Mathematical modeling and optimization provide decision-support toolsof increasing popularity to the management of invasive species. In this paper weinvestigate problems formulated in terms of optimal control theory. A free terminaltime optimal control problem is considered for minimizing the costs and the durationof an abatement program. Here we introduce a discount term in the objectivefunction that destroys the non-autonomous nature of the state-costate system. Weshow that the alternative state-control optimality system is autonomous and its analysisprovides the complete qualitative description of the dynamics of the discountedoptimal control problem. By using the expression of its invariant we deduce severalinsights for detecting the optimal control solution for an invasive species obeying alogistic growth.

Protected areas are continuously subjected to ecological change due to anthropic pressures. Analyses of changes in the extent and intensity of pressures over time are essential for adaptive management, yet such analyses are rarely conceptualized or performed in a well-defined, standardized way, with a frequent lack of clarity in development, definition and measurement. Over-time remote sensing data has great potential for mapping spatial pattern of pressures and their impacts. Some pressures can be mapped directly (e.g. land use dynamics, some invasive species), whereas for others the nature, intensity and spatial pattern of impacts can be used to infer on pressures. This paper develops a framework within which remote sensing datasets in combination with GIS and ecological modeling may be used to identify potential pressure growth through either direct detection or indirect monitoring of impacts on landscapes, land cover/habitat types, communities and species through multi-temporal remote sensing image series. The use of this framework is illustrated through categorization of pressure-derived impacts on protected areas in six countries - Greece, Italy, Portugal, Wales, The Netherlands, and India - located in diverse biogeographic, environmental and social-ecological contexts, and facing a different range of pressures. The framework is conceptually robust, geographically invariant, scalable and spatially-explicit, connecting to the growing data sets from remote sources, and we urge it to be tested over a wide range of pressures and social-ecological settings.

Protected Areas are subject to long-term modifications associated with climate and environmental change, enhancing the risk of ecosystem collapse, tipping points and unexpected responses to droughts, fires, floods and other individual events. One of the goals of the EU H2020 Project ECOPOTENTIAL and of the GEO ECO Initiative of the Group on Earth Observations (GEO) is to quantify ongoing and expected changes in the drivers and the characteristics of Protected Areas in Europe and beyond, using gridded climatic datasets, in situ meteo-climatic and biological data and Remote Sensing observations. Several statistical approaches are used to this goal, with the aim of determining the patterns and properties of the changes currently affecting Protected Areas. Use of suitably downscaled climate scenarios allows for estimating how such changes are projected into the next decades. Here we report the results on the changes in meteo-climatic drivers and in some remotely-sensed variables for the set of Protected Areas participating in the ECOPOTENTIAL project, focusing on a few specific examples encompassing mountain, arid/semi-arid and coastal ecosystems.

One of the core European Union environmental policies is the creation and monitoring of the Natura 2000 network of protected areas. This network has been explicitly established for the preservation of conservation priority habitat types and species. Still the concept of habitat is a key concept for ecologists that remains ill defined and is notoriously hard to quantify and measure. Several classification schemes have been put forward, but their relative strengths and weaknesses remain less well examined. In this study we analyzed 8 different Natura 2000 sites (3 Greek, 2 Italian, 2 Portuguese, 1 British). Our study sites reflect a variety of ecosystems, most of them are Mediterranean (7 of the 8) and most of them are wetlands (6 of the 8). In each site, we classified habitats according to 4 different classification schemes (Annex I of the Habitats Directive, Corine Biotopes, EUNIS and General Habitat categories). Also, we used three other widely used land cover classification schemes (namely Corine Land Cover, FAO Land Cover Classification System and IGBP DIS scheme). We found that the different schemes produced considerably different values of landscape diversity leading even to different ranking of the sites according to their diversity. Furthermore, when comparing the landscape composition among sites according to the different schemes, they led to different inferences. Our results imply that the classification scheme used for estimating habitat composition plays an important role for the monitoring of protected areas, perhaps more important than previously assumed.

