Wetlands are essential for global biogeochemical cycles and ecosystem services, with the dynamics of soil organic carbon (SOC) serving as the critical regulatory mechanism for these processes. However, accurately modeling carbon dynamics in wetlands presents challenges due to their complexity. Traditional approaches often fail to capture spatial variations, long-range transport, and periodical flooding dynamics, leading to uncertainties in carbon flux predictions. To tackle these challenges, we introduce a novel extension of the fractional RothC model, integrating temporal fractional-order derivatives into spatial dimensions. This enhancement allows for the creation of a more adaptive tool for analyzing SOC dynamics. Our differential model incorporates Richardson–Richard's equation for moisture fluxes, a diffusion–advection–reaction equation for fractional-order dynamics of SOC compounds, and a temperature transport equation. We examine the influence of diffusive movement and sediment moisture content on model solutions, as well as the impact of including advection terms. Finally, we validated the model on a restored wetland scenario at the Ebro Delta site, aiming to evaluate the effectiveness of flooding strategies in enhancing carbon sequestration and ecosystem resilience.

Models of soil organic carbon (SOC) frequently overlook the effects of spatial dimensions and microbiological activities. In this paper, we focus on two reaction-diffusion chemotaxis models for SOC dynamics, both supporting chemotaxis-driven instability and exhibiting a variety of spatial patterns as stripes, spots and hexagons when the microbial chemotactic sensitivity is above a critical threshold. We use symplectic techniques to numerically approximate chemotaxis-driven spatial patterns and explore the effectiveness of the piecewise Dynamic Mode Decomposition (pDMD) to reconstruct them. Moreover, we analyse the predictive performance of the pDMD for moderate time horizons. Our findings show that pDMD is effective at precisely recreating and predicting chemotaxis-driven spatial patterns, therefore broadening the range of application of the method to classes of solutions different than Turing patterns. By validating its efficacy across a wider range of models, this research lays the groundwork for applying pDMD to experimental spatiotemporal data, advancing predictions crucial for soil microbial ecology and agricultural sustainability.

We study pattern formation in a chemotaxis model of bacteria and soil carbon dynamics as an example system where transient dynamics can give rise to pattern formation outside of Turing unstable regimes. We use a detailed analysis of the reactivity of the non-spatial and spatial dynamics, stability analyses, and numerical continuation to uncover detailed aspects of this system’s pattern-forming potential. In addition to patterning in Turing unstable parameter regimes, reactivity of the spatial system can itself lead to a range of parameters where a spatially uniform state is asymptotically stable, but exhibits transient growth that can induce pattern formation. We show that this occurs in the bistable region of a subcritical Turing bifurcation. Intriguingly, such bistable regions appear in two spatial dimensions, but not in a one-dimensional domain, suggesting important interplays between geometry, transient growth, and the emergence of multistable patterns. We discuss the implications of our analysis for the bacterial soil organic carbon system, as well as for reaction-transport modeling more generally.

Ecological systems are subject to environmental variability and fluctuations: understanding the role of such stochastic perturbations in inducing on–off intermittency is the central motivation for this study. This research extends the exploration of parameters leading to the emergence of on–off intermittency within a discrete Beddington-Free-Lawton host-parasitoid model. We introduce random perturbation factors that impact both the grazing intensity and the growth rate of the host population. An intriguing aspect of this study is the numerical evidence of the reactivity of the free-parasitoid fixed point as a route to on–off intermittency. This finding is significant because it sheds light on how stable ecological equilibria can transition into intermittency before progressing toward chaotic behaviour. Moreover, our study explores the host-parasitoid coupling within the Beddington-Free-Lawton model when it is applied to a complex network, a significant framework for modelling ecological interactions. The paper reveals that such network-based interactions induce parasitoid bursts that are not observed in a single population scenario.

