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

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.

In this paper we investigate the stability properties of the so-called gBBKS and GeCo methods, which belong to the class of nonstandard schemes and preserve the positivity as well as all linear invariants of the underlying system of ordinary differential equations for any step size. A stability investigation for these methods, which are outside the class of general linear methods, is challenging since the iterates are always generated by a nonlinear map even for linear problems. Recently, a stability theorem was derived presenting criteria for understanding such schemes. For the analysis, the schemes are applied to general linear equations and proven to be generated by C1-maps with locally Lipschitz continuous first derivatives. As a result, the above mentioned stability theorem can be applied to investigate the Lyapunov stability of non-hyperbolic fixed points of the numerical method by analyzing the spectrum of the corresponding Jacobian of the generating map. In addition, if a fixed point is proven to be stable, the theorem guarantees the local convergence of the iterates towards it. In the case of first and second order gBBKS schemes the stability domain coincides with that of the underlying Runge–Kutta method. Furthermore, while the first order GeCo scheme converts steady states to stable fixed points for all step sizes and all linear test problems of finite size, the second order GeCo scheme has a bounded stability region for the considered test problems. Finally, all theoretical predictions from the stability analysis are validated numerically.

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.

We analyze a second-order accurate implicit-symplectic (IMSP) scheme for reaction-diffusion systems modeling spatiotemporal dynamics of predator-prey populations. We prove stability and errors estimates of the semi-discrete-in-time approximations, under positivity assumptions. The numerical simulations confirm the theoretically derived rates of convergence and show an improved accuracy in the second-order IMSP in comparison with the first-order IMSP, at same computational cost.

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.

We generalize the nonstandard Euler and Heun schemes in order to provide explicit geometric numerical integrators for biochemical systems, here denoted as GeCo schemes, that preserve both positivity of the solutions and linear invariants. We relax the request on the order convergence of the denominator function for the first-order approximation and we let it depend on the step size also throughout the solution approximating values. The first-order variant is exact on a two-dimensional linear test problem. Moreover, we introduce a class of modified mGeCo(alpha) schemes and, by tuning the parameter alpha >= 1, we improve the numerical performance of GeCo integrators on some examples taken from the literature. A numerical comparison with BBKS and mBBKSschemes, which are implicit integrators, positive and conserve linear invariants, show the gain in efficiency of both GeCo and mGeCo(alpha) procedures as they generate similar errors with an explicit functional form. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.

Invasive species cause huge amounts of environmental, economic, social and cultural damage in Europe and worldwide. Improving measures to control them is an ongoing challenge, and mathematical modeling and optimization are becoming increasingly popular as a tool to assist management (1; 2; 4). We analyse an optimal control model for the control of invasive species which aims to find the best temporal resource allocation strategy for the population reduction, under a budget constraint (3). We derive the optimality system in the state and control variables and we use the phase-space analysis to provide qualitative insights about the behaviour of the optimal solution. We pay special attention to the nature of the optimal trajectories in long time intervals and the explore the Turnpike property of the problem (5). Finally, we introduce a numerical scheme for the solution of the state-costate nearly-Hamiltonian system, based on exponential-Lawson symplectic Runge-Kutta schemes applied in a forward-backward procedure.

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).

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

La proprietà di conservazione della positività dei metodi numerici applicati ai sistemi differenziali di tipo ODE e PDE a valori iniziali e/o ai bordi, è un argomento di ricerca di notevole interesse. La positività del flusso numerico è un aspetto fondamentale in numerose applicazioni che vanno dalla biologia computazionale, alla dinamica molecolare, alla modellistica in ambito ecologico, dovunque risulti fondamentale che le grandezze in gioco (popolazioni, densità, concentrazioni) non assumano valori negativi. Tale condizione, fatta eccezione per lo schema di Eulero Implicito, non è verificata dai metodi standard (Runge-Kutta o multistep), a meno di imporre restrizioni sul passo di integrazione talvolta molto significative. Tuttavia, le restrizioni sul passo di integrazione diminuiscono sensibilmente l'efficienza dei metodi numerici a tal punto da renderli di fatto inutilizzabili nelle applicazioni reali. La letteratura più recente si è quindi focalizzata sulla costruzione di integratori numerici che garantiscono la positività del flusso numerico per costruzione. Tra i lavori su questo argomento, citiamo [10, 4] in cui vengono proposte tecniche di splitting and composition per la soluzione di modelli differenziali.

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.