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).

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).

We investigate how the Z-type dynamic approach can be applied to control backward bifurcation phenomena in epidemic models. Because of its rich phenomenology, that includes stationary or oscillatory subcritical persistence of the disease, we consider the SIR model introduced by Zhou & Fan in [Nonlinear Analysis: Real World Applications, 13(1), 312-324, 2012] and apply the Z-control approach in the specific case of indirect control of the infective population. We derive the associated Z-controlled model both when the desired Z-controlled equilibrium is an endemic equilibrium with a very low number of infectives and when the Z-controlled equilibrium is a disease-free equilibrium. We investigate the properties of these Z-controlled models from the point of view of the dynamical system theory and elucidate the key role of the design parameter lambda. Numerical investigations on the model also highlight the impacts of the Z-control method on the backward scenario and on a variety of dynamical regimes emerging from it.

Effectively dealing with invasive species is a pervasive problem in environmental management. The damages that stem from invasive species are well known. However, controlling them cost-effectively is an ongoing challenge, and mathematical modeling and optimization are becoming increasingly popular as a tool to assist management. In this paper we investigate problems where optimal control theory has been implemented. We show that transforming these problems from state-costate systems to state-control systems provides the complete qualitative description of the optimal solution and leads to its theoretical expression for free terminal time problems. We apply these techniques to two case studies: one of feral cats in Australia, where we use logistic growth; and the other of wild-boars in Italy, where we include an Allee effect. (C) 2019 The Authors. Published by Elsevier Ltd.

Routine in linguaggio R per il controllo ottimo delle specie invasive

We present a modified version of an existing lake ecosystem model, describing a trophic chain generated by nutrients, phytoplankton and zooplankton (NPZ model). The NPZ model takes into account the vertical dynamics of the biomasses of the main species. We tailor the model to specific ecosystems by including seasonality in the dynamics of the various compartments. Moreover, different species exhibit a different behaviour with respect to diffusion and to the rate of vertical movement. With this model, we simulate the ecosystem dynamics of Alpine lakes located in study sites of the H2020 ECOPOTENTIAL project.

We develop a modelling approach for the optimal spatiotemporal control of invasive species in natural protected areas of high conservation value. The proposed approach, based on diusion 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 eort allocation for the minimization of the invasive species density. Both the initial density map and the land cover map used to estimate the habitat suitability to the species diusion, 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, one of the study site of the on-going H2020 project ECOPOTENTIAL: Improving Future Ecosystem Benets Through Earth Observations' (http://www.ecopotential-project.eu) which has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 641762. All the 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+ nancial instrument of the European Commission.

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).

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 beneficial effects of physical activity for the prevention and management of several chronic diseases are widely recognized. Mathematical modeling of the effects of physical exercise in body metabolism and in particular its influence on the control of glucose homeostasis is of primary importance in the development of eHealth monitoring devices for a personalized medicine. Nonetheless, to date only a few mathematical models have been aiming at this specific purpose. We have developed a whole-body computational model of the effects on metabolic homeostasis of a bout of physical exercise. Built upon an existing model, it allows to detail better both subjects' characteristics and physical exercise, thus determining to a greater extent the dynamics of the hormones and the metabolites considered.

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.

The numerical solution of reaction-diffusion systems modelling predator-prey dynamics using implicit-symplectic (IMSP) schemes is relatively new. When applied to problems with chaotic dynamics they perform well, both in terms of computational effort and accuracy. However, until the current paper, a rigorous numerical analysis was lacking. We analyse the semi-discrete in time approximations of a first-order IMSP scheme applied to spatially extended predator-prey systems. We rigorously establish semi-discrete a priori bounds that guarantee positive and stable solutions, and prove an optimal a priori error estimate. This analysis is an improvement on previous theoretical results using standard implicit-explicit (IMEX) schemes. The theoretical results are illustrated via numerical experiments in one and two space dimensions using fully-discrete finite element approximations.

Effectively dealing with invasive species is a pervasive problem in environmental management. The damages, and associated costs, that stem from invasive species are well known, as is the benefit from their removal. We investigate problems where optimal control theory has been implemented, and we show that these problems can easily become hypersensitive, making their numerical solutions unstable. We show that transforming these problems from state-adjoint systems to state-control systems can provide useful insights into the system dynamics and simplify the numerics. We apply these techniques to two case studies: one of feral cats in Australia, where we use logistic growth; and the other of wild-boars in Italy, where we include an Allee effect. A further development is to optimize the control strategy by taking into 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 allocation of 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 and innovation programme (grant agreement No 641762).

A challenging task in the management of Protected Areas is the conservation of natural habitats and native endangered species through the optimization of control strategies for invasive plant or animal species, typically competing for the use of resources in a fragmented habitat [1]. We review two cases of control strategies on the wolf-wild boar populations in a Southern Italy Protected Area belonging to the Natura 2000 network [2,3]. The challenge for the regional authorities is to plan conservation policies able to maintain the population of wolves while limiting the presence of wild boars, here considered invasive because of their harmfulness on cultivated areas. The first strategy [2] considers the impact of control policies on predator-prey dynamics in fragmented habitats by simulating different dynamical scenarios theoretically analysed with the aggregation method. The key warning from the model is that a very careful combination of control - through proper planning programs - and migration processes among patches of habitats - through the existing suitable ecological corridors - must be used in order to limit the wild-boar population while preserving wolves from extinction. The second strategy has been developed to apply the Z-control approach to a generalized predator-prey system [3]. It considers the specific case of indirect control of the prey (invasive) population. The key role of the model design parameter is stressed and the critical values of the design parameter are found, delimiting the parameter range for the successful application of the Z-method. A further development is the optimization of a control strategy by taking into account the spatio-temporal data related to the control problem of an invasive species over a wide natural protected area. That approach will be applied to the Alta Murgia National Park, where a EU LIFE+ project is underway to eradicate Ailanthus altissima, included in the list of the most invasive alien plant species in Europe causing serious damages both in protected and urban areas [4]. The Alta Murgia National Park is one of the study site of an on-going H2020 project (ECOPOTENTIAL). This work has been carried out within the H2020 project `ECOPOTENTIAL: Improving Future Ecosystem Benefits Through Earth Observations', coordinated by CNR-IGG (http://www.ecopotential-project.eu). The project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 641762. Bibliography 1)Baker, C. M., Target the Source: Optimal Spatiotemporal Resource Allocation for Invasive Species Control, CONS. LETTERS, pp 1-8, 2016, doi: 10.1111/conl.12236 2)Lacitignola, D.; Diele, F.; Marangi, C., Dynamical scenarios from a two-patch predator-prey system with human control - Implications for the conservation of the wolf in the Alta Murgia National Park ECOLOGICAL MODELLING, Vol. 316, pp 28-40, 2015, doi: 10.1016/j.ecolmodel.2015.07.027 3)Lacitignola, D.; Diele, F.; Marangi, C.; Provenzale A., On the dynamics of a generalized predator-prey system with Z-type control, MATHEMATICAL BIOSCIENCES, vol. 280, pp 10-23, 2016, doi: 10.1016/j.mbs.2016.07.011 4)Casella F., Vurro M. , Ailanthus altissima (tree of heaven): Spread and harmfulness in a case-study urban area, Arboricultural Journal: The International Journal of Urban Forestry, 35(3), pp 172-181, 2013, doi: 10.1080/03071375.2013.852352

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.

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 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.