Pattern formation of a predator-prey system with Ivlev-type functional response

In this paper, we investigate the spatial pattern formation of a predator-prey system with prey-dependent functional response Ivlev-type and reaction-diffusion. The Hopf bifurcation of the model is discussed, and the sufficient conditions for the Turing instability with zero-flux boundary conditions are obtained. Based on this, we perform the spiral and the chaotic spiral patterns via numerical simulation, i.e., the evolution process of the system with the initial conditions which was small amplitude random perturbation around the steady state. For the sake of learning the pattern formation of the model further, we perform three categories of unsymmetric initial condition, and find that with these special initial conditions the system can emerge not only spiral pattern but also target pattern and so on, and the effect of these special conditions on the formation of spatial patterns is less and less with more and more iterations but the effect does not decay forever. This indicates that for prey-dependent type predator-prey system, pattern formations do depend on the initial conditions, while for predator-dependent type they do not.     

Neural network modelling for the analysis of forcings/temperatures relationships at different scales in the climate system

A fully non-linear analysis of forcing influences on temperatures is performed in the climate system by means of neural network modelling. Two case studies are investigated, in order to establish the main factors that drove the temperature behaviour at both global and regional scales in the last 140 years. In particular, our neural network model shows the ability to catch non-linear relationships among these variables and to reconstruct temperature records with a high degree of accuracy. In this framework, we clearly show the need of including anthropogenic inputs for explaining the temperature behaviour at global scale and recognise the role of El Niño southern oscillation for catching the inter-annual variability of temperature data. Furthermore, we analyse the relative influence of global forcing and a regional circulation pattern in determining the winter temperatures in Central England, showing that the North Atlantic oscillation represents the driven element in this case study. Our modelling activity and results can be very useful for simple assessments of relationships in the complex climate system and for identifying the fundamental elements leading to a successful downscaling of atmosphere–ocean general circulation models.     

Pattern formation and individual-based models: The importance of understanding individual-based movement

This paper demonstrates that while pattern formation can stabilize individual-based models of predator–prey systems, the same individual-based models also allow for stabilization by alternate mechanisms, particularly localized consumption or diffusion limitation. The movement rules of the simulation are the critical feature which determines which of these mechanisms stabilizes any particular predator–prey individual-based model. In particular, systems from well-connected subpopulations, in each of which a predator can attack any prey, generally exhibit stabilization by pattern formation. In contrast, when restricted movement within a (sub-)population limits the ability of predators to consume prey, localized consumption or diffusion limitation can stabilize the system. Thus while the conclusions from differential equations on the role of pattern formation for stability may apply to discrete and noisy systems, it will take a detailed understanding of movement and scales of interaction to examine the role of pattern formation in real systems. Additionally, it will be important to link an understanding of both foraging and inter-patch movement, since by analogy to the models, both would be critical for understanding how real systems are stabilized by being discrete and spatial.     

Idealized tracer transport models with time-varying transport: applications to ocean boundary currents

One-dimensional advection–diffusion and advection–diffusion–dilution (or “leaky-pipe”) models have been widely used to interpret a variety of geophysical phenomena. For example, in the ocean these tools have been used to interpret the penetration and spreading of tracers such as Chlorofluorocarbons(CFCs) along the Deep Western boundary current (DWBC). Usually, the transport coefficients of such models are taken to be constant in time, thus assuming the transport to be in steady state. Here, we relax this assumption and calculate tracer-signal variability in two simple 1D models for the boundary current having low-amplitude time-varying coefficients. Given a background tracer gradient due, for example, to a steady-state source in a boundary region, the resulting tracer field exhibits fluctuations due to the transport acting on the gradients. We compare the transport-induced tracer fluctuations to propagated fluctuations occurring in steady-state models with a periodic source in the boundary region. Using coefficients fitted to DWBC tracer observations, we find that in the North Atlantic propagated tracer fluctuations are larger, while in the sub-tropics transport-induced fluctuations dominate. This contrasts a common view that subtropical and tropical DWBC fluctuations in tracers such as CFCs, temperature and salinity anomalies are propagated signals from the northern formation region. However, the predicted transport-induced fluctuations in these models are still smaller than the observed fluctuations.     