Vitamin D has been proven to be a strong stimulator of mechanisms associated with the elimination of pathogens. Because of its recognized effectiveness against viral infections, during SARS-CoV-2 infection, the effects of Vitamin D supplementation have been the object of debate. This study aims to contribute to this debate by the means of a qualitative phenomenological mathematical model in which the role of Vitamin D and its interactions with the innate immune system are explicitly considered. We show that Vitamin D influx and degradation can be considered as possible control parameters for the disease evaluation and recovery. By varying Vitamin D influx, three dynamical scenarios have been found with different modalities of recovery from the disease. Inside each scenario, Vitamin D degradation has been related to different degrees of severity in disease development. Interestingly, the emergence of hysteretic phenomenologies when Vitamin D influx is too low can be related to the onset of Long-COVID syndrome, confirming clinical evidence from recent studies on the topic.

We tackle the problem of coupling a spatiotemporal model for simulating the spread and control of an invasive alien species with data coming from image processing and expert knowledge. In this study, we implement a spatially explicit optimal control model based on a reaction-diffusion equation which includes an Holling II type functional response term for modeling the density control rate. The model takes into account the budget constraint related to the control program and searches for the optimal effort allocation for the minimization of the invasive alien species density. Remote sensing and expert knowledge have been assimilated in the model to estimate the initial species distribution and its habitat suitability, empirically extracted by a land cover map of the study area. The approach has been applied to the plant species Ailanthus altissima (Mill.) Swingle within the Alta Murgia National Park. This area is one of the Natura 2000 sites under the study of the ongoing National Biodiversity Future Center (NBFC) funded by the Italian National Recovery and Resilience Plan (NRRP), and pilot site of the finished H2020 project ECOPOTENTIAL, which aimed at the integration of modeling tools and Earth Observations for a sustainable management of protected areas. Both the initial density map and the land cover map have been generated by using very high resolution satellite images and validated by means of ground truth data provided by the EU Life Alta Murgia project (LIFE12 BIO/IT/000213), a project aimed at the eradication of Ailanthus altissima in the Alta Murgia National Park

A new fractional q-order variation of the RothC model for the dynamics of soil organic carbon is introduced. A computational method based on the discretization of the analytic solution along with the finite-difference technique are suggested and the stability results for the latter are given. The accuracy of the scheme, in terms of the temporal step size h, is confirmed through numerical testing of a constructed analytic solution. The effectiveness of the proposed discrete method is compared with that of the classical discrete RothC model. Results from real-world experiments show that, by adjusting the fractional order q and the multiplier term ?(t,q), a better match between simulated and actual data can be achieved compared to the traditional integer-order model.

To evaluate changes in the Soil Organic Carbon (SOC) index, one of the key indicators of land degradation neutrality, soil carbon modeling is of primary importance. In litera-ture, the analysis has been focused on the stability characterization of soil carbon steady states and in the calculation of the resilience of the stable equilibria. Neither stability nor resilience, however, provide any information about transient dynamics, and models with highly resilient equilibria can exhibit dramatic transient responses to perturbations. To trace how environmental changes affect the transient dynamics of SOC indicator, we use the concept of generalized reactivity (g-reactivity) to models belonging to two main classes: the first-order, linear and semilinear carbon transfer models and fully nonlinear microbe-explicit models. A novel formulation of a general two-dimensional model allows to deal with different functional forms and to perform a systematic analysis of both stabil-ity of soil carbon equilibria and SOC-reactivity. Using temperatures and Net Primary Pro-duction (NPP) data of Alta Murgia National Park, the RothC, MOMOS and the fully implicit dynamical planar system are compared in predicting the impact of increased temperatures in the years 2005-2019 on the asymptotic stability of carbon steady states and in increas-ing the SOC-reactivity.(c) 2023 Elsevier Inc. All rights reserved.

Controlling and planning the removal of invasive species are topics of outmost importance in management of natural resources because of the severe ecological damages and economic losses caused by non-native alien species. Optimal management strategies often rely on coupling population dynamics models with optimization procedures to achieve an effective allocation of limited resources for removing invasive species from hosting ecosystems. We analyse a parabolic optimal control model to simulate the best spatiotemporal strategy for the removal of the species when a budget constraint is applied. The model also predicts the species spread under the control action. We improve the capability of the model to reproduce realistic scenarios by introducing an advection term in the state equation. That allows to model the action of external forces, like currents or winds, which might bias dispersal in certain directions. The analytical properties of the model are discussed under suitable boundary conditions. As a further original contribution, we introduce a novel numerical procedure for approximating the solution reducing the computational costs in view of its implementation as a support decision tool. Then we test the approach by simulating the spread and the control of a hypothetical invasive plant in the territory of the Italian Sardinia island. To reproduce the anisotropy of the diffusion we include the effect of the altitude in the habitat suitability of the species.

