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.

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.

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.

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).

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.

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).

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.

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.

The non-Newtonian calculi constitute innitely many alternatives to the classical calculus. Among themultiplicative non-Newtonian calculi, the geometric calculus provides a natural framework for problemsinvolving positivity preserving operators. Indeed, some existing positive symplectic schemes for integratingpopulation dynamics can be reinterpreted in the light of the geometric calculus framework. Anovel application in the eld of chemical kinetics is proposed. Mass action chemical systems are massconservative and the solutions are required to remain positive. Hence, numerical methods applied tosuch kind of equations have to maintain unconditional positivity as well as conservativity in a discretesense. Composition of geometric Euler scheme with a positive integrator is proposed for integratingproduction-destruction systems. This work has been supported by GNCS-INDAM.

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.