A three-trophic level estuarine model: Synergism of two mechanistic simulations

Few numerical simulations have attempted to include a high degree of biological detail for several trophic levels. Typically, in planktonic ecosystem models, if the dynamics of nutrients, phytoplankton and herbivorous zooplankton are formulated with ecological complexity, then carnivores are ignored, forced or modeled in an extremely simplified manner. Extensive mechanistic detail for important carnivores is difficult to represent because reliable and relevant ecological data are rarely available for appropriate species and local populations. Further, the wide temporal and spatial differences between life histories of lower plankton and carnivores may be technically difficult to model.In Narragansett Bay, Rhode Island, the ctenophore Mnemiopsis leidyi is an important carnivore to which these objections do not apply. A detailed carbon-based simulation model of this population of ctenophores was developed independently from an ecosystem model of Narragansett Bay which included detailed interactions between phytoplankton, primarily herbivorous zooplankton and nutrients. The interfacing of these two models without changing any of the formulations or values of the coefficients provided a test of the commonly used practice of forcing certain components. Both models were originally constructed with the biomass of a critical compartment forced according to observed data; in the plankton model, ctenophores were forced, and in the ctenophore model, zooplankton were forced.Predicted biomasses for zooplankton and ctenophores in the combined model were similar to the results of the two parent models, but improved relative to the actual field observations. From the findings it appears that the strategy of forcing is valid provided the forced patterns are appropriate and reasonable.     

Periodically driven circulation near the shore of a lake

Solutions are found for a linear model of the circulation near the shore of a lake that is subject to two diurnal forcing mechanisms. The first is the day/night heating/cooling induced horizontal pressure gradient. The second is an unsteady surface stress modelling a sea breeze/gully wind pattern. The two forcing mechanisms can oppose or reinforce each other depending on their relative phase. The interplay of different dynamic balances at different times and locations in the domain lead to complex circulation patterns especially during the period of flow reversal.     

The flow structure in the wake of a fractal fence and the absence of an “inertial regime”

Recent theoretical work has highlighted the importance of multi-scale forcing of the flow for altering the nature of turbulence energy transfer and dissipation. In particular, fractal types of forcing have been studied. This is potentially of real significance in environmental fluid mechanics where multi-scale forcing is perhaps more common than the excitation of a specific mode. In this paper we report the first results studying the detail of the wake structure behind fences in a boundary layer where, for a constant porosity, we vary the average spacing of the struts and also introduce fractal fences. As expected, to first order, and in the far-wake region, in particular, the response of the fences is governed by their porosity. However, we show that there are some significant differences in the detail of the turbulent structure between the fractal and non-fractal fences and that these override differences in porosity. In the near wake, the structure of the fence dominates porosity effects and a modified wake interaction length seems to have potential for collapsing the data. With regards to the intermittency of the velocities, the fractal fences behave more similarly to homogeneous, isotropic turbulence. In addition, there is a high amount of dissipation for the fractal fences over scales that, based on the energy spectrum, should be dominated by inter-scale transfers. This latter result is consistent with numerical simulations of flow forced at multiple scales and shows that what appears to be an “inertial regime” cannot be as production and dissipation are both high.     

A structural equation model to integrate changes in functional strategies during old-field succession

From a functional perspective, changes in abundance, and ultimately species replacement, during succession are a consequence of integrated suites of traits conferring different relative ecological advantages as the environment changes over time. Here we use structural equations to model the interspecific relationships between these integrated functional traits using 34 herbaceous species from a Mediterranean old-field succession and thus quantify the notion of a plant strategy. We measured plant traits related to plant vegetative and reproductive size, leaf functioning, reproductive phenology, seed mass, and production on 15 individuals per species monitored during one growing season. The resulting structural equation model successfully accounts for the pattern of trait covariation during the first 45 years post-abandonment using just two forcing variables: time since site abandonment and seed mass; no association between time since field abandonment and seed mass was observed over these herbaceous stages of secondary succession. All other predicted traits values are determined by these two variables and the cause-effect linkage between them. Adding pre-reproductive vegetative mass as a third forcing variable noticeably increased the predictive power of the model. Increasing the time after abandonment favors species with increasing life span and pre-reproductive biomass and decreasing specific leaf area. Allometric coefficients relating vegetative and reproductive components of plant size were in accordance with allometry theory. The model confirmed the trade-off between seed mass and seed number. Maximum plant height and seed mass were major determinants of reproductive phenology. Our results show that beyond verbal conceptualization, plant ecological strategies can be quantified and modeled.     