The SOC change index, defined as the normalized difference between the actual Soil Organic Carbon and the value assumed at an initial reference year, is here tailored to the RothC carbon model dynamics. It assumes as a baseline the value of the SOC equilibrium under constant environmental conditions. A sensitivity analysis is performed to evaluate the response of the model to changes in temperature, Net Primary Production (NPP), and land use soil class (forest, grassland, arable). A non-standard monthly time-stepping procedure has been proposed to approximate the SOC change index in the Alta Murgia National Park, a protected area in the Italian Apulia region, selected as a test site. The SOC change index exhibits negative trends for all the land use considered without fertilizers. The negative trend in the arable class can be inverted by a suitable organic fertilization program here proposed.

Progress in understanding, managing, and securing current and future ecosystem functions and services is challenged by fragmented and dispersed ecosystem research. As the topic is often approached using narrow disciplinary perspectives, a holistic understanding of complex eco- and socio-ecological systems is hampered and prevented. The emerging European Long-Term Ecosystem, critical zone and socio-ecological systems Research Infrastructure (eLTER) aims to overcome this challenge by addressing this issue in the ecosystem and biodiversity domain and thereby closing this gap in the European RI landscape. With its concept of the 'Information Clusters' eLTER aims to provide a framework to lower the barrier to information access and exchange. The main idea behind the concept is to simplify the harvesting and user uptake of data from multiple information sources, facilitating the integration with eLTER data by making use of existing services, like Copernicus or statistical information. The selection of sources and content of relevant data layers is the result of an internal discussion where the Research Challenges (RC) play the main role by identifying the current requirements for environmental research and the ensuing demand for external data. The overarching framework of the eLTER Standard Observations informs this process. In order to achieve the implementation of 'Information Clusters', three different data sources have been identified to complement eLTER observations and analysis: (a) in-situ legacy and third party data, (b) data from official statistics, and (c) remote sensing data and products. The activities described in the report focus on the collection and exemplary retrieval of relevant in-situ legacy data, which we identified as complementary data sources and could play an important role within the planned eLTER data analysis workflows. This is relevant to (a) get additional data for data analysis or visualisation, (b) retrieve data from eLTER sites provided by national level catalogues, and (c) retrieve data from eLTER sites provided to other relevant RIs or monitoring networks. The aim of task 4.1 was to develop and test workflows for access and basic level harmonisation of relevant in-situ data sources on global, continental and national scale. We focused on data requirements defined both by the RC addressed in the eLTER PLUS project as well as the needs for supporting the implementation of data flows defined by the eLTER SOs. We identified 176 legacy and third party data sources which could be assigned to a respective eLTER SO and which sufficiently cover each component of the Ecological Integrity concept. Based on a generic workflow described in the report we tested through demonstrators exemplary data extraction workflows being of relevance in the project context. This demonstrators focused on: (a) retrieve occurrence biodiversity data based on API access, (b) retrieve harmonised site gas flux observation data based on downloads, (c) retrieve data from E-OBS historic data (Copernicus Climate Change Service, 2020) to calculate climate diagrams for sites, (d) retrieve data from gridded and modelled data (e.g. E-OBS) based on the site extent, and (e) retrieve earth observation data products based on site extent. It could be shown that the selected workflows are, at least on a prototype level, operational and are useful for the eLTER PLUS users. We applied a co-design process including the respective RC leads and Science Case (SC) contributors in the design and implementation phase on a regular basis. However, eLTER needs to decide if eLTER Information Clusters focus on on-demand services for extracting information sources or pre-calculated datasets. The results of the work done in task 4.1 provide input to the design and architecture of the extended eLTER Information System led by WP11 and the further definition of workflows towards the eLTER Standard Data Products led by WP10. The report summarises the work done with respect to define and prototype workflows for the retrieval and harmonisation of legacy data. It specifically focuses on priority variables defined by the eLTER SO and aims to support Research Challenge related Science Cases at both, site and network scale. The first section describes the context of the work done, also in relation to the 'Information Clusters' concept, which aims to enhance findability and accessibility of relevant data sources in the eLTER context. The second section lists identified relevant data sources relevant in this context and provides demonstrators for data retrieval and harmonisation in the third part. We finally discuss and provide recommendations for the eLTER Information Clusters that focus on thematic prioritisation, structural and legal interoperability as well as outline next steps for the implementation. The annexes provide detailed information shown in the report only in aggregated format.

