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We apply the Z-control approach to a SEIR model including a overexposure mechanism and consider awareness as a time-dependent variable whose dynamics is not assigned a priori. Exploiting the potential of awareness to produce social distancing and self-isolation among susceptibles, we use it as an indirect control on the class of infective individuals and apply the Z-control approach to detect what trend awareness must display over time in order to eradicate the disease. To this aim, we generalize the Z-control procedure to appropriately treat an uncontrolled model with more than two governing equations. Analytical and numerical investigations on the resulting Z-controlled system show its capability in controlling some representative dynamics within both the backward and the forward scenarios. The awareness variable is qualitatively compared to Google Trends data on Covid-19 and qualitative indications are inferred in view of the disease control.

This paper focuses on the impact of cross-diffusion for Turing-Hopf instability in a morphochemical model for electrodeposition (DIB) and completes the analysis on the role of cross-diffusion on pattern formation in electrodeposition we recently carried out in [Lacitignola et al. 2018]. We derive and discuss conditions for the destabilization a ´ la Turing of the Hopf limit cycle in the neighborhood of the codimension-two Turing-Hopf bifurcation point, resulting in oscillatory Turing patterns. Numerical validation of the above theoretical findings is provided both in the case of Weak and Strong Turing-Hopf instability. Experimental evidence of this kind of phenomenology is then discussed also in relation to the charging and discharging process of a battery.

In this paper, we use the Z-control approach to get further insight on the role of awareness in the management of epidemics that, just like Covid-19, display a high rate of overexposure because of the large number of asymptomatic people. We focus on a SEIR model including a overexposure mechanism and consider awareness as a time-dependent variable whose dynamics is not assigned a priori. Exploiting the potential of awareness to produce social distancing and self-isolation among susceptibles, we use it as an indirect control on the class of infective individuals and apply the Z-control approach to detect what trend must awareness display over time in order to eradicate the disease. To this aim, we generalize the Z-control procedure to appropriately treat an uncontrolled model with more than two governing equations. Analytical and numerical investigations on the resulting Z-controlled system show its capability in controlling some representative dynamics within both the backward and the forward scenarios. The awareness variable is qualitatively compared to Google Trends data on Covid-19 that are discussed in the perspective of the Z-control approach, inferring qualitative indications in view of the disease control. The cases of Italy and New Zealand in the first phase of the pandemic are analyzed in detail. The theoretical framework of the Z-control approach can hence offer the chance to reflect on the use of Google Trends as a possible indicator of good management of the epidemic.

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).

In this study, the formation of the adult sea urchin shape is rationalized within the Turing's theory paradigm. The emergence of protrusions from the expanding underlying surface is described through a reaction-diffusion model with Gray-Scott kinetics on a growing oblate spheroid. The case of slow exponential isotropic growth is considered. The model is first studied in terms of the spatially homogenous equilibria and of the bifurcations involved. Turing diffusion-driven instability is shown to occur and the impact of the slow exponential growth on the resulting Turing regions adequately discussed. Numerical investigations validate the theoretical results showing that the combination between an inhibitor and an activator can result in a distribution of spot concentrations that underlies the development of ambulacral tentacles in the sea urchin's adult stage. Our findings pave the way for a model-driven experimentation that could improve the current biological understanding of the gene control networks involved in patterning.

This paper focuses on the emergence of spiral waves in a specific morpho-electrochemical reaction-diffusion model on a sphere. This study fits in the framework of the morphological control of material electrodeposited onto spherical particles that is crucial to the energetic efficiency of the recharge process as well as to the durability of energy storage devices. The spherical geometry for the electrode surface is of notable practical interest since spheres are the shape of choice for flow batteries and metal-air devices [35]. Motivated by this technological framework, in this paper we extend the results on pattern formation in [27] to include investigations on the spiral wave phenomenology. We show that spiral waves emerge because of the interplay between two specific model parameters: one regulating the oscillatory dynamics in the kinetics and the other one related to the domain size. We present systematic numerical simulations based on the finite element method LSFEM [45,50] accompanied by the computation of suitable indicators that allow to characterize and compare the spatio-temporal features. Interestingly, the model also supports a mechanism of spirals break up leading to a complex spatio-temporal phenomenology. The findings of our study have been validated with experimental results on Ag-In and Ag-Co electrodeposition.

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.