At a global level, protected sites have been established for the primary purpose of conserving biodiversity, with survey and monitoring of habitats undertaken largely within their boundaries. However, because of increasing human populations with greater access to resources, there is a need to now consider monitoring anthropic activities in the surrounding landscapes as pressures and disturbances are impacting on the functioning and biodiversity values of many protected sites. Earth Observation (EO) data acquired across a range of spatial and temporal scales offer new opportunities for monitoring biodiversity over varying time-scales, either through direct or indirect mapping of species or habitats. However, Land Cover (LC) and/or Land Use (LU), rather than habitat maps are generated in many national and international programs and, whilst the translation from one classification to the other is desirable, differences in definitions and criteria have so far limited the establishment of a unified approach. Focusing on both natural and non-natural environments associated with Natura 2000 sites in the Mediterranean, this paper considers the extent to which three common LC/LU taxonomies (CORINE, the Food and Agricultural Organisation (FAO) Land Cover Classification System (FAO-LCCS) and the IGBP) can be translated to habitat taxonomies with minimum use of additional environmental attributes and/or in situ data. A qualitative and quantitative analysis based on the Jaccard's index established the FAOLCCS as being the most useful taxonomy for harmonizing LC/LU maps with different legends and dealing with the complexity of habitat description and as a framework for translating EO-derived LC/LU to habitat categories. As demonstration, a habitat map of a wetland site is obtained through translation of the LCCS taxonomy.

We propose spatially implicit models described by ordinary differential equations which inherit the information of spatial explicit metapopulation models described by reaction-diffusion partial differential equations. Numerical simulations confirm that the proposed implicit models can capture the qualitative features of the explicit ones and may reveal as an effective tool to extract predictive information through a further theoretical analysis.

We evaluate a mathematical model of the predator-prey population dynamics in a fragmented habitat where both migration processes between habitat patches and prey control policies are taken into account. The considered system is examined by applying the aggregation method and different dynamical scenarios are generated. The resulting implications are then discussed, their primary aim being the conservation of the wolf population in the Alta Murgia National Park, a protected area situated in the Apulian Foreland and also part of the Natura 2000 network. The Italian wolf is an endangered species and the challenge for the regional authorities is how to formulate conservation policies which enable the maintenance of the said wolf population while at the same time curbing that of the local wild boars and its negative impact on agriculture. We show that our model provides constructive suggestions in how to combine wild boar abatement programs awhile maintaining suitable ecological corridors which ensure wolf migration, thus preserving wolves from extinction.

Sustainable development demands for an enhanced-evidence-and science-based policy and management of the environment. The ECOPOTENTIALway to address the issue is to improve the capability of monitoring and modelling the ecosystems and ecosystem services by starting fromProtected Areas. The PAs thus act as open labs for building the necessary knowledge base to face the challenge of global change.A major asset of the ECOPOTENTIAL's approach to enhance predictive capacity of the state and temporal evolution of ecosystems is to focus on theintegration of modelling techniques with EO data, both remote and in-field. The goal is to improve our knowledge on ecosystem nonlinearity,complexity and uncertainties and to predict ecosystem changes in key PAs.ECOPOTENTIAL develops and applies a range of EO-data based conceptual, correlative and process-based models on mountains, arid and coastal/marine ecosystems and prototypes online data services for ecosystem indicators and potential supply of ecosystem services using data-assimilation.The poster showcases a few examples.

