The Influence of Climate, Soils, Weather, and Land Use on Primary Production and Biomass Seasonality in the US Great Plains

Identifying the conditions and mechanisms that control ecosystem processes, such as net primary production, is a central goal of ecosystem ecology. Ideas have ranged from single limiting-resource theories to colimitation by nutrients and climate, to simulation models with edaphic, climatic, and competitive controls. Although some investigators have begun to consider the influence of land-use practices, especially cropping, few studies have quantified the impact of cropping at large scales relative to other known controls over ecosystem processes. We used a 9-year record of productivity, biomass seasonality, climate, weather, soil conditions, and cropping in the US Great Plains to quantify the controls over spatial and temporal patterns of net primary production and to estimate sensitivity to specific driving variables. We considered climate, soil conditions, and long-term average cropping as controls over spatial patterns, while weather and interannual cropping variations were used as controls over temporal variability. We found that variation in primary production is primarily spatial, whereas variation in seasonality is more evenly split between spatial and temporal components. Our statistical (multiple linear regression) models explained more of the variation in the amount of primary production than in its seasonality, and more of the spatial than the temporal patterns. Our results indicate that although climate is the most important variable for explaining spatial patterns, cropping explains a substantial amount of the residual variability. Soil texture and depth contributed very little to our models of spatial variability. Weather and cropping deviation both made modest contributions to the models of temporal variability. These results suggest that the controls over seasonality and temporal variation are not well understood. Our sensitivity analysis indicates that production is more sensitive to climate than to weather and that it is very sensitive to cropping intensity. In addition to identifying potential gaps in out knowledge, these results provide insight into the probable long- and short-term ecosystem response to changes in climate, weather, and cropping.     

Traveling wave solutions in a plant population model with a seed bank

We propose an integro-difference equation model to predict the spatial spread of a plant population with a seed bank. The formulation of the model consists of a nonmonotone convolution integral operator describing the recruitment and seed dispersal and a linear contraction operator addressing the effect of the seed bank. The recursion operator of the model is noncompact, which poses a challenge to establishing the existence of traveling wave solutions. We show that the model has a spreading speed, and prove that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions by using an asymptotic fixed point theorem. Our numerical simulations show that the seed bank has the stabilizing effect on the spatial patterns of traveling wave solutions.     

Disentangling spatio‐temporal processes in a hierarchical system: a case study in fisheries discards

In the last decade, various spatial and temporal methodologies were developed to investigate the processes that drive ecological and evolutionary patterns. However, these methods frequently fail to acknowledge that the observed patterns result from the overlap of different underlying processes. In order to understand how the patterns are formed, we must have recourse to methods that allow us to disentangle these simultaneous processes. Here we develop a hierarchical spatial predictive process (PP) combined with a separable temporal PP to disentangle and describe those overlapping processes in one very frequent setting in ecology and evolution: multilevel spatio‐temporally indexed data. We present our methodology through a case study of fisheries discards and investigate for example whether the inclusion of the hierarchical structure and the temporal processes of the system alter the observed spatial patterns. Recently it is recognized that understanding the processes driving discards is essential to sustainably manage and conserve marine resources. The results show that consideration of multiple underlying processes dramatically changes the pattern and characteristics of the discards hot‐ and coldspots. In the Irish Sea, the inclusion of the hierarchical structure of the system leads to the reduction of the hot‐ and coldspots. Simultaneously, our model identifies key bi‐annual fluctuations in the temporal process which, together with the variance associated at the level of individual fishing trips in the hierarchical structure of the data explained most of the variance driving discards. Whether the hierarchical, spatial and temporal processes are considered together or not can profoundly alter our understanding of what constitutes an appropriate mitigation measure. Misidentification of hotspots can culminate in inappropriate mitigation practices which can sometimes be irreversible. As the proposed method offers a unified approach for understanding the processes that drive observed patterns, many areas in ecology such as conservation and epidemiological studies can benefit from its use, increasing the effectiveness of management plans.