Alloy electrodeposition processes have been experimentally shown to exhibit electrokinetic instability that can lead to compositional heterogeneity in the electrodeposit bulk. Inspired by these experimental evidences we consider the DIB model, a morphochemical reaction-diffusion system introduced in [Bozzini et al., J. Solid State Electr.17, 467–479 (2013)] as a mean of understanding the formation of morphological patterns found in the electrodeposition processes. We discuss its rich spatiotemporal dynamics from both the analytical and numerical points of view, revealing interesting classes of morphogenetic scenarios in good qualitative accordance with experiments. The DIB model can in fact undergo to Turing and Hopf instabilities and classical Turing patterns as well as oscillating Turing patterns can be detected. In the Hopf region, we prove the existence of spiral wave behaviour and observe transition to spatiotemporal chaos via spirals break up. The role of cross-diffusion in pattern formation and selection is also discussed as well as some recent extensions of the model to spherical domains. References [1] Bozzini, B., Lacitignola, D., Sgura, I., 2013. Spatio-temporal organization in alloy electrodeposition: a morphochemical mathematical model and its experimental validation. Journal of Solid State Electrochemistry 17 (2), 467–479. [2] Lacitignola, D., Bozzini, B., Sgura, I., 2015. Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay. European Journal of Applied Mathematics 26 (2), 143–173. [3] Lacitignola, D., Bozzini, B., Sgura, I., 2014. Spatio-temporal organization in a morphochemical electrodeposition model: Analysis and numerical simulation of spiral waves. Acta Applicandae Mathematicae 132, 377–389. [4] Lacitignola, D., Bozzini, B., Frittelli, M., Sgura, I., 2017. Turing pattern formation on the sphere for a morphochemical reaction-diffusion model for electrodeposition. Communications in Nonlinear Science and Numerical Simulation 48, 484 – 508. [5] Lacitignola, D., Bozzini, B., Peipmann, R., Sgura, I., 2018. Cross-diffusion effects on a morphochemical model for electrodeposition, Applied Mathematical Modelling 57, 492– 513. [6] Lacitignola, D., Sgura I., Bozzini, B., Dobrovolska Ts., Krastev I., 2019. Spiral waves on the sphere for an alloy electrodeposition model, Communications in Nonlinear Science and Numerical Simulation 79, 104930.

We analyze the effects of cross-diffusion on pattern formation in a PDE reaction-diffusion system introduced in Bozzini et al. 2013 to describe metal growth in an electrodeposition process. For this morphochemical model - which refers to the physico-chemical problem of coupling of growth morphology and surface chemistry - we have found that negative cross-diffusion in the morphological elements as well as positive cross-diffusion in the surface chemistry produce larger Turing parameter spaces and favor a tendency to stripeness that is not found in the case without cross-diffusion. The impact of cross-diffusion on pattern selection has been also discussed by the means of a stripeness index. Our theoretical findings are validated by an extensive gallery of numerical simulations that allow to better clarify the role of cross-diffusion both on Turing parameter spaces and on pattern selection. Experimental evidence of cross-diffusion in electrodeposition as well as a physico-chemical discussion of the expected impact of cross diffusion-controlled pattern formation in alloy electrodeposition processes complete the study.

Alcohol dependence is among the main healthy risky behavior due to the high relevance of negative health and social effect. We study a mathematical model, given by nonlinear ordinary differential equations, describing the spread of high–risk alcohol consumption behavior within a community of individuals. We describe the peer-influence effects on alcohol addiction by assuming that susceptibles become heavy drinkers through the mechanism of imitation. We show that the model may exhibit the phenomenon of backward bifurcation. This means that alcohol problems may persist in the population even if the basic reproduction number is less than one. Nonlinear stability analysis of equilibria is also provided.

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.

In this paper, we report on the use of high-space resolution soft X-ray fluorescence microspectroscopy for the study of electrodeposited composites containing catalytic ternary metal nanostructures. X-ray fluorescence maps are interpreted in terms of a dynamic mathematical model of the electrode morphology and metal space distribution, allowing to reproduce the observed space patterns and electrochemical transients by assigning an appropriate set of electrokinetic parameters. The discussed materials-science case is the electrochemical growth of a Mn-Mg-Cu-polypyrrole nanocomposite electrocatalyst material – free of expensive Pt and environmentally unfriendly Co – with promising performance for fuel-cell oxygen electrodes. The synergy of high-resolution compositional mapping with electrokinetic modelling not only provides the general rationale for quantitative use of potentially large compositional distribution datasets but also yields unprecedented insight into the specific catalyst synthesis process. The expounded application is just a prototypical case study of a more general approach, which can be employed for the understanding of electrochemical material science processes, both in situ and ex situ, as well as for the characterisation of the corresponding products, with no other limitations in principle than X-ray transmission and beam damage. Copyright © 2015 John Wiley & Sons, Ltd.