Testing the robustness of primary production models in shallow coastal areas: a case study

In this paper we investigate the robustness of a dynamic model, which describes the dynamic of the seagrass Zostera marina, with respect to the inter-annual variability of the two main forcing functions of primary production models in eutrophicated environments. The model was previously applied to simulate the seasonal evolution of this species in the Lagoon of Venice during a specific year and calibrated against time series of field data. In the this paper, we present and discuss the results which were obtained by forcing the model using time series of site-specific daily values concerning the solar radiation intensity and water temperature. The latter was estimated by means of a regression model, whose input variable was a site-specific time series of the air temperature. The regression model was calibrated using a year-long time series of hourly observations. The Z. marina model was first partially recalibrated against the same data set that was used in the original paper. Subsequently, the model was forced using a 7-year-long time series of the driving functions, in order to check the reliability of its long-term predictions. Even though the calibration gave satisfactory results, the multi-annual trends of the output variables were found to be in contrast with the observed evolution of the seagrass biomasses. Since detailed information about the air temperature and solar radiation are often available, these findings suggest that the testing of the ecological consistency of the evolution of primary production models in the long term would provide additional confidence in their results, particularly in those cases in which the scarcity of field data does not allow one to perform a formal corroboration/validation of these models.     

Which forcing factors fit? Using ecosystem models to investigate the relative influence of fishing and changes in primary productivity on the dynamics of marine ecosystems

Fishing mortality and primary production (or proxy for) were used to drive the dynamics of fish assemblages in 9 trophodynamic models of contrasting marine ecosystems. Historical trends in abundance were reconstructed by fitting model predictions to observations from stock assessments and fisheries independent survey data. The model fitting exercise derives values for otherwise unknown parameters that specify the relative strength of trophic interactions and, in some instances, a time series anomaly for changes in primary production. We measured how much better or worse were model predictions when bottom-up forcing by primary production were added to top-down forcing by fishing. Searching for cross system patterns, the relative contribution of fishing and changes in primary production, mediated through trophic interactions, are evaluated for the ecosystems as a whole and for selected similar species in different ecosystems. The analysis provides a simple qualitative way to explain which forcing factors have most influence on modeled dynamics. Both fishing and primary production forcing were required to obtain the best model fits to data. Fishing effects more strongly influenced 6 of 9 of the ecosystems, but primary production was more often found to be the main factor influencing the selected pelagic and demersal fish stock trends. Examination of sensitivity to ecological and model parameters suggests that the results are the product of complex food-web interactions rather than simple deterministic responses of the models.     

Removing the boundary influence on negatively buoyant jets

A comprehensive laboratory study of negatively buoyant discharges is presented. Unlike previous studies, here the focus is on generating data sets where influences of the bottom boundary have been eliminated. There are significant discrepancies in the published dilution data for these flows and a contributing factor is the large variation in the bottom boundary condition. A Laser-induced Fluorescence system is employed to gather flow spread, peak concentration (minimum dilution) and trajectory data for a wide range of densimetric Froude numbers and initial discharge angles. Data from these experiments are compared with previously published data, along with predictions from integral models and a revised form of the previously published semi-analytical solutions. The new data sets are not distorted by mixing processes associated with the bottom boundary and therefore provide the basis for more meaningful assessments of the predictive capabilities of existing models, given that the influences of the bottom boundary on contaminant mixing are not incorporated into these models. In general the models assessed are able to predict key geometric quantities with reasonable accuracy, but their minimum dilution predictions are conservative. Importantly dilution at the return point shows a strong dependence on the initial discharge angle and this could have important implications for the design of discharge systems.     

A new model for the CBL growth based on the turbulent kinetic energy equation

Starting from the evolution equation for the turbulent energy density spectrum (EDS), we develop a new model for the growth of the Convective boundary layer (CBL). We apply dimensional analysis to parameterize the unknown inertial transport and convective source term in the dynamic equation for the three-dimensional (3-D) spectrum and solve the 3-D EDS equation. The one-dimensional vertical spectrum is derived from the 3-D spectrum, employing a weight function. This allows us to select the magnitude of the vertical spectral component for the construction of the growing 3-D EDS. Furthermore, we employ the vertical component of the energy spectrum to calculate the eddy diffusivity (required in dispersion models). Currently there are no available experimental data to directly verify our EDS model.     

Cyclo-stationary maximum cross-covariance analysis for exogenously forced environmental systems

The response of an ecological system to external forcings shows marked seasonality and complex time delays. In order to take seasonality into account, the assumption of stationarity inherent in most statistical models for time series is replaced here by more flexible cyclo-stationarity. To describe the dominant modes of the cross-covariance structure of the system and the forcing, temporal (orthonormal) patterns are derived. They are estimated by a singular value decomposition of the cyclo-stationary sample cross-covariance matrix. These patterns allow to formulate a prediction scheme for the system once the forcing is known. The method has comparatively little tendency for multi-collinearity, so that it is well applicable also for multivariate settings and shorter data records. The method is illustrated for an aquatic ecosystem with distinct phases of a biological succession.     