Monitoring biodiversity at the level of habitats and landscape is becoming widespread in Europe and elsewhere as countries establish international and national habitat conservation policies and monitoring systems. Earth Observation (EO) data offers a potential solution to long-term biodiversity monitoring through direct mapping of habitats or by integrating Land Cover/Use (LC/LU) maps with contextual spatial information and in situ data. Therefore, it appears necessary to develop an automatic/ semi-automatic translation framework of LC/ LU classes to habitat classes, but also challenging due to discrepancies in domain definitions. In the context of the FP7 BIO_SOS (www.biosos.eu) project, the authors demonstrated the feasibility of the Food and Agricultural Organization Land Cover Classification System (LCCS) taxonomy to habitat class translation. They also developed a framework to automatically translate LCCS classes into the recently proposed General Habitat Categories classification system, able to provide an exhaustive typology of habitat types, ranging from natural ecosystems to urban areas around the globe. However discrepancies in terminology, plant height criteria and basic principles between the two mapping domains inducing a number of one-to-many and many-to-many relations were identified, revealing the need of additional ecological expert knowledge to resolve the ambiguities. This paper illustrates how class phenology, class topological arrangement in the landscape, class spectral signature from multi-temporal Very High spatial Resolution (VHR) satellite imagery and plant height measurements can be used to resolve such ambiguities.Concerning plant height, this paper also compares the mapping results obtained by using accurate values extracted from LIght Detection And Ranging (LIDAR) data and by exploiting EO data texture features (i.e. entropy) as a proxy of plant height information, when LIDAR data are not available. An application for two Natura 2000 coastal sites in Southern Italy is discussed.

We consider explicit symplectic partitioned Runge-Kutta (ESPRK) methods for the numerical integration of non-autonomous dynamical systems. It is known that, in general, the accuracy of a numerical method can diminish considerably whenever an explicit time dependence enters the differential equations and the order reduction can depend on the way the time is treated. In the present paper, we demonstrate that explicit symplectic partitioned Runge-Kutta-Nyström (ESPRKN) methods specifically designed for second order differential equations , undergo an order reduction when M=M(t), independently of the way the time is approximated. Furthermore, by means of symmetric quadrature formulae of appropriate order, we propose a different but still equivalent formulation of the original non-autonomous problem that treats the time as two added coordinates of an enlarged differential system. In so doing, the order reduction is avoided as confirmed by the presented numerical tests.

We are concerned with the discretization of optimal control problems when a Runge-Kutta scheme is selected for the related Hamiltonian system. It is known that Lagrangian's first order conditions on the discrete model, require a symplectic partitioned Runge-Kutta scheme for state-costate equations. In the present paper this result is extended to growth models, widely used in Economics studies, where the system is described by a current Hamiltonian. We prove that a correct numerical treatment of the state-current costate system needs Lawson exponential schemes for the costate approximation. In the numerical tests a shooting strategy is employed in order to verify the accuracy, up to the fourth order, of the innovative procedure we propose.

Spatially explicit models consisting of reaction-diffusion partial differential equations are considered in order to model prey-predator interactions, since it is known that the role of spatial processes reveals of great interest in the study of the effects of habitat fragmentation on biodiversity. As almost all of the realistic models in biology, these models are nonlinear and their solution is not knwon is closed form. Our aim is approximating the solution itself by means of exponential Runge-Kutta integrators. Moreover, we apply the shift-and-invert Krylov approach in order to evaluate the entire functions needed for implementing the exponential method. This numerical procedure reveals to be very efficient in avoiding numerical instability during the simulation, since it allows us to adopt high order in the accuracy.

The large H2020 project ECOPOTENTIAL (2015-2019, 47 partners, contributing to GEO and GEOSS http://www.ecopotential-project.eu/) is devoted to making best use of remote sensing and in situ data to improve future ecosystem benefits, adopting the view of ecosystems as one physical system with their environment, focusing on geosphere-biosphere interactions, Earth Critical Zone dynamics, Macrosystem Ecology and cross-scale interactions, the effect of extreme events and using Essential (Climate, Biodiversity and Ocean) Variables as descriptors of change. In ECOPOTENTIAL, remote sensing and in situ data are collected, processed and used for a betterunderstandingoftheecosystemdynamics,analysingandmodellingtheeffectsofglobalchangesonecosystem functions and services, over an array of different ecosystem types, including mountain, marine, coastal, arid and semi-arid ecosystems. The project focuses on a network of Protected Areas of international relevance, that is representative of the range of environmental and biogeographical conditions characterizing Europe. Some of the activitiesoftheprojectaredevotedtodetectandquantifythechangestakingplaceintheProtectedAreas,through the analysis of remote sensing observations, in-situ data and gridded climatic datasets. Likewise, the project aims atprovidingestimatesofthefutureecosystemconditionsindifferentclimateandenvironmentalchangescenarios. In all such endeavours, one is faced with cross-scale issues: downscaling of climate information to drive ecosystem response, and upscaling of local ecosystem changes to larger scales. So far, the analysis has been conducted mainly by using traditional methods, but there is wide room for improvement by using more refined approaches. In particular, a crucial question is how to upscale the information gained at single-site scale to larger, regional or continental scale, an issue that could benefit from using, for example, complex network analysis.