Soil Organic Carbon (SOC) is one of the key indicators of land degradation. SOC positively affects soil functions with regard to habitats, biological diversity and soil fertility; therefore, a reduction in the SOC stock of soil results in degradation, and it may also have potential negative effects on soil-derived ecosystem services. Dynamical models, such as the Rothamsted Carbon (RothC) model, may predict the long-term behaviour of soil carbon content and may suggest optimal land use patterns suitable for the achievement of land degradation neutrality as measured in terms of the SOC indicator. In this paper, we compared continuous and discrete versions of the RothC model, especially to achieve long-term solutions. The original discrete formulation of the RothC model was then compared with a novel non-standard integrator that represents an alternative to the exponential Rosenbrock-Euler approach in the literature. Soil Organic Carbon (SOC) is one of the key indicators of land degradation. SOC positively affects soil functions with regard to habitats, biological diversity and soil fertility; therefore, a reduction in the SOC stock of soil results in degradation, and it may also have potential negative effects on soil-derived ecosystem services. Dynamical models, such as the Rothamsted Carbon (RothC) model, may predict the long-term behaviour of soil carbon content and may suggest optimal land use patterns suitable for the achievement of land degradation neutrality as measured in terms of the SOC indicator. In this paper, we compared continuous and discrete versions of the RothC model, especially to achieve long-term solutions. The original discrete formulation of the RothC model was then compared with a novel non-standard integrator that represents an alternative to the exponential Rosenbrock-Euler approach in the literature.

Trends of soil organic carbon (SOC) are significant indicators for land and soil degradation. Decrease in SOC compromises the efforts to achieve by 2030, a land degradation neutral world, as required by Target 15.3 of the Seventeen Sustainable Development Goals (SDGs) adopted by United Nations in September 2015. Differential models, as the Rothamsted Carbon model (RothC) [1], can be useful tools to predict SOC changes, taking into account the interactions among climate, soil and land use management. In this talk, we illustrate some results on the application of a novel nonstandard discretization [2] of the continuous RothC model [3] for assessing the SOC indicator in Alta Murgia National Park, a protected area in Apulia region in the south of Italy. A procedure for determining the initial plant input necessary to run the model is described. Moreover, in order to detect the factors that determine the size and direction of SOC changes, a local sensitivity analysis based on the so-called direct method is performed. This work received fundings from the REFIN project N.0C46E06B (Regione Puglia, Italy) and from the European Union's Horizon 2020 research and innovation programme under grant agreement No 871128 (H2020-eLTER-PLUS project).

A novel model is here introduced for theSOC change indexdefinedas the normalized difference between the actual Soil Organic Carbon and thevalue assumed at an initial reference year. It is tailored on the RothC carbonmodel dynamics and assumes as baseline the value of the SOC equilibriumunder constant environmental conditions. A sensitivity analysis is performedto evaluate the response of the model to changes of temperature, Net PrimaryProduction (NPP), and land use soil class (forest, grassland, arable). A non-standard monthly time-stepping procedure has been proposed to approximatethe SOC change index in the Alta Murgia National Park, a protected areain the Italian Apulia region, selected as test site. In the case of arable class,the SOC change index exhibits a negative trend which can be inverted by asuitable organic fertilization program here proposed.

A major neglected weakness of many ecological models is the numerical method used to solve the governing systems of differential equations. Indeed, the discrete dynamics described by numerical integrators can provide spurious solution of the corresponding continuous model. The approach represented by the geometric numerical integration, by preserving qualitative properties of the solution, leads to improved numerical behaviour expecially in the long-time integration. Positivity of the phase space, Poisson structure of the flows, conservation of invariants that characterize the continuous ecological models are some of the qualitative characteristics well reproduced by geometric numerical integrators. In this paper we review the benefits induced by the use of geometric numerical integrators for some ecological differential models.