Scaling-up spatially-explicit ecological models using graphics processors

How the properties of ecosystems relate to spatial scale is a prominent topic in current ecosystem research. Despite this, spatially explicit models typically include only a limited range of spatial scales, mostly because of computing limitations. Here, we describe the use of graphics processors to efficiently solve spatially explicit ecological models at large spatial scale using the CUDA language extension. We explain this technique by implementing three classical models of spatial self-organization in ecology: a spiral-wave forming predator-prey model, a model of pattern formation in arid vegetation, and a model of disturbance in mussel beds on rocky shores. Using these models, we show that the solutions of models on large spatial grids can be obtained on graphics processors with up to two orders of magnitude reduction in simulation time relative to normal pc processors. This allows for efficient simulation of very large spatial grids, which is crucial for, for instance, the study of the effect of spatial heterogeneity on the formation of self-organized spatial patterns, thereby facilitating the comparison between theoretical results and empirical data. Finally, we show that large-scale spatial simulations are preferable over repetitions at smaller spatial scales in identifying the presence of scaling relations in spatially self-organized ecosystems. Hence, the study of scaling laws in ecology may benefit significantly from implementation of ecological models on graphics processors.     

A numerical model of periodically forced circulation near the shore of a lake

Numerical calculations for a model of the near-shore circulation in a lake subject to two diurnal forcing mechanisms are presented. The first mechanism is a heating/cooling term in the heat equation representing the daytime heating and nighttime cooling of the diurnal cycle. The second is a periodic surface stress modelling a sea-breeze/gully wind system typical of some coastal regions. The two forcing mechanisms can either act together or against each other depending on their relative phase. The numerical solutions are compared with previously published analytical solutions and used to explore the extra dynamics associated with non-linear effects (specifically advection). The latter dynamics include the formation of gravity currents and unstable density profiles leading to secondary circulation.     

Combined integral and particle model for describing the dispersion,dilution, terminal layer formation and influence area from a point source discharge into a water body

Contamination of coastal water is a persistent threat to ecosystems around the world. In this study, a novel model for describing the dispersion, dilution, terminal layer formation and influence area from a point source discharge into a water body is presented and compared with field measured data. The model is a Combined Integral and Particle model (CIPMO). In the initial stage, the motion, dispersion and dilution of a buoyant jet are calculated. The output from the buoyant jet model is then coupled with a Lagrangian Advection and Diffusion model describing the far-field. CIPMO ensures that both the near- and far-field processes are adequately resolved. The model either uses empirical data or collects environmental forcing data from open source hydrodynamic models with high spatial and temporal resolution. The method for coupling the near-field buoyant jet and the particle tracking model is described and the output is discussed. The model shows good results when compared with measurements from a field study.

     

Adding complex trophic interactions to a size-spectral plankton model: Emergent diversity patterns and limits on predictability

A new model in the NPZ (nutrient-phytoplankton-zooplankton) style is presented, mechanistically simple but with 40 size classes each of phytoplankton (1-20 μm) and small zooplankton (2.1-460 μm), in order to resolve one level of trophic interactions in detail. General, empirical allometric relationships are used to parameterize both the optimal prey size and size selectivity for each grazer class, as is rarely done. This inclusion of complex predator-prey linkages and realistic prey preferences yields a system with an emergent pattern of phytoplankton diversity consistent with global ocean observations, i.e., a parabolic relationship between diversity (as measured by the Shannon evenness) and biomass. It also yields significant long-term time evolution, which places limits on the extent to which the community response to nutrient forcing can be predicted from forcing in a pragmatic sense. When a simple annual cycle in nutrient supply is repeated exactly for many years, transient fluctuations up to a factor of two in spring bloom magnitude persist for 10-20 years before a stable seasonal biomass cycle is achieved. When the amplitude of the nutrient-supply annual cycle is given a random interannual modulation, these long-lived transients add significant noise to a 100-year correlation between annual-mean nutrient supply and annual-mean biomass. This noise is 20% of total interannual variance in the model base case, and ranges from 0% to 40% depending on the grazer size selectivity. In general, unpredictability on the bloom timescale is damped when food-web complexity is increased by making grazers less selective, while unpredictability on the interannual scale shows the opposite pattern, increasing with increasing food-web complexity up to a high threshhold, past which community structure and biomass time evolution both suddenly simplify. These results suggests a new strategy for ensemble ecosystem forecasting and uncertainty estimation, analogous to methods common in circulation and climate modeling, in which internal variability (predator-prey interactions in the biological case; eddies and climate-system oscillations in the physical case) are resolved and quantified, rather than suppressed.     