The full text can be downloaded at http://journal.embnet.org/index.php/embnetjournal/article/view/205/460

Effectively dealing with invasive species is a pervasive problem in environmental management. The damages, andassociated costs, that stem from invasive species are well known, as is the benefit from their removal. We investigateproblems where optimal control theory has been implemented, and we show that these problems can easily becomehypersensitive, making their numerical solutions unstable. We show that transforming these problems from state-adjointsystems to state-control systems can provide useful insights into the system dynamics and simplify the numerics. Weapply these techniques to two case studies: one of feral cats in Australia, where we use logistic growth; and the other ofwild-boars in Italy, where we include an Allee effect. A further development is to optimize the control strategy by takinginto account the spatio-temporal features of the invasive species control problems over large and irregular environments.The approach is used in a management scenario where the invasive species to be controlled with an optimal allocationof resources is the deciduous tree Ailanthus Altissima, infesting the Alta Murgia National Park in the south of Italy.This work has been carried out within the H2020 project ECOPOTENTIAL (http://www.ecopotential-project.eu),coordinated by CNR-IGG. The project has received funding from the European Union's Horizon 2020 research andinnovation programme (grant agreement No 641762).

We examine spatially explicit models described by reaction-diffusion partial differential equations for the study of predator-prey population dynamics. The numerical methods we propose are based on the coupling of a finite difference/element spatial discretization and a suitable partitioned Runge-Kutta scheme for the approximation in time. The RK scheme here implemented uses an implicit scheme for the stiff diffusive term and a partitioned RK symplectic scheme for the reaction term (IMSP schemes). We revisit some results provided in the literature for the classical Lotka-Volterra system and the Rosenzweig-MacArthur model. We then extend the approach to metapopulation dynamics in order to numerically investigate the effect of migration through a corridor connecting two habitat patches. Moreover, we analyze the synchronization properties of subpopulation dynamics, when the migration occurs through corridors of variable sizes.

A challenging task in the management of Protected Areas is to control thespread of invasive species, either floristic or faunistic, and the preservation of indigenousendangered species, tipically competing for the use of resources in a fragmentedhabitat. In this paper, we present some mathematical tools that have beenrecently applied to contain the worrying diffusion of wolf-wild boars in a SouthernItaly Protected Area belonging to the Natura 2000 network. They aim to solve theproblem 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 intoaccount the spatio-temporal features of the invasive species over large and irregularenvironments. This approach can be successfully applied, with an optimal allocationof resources, to control an invasive alien species infesting the Alta Murgia NationalPark: Ailanthus altissima. This species is one of the most invasive species in Europeand its eradication and control is the object of research projects and biodiversityconservation actions in both protected and urban areas [11]. We lastly present, as afurther example, the effects of the introduction of the brook trout, an alien salmonidfrom North America, in naturally fishless lakes of the Gran Paradiso National Park,study site of an on-going H2020 project (ECOPOTENTIAL).