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

The present paper deals with the pattern formation properties of a specific morpho- electrochemical reaction-diffusion model on a sphere. The physico-chemical background to this study is the morphological control of material electrodeposited onto spherical parti- cles. The particular experimental case of interest refers to the optimization of novel metal- air flow batteries and addresses the electrodeposition of zinc onto inert spherical supports. Morphological control in this step of the high-energy battery operation is crucial to the energetic efficiency of the recharge process and to the durability of the whole energy- storage device. To rationalise this technological challenge within a mathematical modeling perspective, we consider the reaction-diffusion system for metal electrodeposition intro- duced in [Bozzini et al., J. Solid State Electr.17, 467–479 (2013)] and extend its study to spherical domains. Conditions are derived for the occurrence of the Turing instability phe- nomenon and the steady patterns emerging at the onset of Turing instability are investi- gated. The reaction-diffusion system on spherical domains is solved numerically by means of the Lumped Surface Finite Element Method (LSFEM) in space combined with the IMEX Euler method in time. The effect on pattern formation of variations in the domain size is investigated both qualitatively, by means of systematic numerical simulations, and quan- titatively by introducing suitable indicators that allow to assign each pattern to a given morphological class. An experimental validation of the obtained results is finally presented for the case of zinc electrodeposition from alkaline zincate solutions onto copper spheres.

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 λ. 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.

In this paper, we investigate from a theoretical point of view the 2D reaction-diffusion system for electrodeposition coupling morphology and surface chemistry, presented and experimentally validated in Bozzini et al. (2013J. Solid State Electr.17, 467–479). We analyse the mechanisms responsible for spatio-temporal organization. As a first step, spatially uniform dynamics is discussed and the occurrence of a supercritical Hopf bifurcation for the local kinetics is proved. In the spatial case, initiation of morphological patterns induced by diffusion is shown to occur in a suitable region of the parameter space. The intriguing interplay between Hopf and Turing instability is also considered, by investigating the spatio-temporal behaviour of the system in the neighbourhood of the codimension-two Turing--Hopf bifurcation point. An ADI (Alternating Direction Implicit) scheme based on high-order finite differences in space is applied to obtain numerical approximations of Turing patterns at the steady state and for the simulation of the oscillating Turing–Hopf dynamics.

We consider the case of measles in South Africa to show that an high vaccination coverage may be not enough - alone - to ensure measles eradication. The occurrence of certain epidemic episodes may in fact be encouraged by delays in the treatments or by not adequately fast clinical case management, which may be related to the backward bifurcation phenomenon as well as to an intriguing spiking dynamics which appears in the system for specific ranges of parameter values.

We focus on an epidemic model which incorporates a non-linear force of infection and two controls: an imperfect preventive vaccine given to susceptible individuals and therapeutic treatment given to infectious. We study both the cases of constant and non constant controls. In the case of constant controls we perform a qualitative analysis based on Lyapunov stability which allows to integrate the bifurcation analysis performed in a previous paper. The occurrence of a backward bifurcation is discussed in the perspective of disease control. The case of time-dependent controls is studied by means of the optimal control theory. The strategy is to minimize both the disease burden and the intervention costs. We derive the optimality system and solve it numerically. The characterization of the optimal time profile of the controls, together with the qualitative analysis provides a rather complete picture of the possible outcomes of the model.

In this paper we investigate from the numerical point of view the discrete DNA model proposed in Lacitignola and Saccomandi (Bull. Math. Biol., 2014) in order to test the robustness of the parametric resonance condition found in the limit of the continuum approximation. To describe more realistically the binding of RNA polymerase to the DNA macromolecule during the first stage of the transcription process, we here consider a localized DNA-RNA polymerase interaction and a relatively high number of base-pairs. Even with these more realistic assumptions, our findings confirm the ones found in the continuum limit and indicate that the parametric resonance phenomenon can be an intrinsic property of the discrete DNA model.