Identifying landscape scale patterns from individual scale processes

Extrapolating across scales is a critical problem in ecology. Explicit mechanistic models of ecological systems provide a bridge from measurements of processes at small and short scales to larger scales; spatial patterns at large scales can be used to test the outcomes of these models. However, it is necessary to identify patterns that are not dependent on initial conditions, because small scale initial conditions will not normally be measured at large scales. We examined one possible pattern that could meet these conditions, the relationship between mean and variance in abundance of a parasitic tick in an individual based model of a lizard tick interaction. We scaled discrepancies between the observed and simulated patterns with a transformation of the variance–covariance matrix of the observed pattern to objectively identify patterns that are “close”.     

Accounting for rate instability and spatial patterns in the boundary analysis of cancer mortality maps

Boundary analysis of cancer maps may highlight areas where causative exposures change through geographic space, the presence of local populations with distinct cancer incidences, or the impact of different cancer control methods. Too often, such analysis ignores the spatial pattern of incidence or mortality rates and overlooks the fact that rates computed from sparsely populated geographic entities can be very unreliable. This paper proposes a new methodology that accounts for the uncertainty and spatial correlation of rate data in the detection of significant edges between adjacent entities or polygons. Poisson kriging is first used to estimate the risk value and the associated standard error within each polygon, accounting for the population size and the risk semivariogram computed from raw rates. The boundary statistic is then defined as half the absolute difference between kriged risks. Its reference distribution, under the null hypothesis of no boundary, is derived through the generation of multiple realizations of the spatial distribution of cancer risk values. This paper presents three types of neutral models generated using methods of increasing complexity: the common random shuffle of estimated risk values, a spatial re-ordering of these risks, or p-field simulation that accounts for the population size within each polygon. The approach is illustrated using age-adjusted pancreatic cancer mortality rates for white females in 295 US counties of the Northeast (1970–1994). Simulation studies demonstrate that Poisson kriging yields more accurate estimates of the cancer risk and how its value changes between polygons (i.e., boundary statistic), relatively to the use of raw rates or local empirical Bayes smoother. When used in conjunction with spatial neutral models generated by p-field simulation, the boundary analysis based on Poisson kriging estimates minimizes the proportion of type I errors (i.e., edges wrongly declared significant) while the frequency of these errors is predicted well by the p-value of the statistical test.

Interaction structures analysed from water-quality data

Fortnightly observations of water quality parameters, discharge and water temperature along the River Elbe have been subjected to a multivariate data analysis. In a previous study [Petersen, W., Bertino, L., Callies, U., Zorita, E., 2001. Process identification by principal component analysis of river-quality data. Ecol. Model. 138, 193–213] applied principal component analysis (PCA) to show that 60% of variability in the data set can be explained through just two linear combinations of eight original variables. In the present paper more advanced multivariate methods are applied to the same data set, which are supposed to suit better interpretations in terms of the underlying system dynamics.The first method, graphical modelling, represents interaction structures in terms of a set of conditional independence constraints between pairs of variables given the values of all other variables. Assuming data from a multinormal distribution conditional independence constraints are expressed by zero partial correlations. Different graphical structures with nodes for each variable and connecting edges between them can be assessed with regard to their likelihood. The second method, canonical correlation analysis (CCA), is applied for studying the correlation structures of external forcing and water quality parameters.Results of CCA turn out to be consistent with the dominant patterns of variability obtained from PCA. The percentages of variability explained by external forcing, however, are estimated to be smaller. Fitting graphical models allows a more detailed representation of interaction structures. For instance, for given discharge and temperature correlated variations of the concentrations of oxygen and nitrate, respectively, can be modelled as being mediated by variations of pH, which is a representer for algal activity. Considerably simplified graphical models do not much affect the outcomes of both PCA and CCA, and hence it is concluded that these graphical models successfully represent the main interaction structures represented by the covariance matrix of the data. The analysed conditional independence patterns provide constraints to be satisfied by directed probabilistic networks, for instance.