We apply the Z-control approach to a generalized predator prey system and consider the specific case of indirect control of the prey population. We derive the associated Z-controlled model and investigate its properties from the point of view of the dynamical systems theory. The key role of the design parameter A. for the successful application of the method is stressed and related to specific dynamical properties of the Z-controlled model. Critical values of the design parameter are also found, delimiting the lambda-range for the effectiveness of the Z-method. Analytical results are then numerically validated by the means of two ecological models: the classical Lotka-Volterra model and a model related to a case study of the wolf wild boar dynamics in the Alta Murgia National Park. Investigations on these models also highlight how the Z-control method acts in respect to different dynamical regimes of the uncontrolled model. (C) 2016 The Authors. Published by Elsevier Inc.

We develop a modelling approach for the optimal spatiotemporal control of invasive species in naturalprotected areas of high conservation value. The proposed approach, based on diusion equations, isspatially explicit, and includes a functional response (Holling type II) which models the control rateas a function of the invasive species density. We apply a budget constraint to the control programand search for the optimal eort allocation for the minimization of the invasive species density. Boththe initial density map and the land cover map used to estimate the habitat suitability to the speciesdiusion, have been generated by using very high resolution satellite images and validated by means ofground truth data. The approach has been applied to the Alta Murgia National Park, one of the studysite of the on-going H2020 project ECOPOTENTIAL: Improving Future Ecosystem Benets ThroughEarth Observations' (http://www.ecopotential-project.eu) which has received funding from the EuropeanUnion's Horizon 2020 research and innovation programme under grant agreement No 641762. All theground data regarding Ailanthus altissima (Mill.) Swingle presence and distribution are from the EULIFE Alta Murgia Project (LIFE12 BIO/IT/000213) titled Eradication of the invasive exotic plant speciesAilanthus altissima from the Alta Murgia National Park funded by the LIFE+ nancial instrument ofthe European Commission.

Improving strategies for the control and eradication of invasive species is an important aspect of nature conservation, an aspect where mathematical modeling and optimization play an important role. In this paper, we introduce a reaction-diffusion partial differential equation to model the spatiotemporal dynamics of an invasive species, and we use optimal control theory to solve for optimal management, while implementing a budget constraint. We perform an analytical study of the model properties, including the well-posedness of the problem. We apply this to two hypothetical but realistic problems involving plant and animal invasive species. This allows us to determine the optimal space and time allocation of the efforts, as well as the final length of the removal program so as to reach the local extinction of the species.

The threat, impact and management problems associated with alien plant invasions are increasingly becoming a major issue in environmental conservation. Invasive species cause significant damages, and high associated costs. Controlling them cost-effectively is an ongoing challenge, and mathematical models and optimizations are becoming increasingly popular as a tool to assist managers. The aim of this study is to develop a modelling approach for the optimal spatiotemporal control of invasive species in natural protected areas of high conservation value. Typically, control programs are either distributed uniformly across an area, or applied with a given fixed intensity, although there is no guarantee that such a strategy would be cost-effective at the conservation asset. The proposed approach, based on diffusion equations, is spatially explicit, and includes a functional response (Holling type II) which models the control rate as a function of the invasive species density. We apply a budget constraint to the control program and search for the optimal effort allocation for the minimisation of the invasive species density. Remote sensing derived input layers 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. 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. The approach has been applied to the Alta Murgia National Park, where the EU LIFE Alta Murgia Project is underway with the aim to eradicate Ailanthus altissima, one of the most invasive alien plant species in Europe. The Alta Murgia National Park is one of the study site of the on-going H2020 project ECOPOTENTIAL which aims at the integration of modelling tools and Earth Observations for a sustainable management of protected areas. The H2020 project 'ECOPOTENTIAL: Improving Future Ecosystem Benefits Through Earth Observations' (http://www.ecopotential-project.eu) has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 641762. All ground data regarding Ailanthus altissima (Mill.) Swingle presence and distribution are from the EU LIFE Alta Murgia Project (LIFE12 BIO/IT/000213 titled "Eradication of the invasive exotic plant species Ailanthus altissima from the Alta Murgia National Park" funded by the LIFE+ financial instrument of the European Commission).