In this paper we derive Hopf instability conditions for the morphochemical mathematical model for alloy electrodeposition introduced and experimentally validated in Bozzini et al. (J. Solid State Electrochem., 17:467---479, 2013). Using normal form theory we show that in the neighborhood of the Hopf bifurcation, essential features of the system dynamics are captured by a specific Complex Ginzburg-Landau Equation (CGLE). The derived CGLE yields analytical results on the existence and stability of spiral waves. Moreover, the arising of spiral instability is discussed in terms of the relevant system parameters and the related phenomenology is investigated numerically. To face with the numerical approximation of the spiral structures and of their longtime oscillating behavior we apply an Alternating Direction Implicit (ADI) method based on high order finite differences in space.

This paper proposes a novel mathematical model for the formation of spatio-temporal patterns in electrodeposition. At variance with classical modelling approaches that are based on systems of reaction–diffusion equations just for chemical species, this model accounts for the coupling between surface morphology and surface composition as a means of understanding the formation of morphological patterns found in electroplating. The innovative version of the model described in this work contains an original, flexible and physically straightforward electrochemical source term, able to account for charge transfer and mass transport: adsorbate-induced effects on kinetic parameters are naturally incorporated in the adopted formalism. The relevant non-linear dynamics is investigated from both the analytical and numerical points of view. Mathematical modelling work is accompanied by an extensive, critical review of the literature on spatio-temporal pattern formation in alloy electrodeposition: published morphologies have been used as a benchmark for the validation of our model. Moreover, original experimental data are presented—and simulated with our model—on the formation of broken spiral patterns in Ni–P–W–Bi electrodeposition.

We consider a four-compartment tuberculosis model including exogenous reinfection. We derive sufficient conditions, in terms of the parameters of the system, which guarantee the occurrence of backward bifurcation. We also discuss the global stability of the endemic state by using a generalization of the Poincaré-Bendixson criterion. An application is given for the case of Internally Displaced People's Camps in North Uganda. The study suggests how important it is to provide qualitative indications on the threshold value of the population density in the area occupied by the camps, in order to possibly eradicate the disease.

A considerable amount of electrochemical and structural information is available on Au electrodeposition, but its dynamic aspects are still largely ignored. In this paper, an investigation of the microscopic Au electrodeposition dynamics is presented. Extensive sets of SERS spectra during prolonged potentiostatic electrodeposition (∼3 h) are recorded. Two different spot sizes have been used, in order to also obtain some information on the coupling of temporal and spatial fluctuations at the cathodic surface. The time evolution of the SERS background and the intensity of the stretching band of adsorbed CN− have been used as markers of surface morphology and chemistry respectively. Statistical analyses of these quantities allowed extraction of detailed information at the coupled dynamics of SERS enhancement and surface concentration of CN−. The spectroelectrochemical investigation is complemented by SEM observations and mathematical modelling of the coupled dynamics of cathode morphology and CN− sur...

An SEIR epidemic model with a nonlinear incidence rate is studied. The incidence is assumed to be a convex function with respect to the infective class of a host population. A bifurcation analysis is performed and conditions ensuring that the system exhibits backward bifurcation are provided. The global dynamics is also studied, through a geometric approach to stability. Numerical simulations are presented to illustrate the results obtained analytically. This research is discussed in the framework of the recent literature on the subject.

We study the global behavior of a non-linear susceptible-infectious-removed (SIR)-like epidemic model with a non-bilinear feedback mechanism, which describes the influence of information, and of information-related delays, on a vaccination campaign. We upgrade the stability analysis performed in d'Onofrio et al. [A. d'Onofrio, P. Manfredi, E. Salinelli, Vaccinating behavior, information, and the dynamics of SIR vaccine preventable diseases, Theor. Popul. Biol. 71 (2007) 301) and, at same time, give a special example of application of the geometric method for global stability, due to Li and Muldowney. Numerical investigations are provided to show how the stability properties depend on the interplay between some relevant parameters of the model.

We focus on the morphochemical reaction–diffusion model introduced in Bozzini et al. (2013) and carry out a nonlinear bifurcation analysis with the aim to characterize the shape and the amplitude of the patterns arising as the result of Turing instability of the physically relevant equilibrium. We perform a weakly nonlinear multiple scales analysis, and derive the normal form equations governing the amplitude of the patterns. These amplitude equations allow us to construct relevant solutions of the model equations and reveal the presence of multiple branches of stable solutions arising as the result of subcritical bifurcations. Hysteretic type phenomena are highlighted also through numerical simulations. We show the occurrence of spatial pattern propagation and derive the Ginzburg–Landau equation describing the envelope of the traveling wavefront.