The containment of the invasive species is a widespread problem in the environmental management, with a significant economic impact. We analyze an optimal control model which aims to find the best temporal resource allocation strategy for the removal of an invasive species. We derive the optimality system in the state and control variables and we use the phase-space analysis to provide qualitative insights about the behavior of the optimal solution. Finally, for the state-costate variables which satisfy a boundary-valued nearly-Hamiltonian system, we propose exponential Lawson symplectic approximations applied in the forward-backward form. The numerical results related to an example of invasive plant considered in Baker, et al. (Nat Resour Model 31(4):e12190, 2018), confirm the qualitative findings provided by the state-control analysis.

A challenging task in the management of Protected Areas is to control the spread of invasive species, either floristic or faunistic, and the preservation of indigenous endangered species, typically competing for the use of resources in a fragmented habitat. In this paper, we present some mathematical tools that have been recently applied to contain the worrying diffusion of wolf-wild boars in a Southern Italy Protected Area belonging to the Natura 2000 network. They aim to solve the problem according to three different and in some sense complementary approaches: (i) the qualitative one, based on the use of dynamical systems and bifurcation theory; (ii) the Z-control, an error-based neural dynamic approach; (iii) the optimal control theory. In the case of the wild-boars, the obtained results are illustrated and discussed. To refine the optimal control strategies, a further development is to take into account the spatio-temporal features of the invasive species over large and irregular environments. This approach can be successfully applied, with an optimal allocation of resources, to control an invasive alien species infesting the Alta Murgia National Park: Ailanthus altissima. This species is one of the most invasive species in Europe and its eradication and control is the object of research projects and biodiversity conservation actions in both protected and urban areas [11]. We lastly present, as a further example, the effects of the introduction of the brook trout, an alien salmonid from North America, in naturally fishless lakes of the Gran Paradiso National Park, study site of an on-going H2020 project (ECOPOTENTIAL).

Questa presentazione descrive le conclusioni finali del progetto ECOPOTENTIAL riguardo al coinvolgimento degli stakeholder delle aree protette nel definire e condurre le attività di ricerca a fianco dei ricercatori, illustrando punti di forza e di debolezza, come risultati da un questionario sottoposto ai gestori delle Aree Protette.

Mathematical modeling and optimization provide decision-support tools of increasing popularity to the management of invasive species. In this paper we investigate problems formulated in terms of optimal control theory. A free terminal time optimal control problem is considered for minimizing the costs and the duration of an abatement program. Here we introduce a discount term in the objective function that destroys the non-autonomous nature of the state-costate system. We show that the alternative state-control optimality system is autonomous and its analysis provides the complete qualitative description of the dynamics of the discounted optimal control problem. By using the expression of its invariant we deduce several insights for detecting the optimal control solution for an invasive species obeying a logistic growth.

A challenging task in the management of Protected Areas is to control the spread of invasive species, either floristic or faunistic, and the preservation of indigenous endangered species, tipically competing for the use of resources in a fragmented habitat. In this paper, we present some mathematical tools that have been recently applied to contain the worrying diffusion of wolf-wild boars in a Southern Italy Protected Area belonging to the Natura 2000 network. They aim to solve the problem according to three different and in some sense complementary approaches: (i) the qualitative one, based on the use of dynamical systems and bifurcation theory; (ii) the Z-control, an error-based neural dynamic approach ; (iii) the optimal control theory. In the case of the wild-boars, the obtained results are illustrated and discussed. To refine the optimal control strategies, a further development is to take into account the spatio-temporal features of the invasive species over large and irregular environments. This approach can be successfully applied, with an optimal allocation of resources, to control an invasive alien species infesting the Alta Murgia National Park: Ailanthus altissima. This species is one of the most invasive species in Europe and its eradication and control is the object of research projects and biodiversity conservation actions in both protected and urban areas [11]. We lastly present, as a further example, the effects of the introduction of the brook trout, an alien salmonid from North America, in naturally fishless lakes of the Gran Paradiso National Park, study site of an on-going H2020 project (ECOPOTENTIAL).