Positive Poisson integrators for approximating Lotka-Volterra predator-prey model will be described. They are based on the trasformatin of the systems into the field of positive solutions by means of a log transformation. Splitting and composition schemes for positive approximation of population dynamics with prey logistic growth and Holling II type functional response will be also introduced.

We propose novel positive numerical integrators for approximating predator-prey models. The schemes are based on suitable symplectic procedures applied to the dynamical system written in terms of the log transformation of the original variables. Even if this approach is not new when dealing with Hamiltonian systems, it is of particular interest in population dynamics since the positivity of the approximation is ensured without any restriction on the temporal step size. When applied to separable M-systems, the resulting schemes are proved to be explicit, positive, Poisson maps. The approach is generalized to predator-prey dynamics which do not exhibit an M-system structure and successively to reaction-diffusion equations describing spatially extended dynamics. A classical polynomial Krylov approximation for the diffusive term joint with the proposed schemes for the reaction, allows us to propose numerical schemes which are explicit when applied to well established ecological models for predator-prey dynamics. Numerical simulations show that the considered approach provides results which outperform the numerical approximations found in recent literature.

tProtected areas are experiencing increased levels of human pressure. To enable appropriate conserva-tion action, it is critical to map and monitor changes in the type and extent of land cover/use and habitatclasses, which can be related to human pressures over time. Satellite Earth observation (EO) data andtechniques offer the opportunity to detect such changes. Yet association with field information and expertinterpretation by ecologists is required to interpret, qualify and link these changes to human pressure.There is thus an urgent need to harmonize the technical background of experts in the field of EO dataanalysis with the terminology of ecologists, protected area management authorities and policy makers inorder to provide meaningful, context-specific value-added EO products. This paper builds on the DPSIRframework, providing a terminology to relate the concepts of state, pressures, and drivers with the appli-cation of EO analysis. The type of pressure can be inferred through the detection of changes in state (i.e.changes in land cover and/or habitat type and/or condition). Four broad categories of changes in stateare identified, i.e. land cover/habitat conversion, land cover/habitat modification, habitat fragmentationand changes in landscape connectivity, and changes in plant community structure. These categories ofchange in state can be mapped through EO analyses, with the goal of using expert judgement to relatechanges in state to causal direct anthropogenic pressures. Drawing on expert knowledge, a set of pro-tected areas located in diverse socio-ecological contexts and subject to a variety of pressures are analysedto (a) link the four categories of changes in state of land cover/habitats to the drivers (anthropogenic pres-sure), as relevant to specific target land cover and habitat classes; (b) identify (for pressure mapping) themost appropriate spatial and temporal EO data sources as well as interpretations from ecologists andfield data useful in connection with EO data analysis. We provide detailed examples for two protectedareas, demonstrating the use of EO data for detection of land cover/habitat change, coupled with expertinterpretation to relate such change to specific anthropogenic pressures. We conclude with a discussionof the limitations and feasibility of using EO data and techniques to identify anthropogenic pressures,suggesting additional research efforts required in this direction

Most physical phenomena are described by time-dependent Hamiltonian systems with qualitative features that should be preserved by numerical integrators used for approximating their dynamics. The initial energy of the system together with the energy added or subtracted by the outside forces, represent a conserved quantity of the motion. For a class of time-dependent Hamiltonian systems [8] this invariant can be defined by means of an auxiliary function whose dynamics has to be integrated simultaneously with the system's equations. We propose splitting procedures featured by a SB3A property that allows to construct composition methods with a reduced number of determining order equations and to provide the same high accuracy for both the dynamics and the preservation of the invariant quantity.

It is known that symplectic algorithms do not necessarily conserve energy even for the harmonic oscillator. However, for separable Hamiltonian systems, splitting and composition schemes have the advantage to be explicit and can be constructed to preserve energy. In this paper we describe and test an integrator built on a one-parameter family of symplectic symmetric splitting methods, where the parameter is chosen at each time step so as to minimize the energy error. For second-degree polynomial Hamiltonian functions as the one describing the linear oscillator, we build up second and fourth order symmetric methods which are symplectic, energy-preserving and explicit. For non-linear examples, it is possible to construct schemes with minimum error on energy conservation. The methods are semi-explicit in the sense that they require, as additional computational effort, the search for a zero of a scalar function with respect to a scalar variable. Therefore, our approach may represent an effective alternative to energy-preserving implicit methods whenever multi-dimensional problems are dealt with as is the case of many applications of interest.

We consider splitting methods for the numerical integration of separable non-autonomous differential equations. In recent years, splitting methods have been extensively used as geometric numerical integrators showing excellent performances (both qualitatively and quantitatively) when applied on many problems. They are designed for autonomous separable systems, and a substantial number of methods tailored for different structures of the equations have recently appeared. Splitting methods have also been used for separable non-autonomous problems either by solving each non-autonomous part separately or after each vector field is frozen properly. We show that both procedures correspond to introducing the time as two new coordinates. We generalize these results by considering the time as one or more further coordinates which can be integrated following either of the previous two techniques. We show that the performance as well as the order of the final method can strongly depend on the particular choice. We present a simple analysis which, in many relevant cases, allows one to choose the most appropriate split to retain the high performance the methods show on the autonomous problems. This technique is applied to different problems and its performance is illustrated for several numerical examples.

This work deals with infinite horizon optimal growth models and uses the results in the Mercenier and Michel (1994a) paper as a starting point. Mercenier and Michel (1994a) provide a one-stage Runge-Kutta discretization of the above-mentioned models which preserves the steady state of the theoretical solution. They call this feature the "steady-state invariance property". We generalize the result of their study by considering discrete models arising from the adoption of s-stage Runge-Kutta schemes. We show that the steady-state invariance property requires two different Runge-Kutta schemes for approximating the state variables and the exponential term in the objective function. This kind of discretization is well-known in literature as a partitioned symplectic Runge-Kutta scheme. Its main consequence is that it is possible to rely on the well-stated theory of order for considering more accurate methods which generalize the first order Mercenier and Michel algorithm. Numerical examples show the efficiency and accuracy of the proposed methods up to the fourth order, when applied to test models.

Periodic monitoring of biodiversity changes at a landscape scale constitutes a key issue for conservation managers. Earth Observation (EO) data offers a potential solution, through direct or indirect mapping of species or habitats. Most national and international programs rely on the use of Land Cover (LC) and/or Land Use (LU) classification systems. Yet, these are not as clearly relatable to biodiversity in comparison to habitat classifications, and provide less scope for monitoring. While a conversion from LC/LU classification to habitat classification can be of great utility, differences in definitions and criteria have so far limited the establishment of a unified approach for such translation between these two classification systems. Focusing on five Mediterranean NATURA 2000 sites, this paper considers the scope for three of the most commonly used global LC/LU taxonomies - CORINE Land Cover (CLC), the Food and Agricultural Organisation (FAO) Land Cover Classification System (LCCS) and the International Geosphere-Biosphere Programme (IGBP) to be translated to habitat taxonomies. Through both quantitative and expert knowledge based qualitative analysis of selected taxonomies, FAO-LCCS turns out to be the best candidate to cope with the complexity of habitat description and provides a framework for EO and in-situ data integration for habitat mapping, reducing uncertainties and class overlaps and bridging the gap between LC/LU and habitats domains for landscape monitoring - a major issue for conservation. This study also highlights the need to modify the FAO-LCCS hierarchical class description process to permit the addition of attributes based on class-specific expert knowledge to select multi-temporal (seasonal) EO data and improve classification. An application of LC/LU to habitat mapping is provided for a coastal Natura 2000 site with high classification accuracy as a resultKey words: Mapping; land cover; land use; habitat; earth observation; taxonomies; Natura 2000; classification schemes