This paper reports on spiral pattern formation in In–Co electrodeposition. We propose an approach to the understanding of this process based on: (i) compositional and chemical-state distribution analysis by high-resolution photoelectron microspectroscopy and (ii) a mathematical model able to capture the morphological features highlighted in the experiments. Microspectroscopy—complemented by electrochemical, structural and morphological characterisations—combined with mathematical modelling, analytical and numerical investigations, converge in pointing out the key role played by intermetallic electrodeposition in spiral formation.

In this paper we study the numerical approximation of Turing patterns corresponding to steady state solutions of a PDE system of reaction–diffusion equations modeling an electrodeposition process. We apply the Method of Lines (MOL) and describe the semi-discretization by high order finite differences in space given by the Extended Central Difference Formulas (ECDFs) that approximate Neumann boundary conditions (BCs) with the same accuracy. We introduce a test equation to describe the interplay between the diffusion and the reaction time scales. We present a stability analysis of a selection of time-integrators (IMEX 2-SBDF method, Crank–Nicolson (CN), Alternating Direction Implicit (ADI) method) for the test equation as well as for the Schnakenberg model, prototype of nonlinear reaction–diffusion systems with Turing patterns. Eventually, we apply the ADI-ECDF schemes to solve the electrodeposition model until the stationary patterns (spots & worms and only spots) are reached. We validate the model by comparison with experiments on Cu film growth by electrodeposition.

In this paper a reaction-diffusion system for electrochemical material growth processes is considered, including an external sinusoidal forcing term for the PDE equation describing the morphology of the electrodeposit surface profile. The numerical approximation by the Alternating Direction Implicit (ADI) method based on Extended Central Difference Formulas (ECDF) of order p = 4 in space is applied to investigate the way the variation of the frequency of the superimposed voltage sinusoid affects Turing pattern scenarios corresponding to steady state solutions of the unforced model. The ADI-ECDF method, introduced in [20] for the approximation of Turing patterns in the unforced case, is shown to be efficient from the computational point of view also to track oscillating Turing patterns for long-time simulations. In particular, the proposed method allows to identify a critical frequency range where the ripple effect arises, that is spots & worms patterns, related to the buildup of roughness in the material growth process, are suppressed and spatially homogeneous steady state solutions are attained. Such results have been validated by comparison with original experimental results on the growth of silver chloride films.

The coupling of surface shape dynamics and surface composition for a material growing by electrodeposition has been found in recent modelling work by the authors, to give rise to a rich morphogenetic scenario. In this paper we concentrate on a systematic description of morphogenesis occurring during metal and alloy electrodeposition. First of all we give a brief presentation of the physical facts relevant to the generation of shapes in electrochemical growth processes, essentially the balance between change-transfer and mass-transport rates. Hence, we review the mathematical modelling work recently published by the authors, based on analytical and numerical studies of a system of reaction-diffusion equations for morphological and adsorption dynamics. Eventually, a selection of examples of experimental validation of the model by means of numerical simulations is proposed.

We study the properties of a n2-dimensional Lotka–Volterra system describing competition among n different species with adaptive skills, i.e. whose interaction coefficients are time averages of the species level of interaction over their past. Starting by the case of adaptive competition among species all having the same carrying capacities, we focus our attention on the model obtained on perturbing the carrying capacity of a fixed species, which is made more or less disadvantaged. We prove the existence of a certain class of invariant subspaces and introduce a seven-dimensional reduced model, where n appears as a parameter, which gives full account of existence and stability of equilibria in the system. The relevance of this reduced model to the complete one has also been found when the time dependent regimes have been investigated. Ecologically relevant questions, i.e. species survival and the time dependent behavior of the system have also been analyzed focusing on the role of behavioral adaptation. In particular, we have found that competitive exclusion cannot occur but coexistence is possible in various forms (i.e. competitive stable equilibria and different periodic oscillations).

We consider a simple mesoscopic model of DNA in which the binding of the RNA polymerase enzyme molecule to the promoter sequence of the DNA is included through a substrate energy term modeling the enzymatic interaction with the DNA strands. We focus on the differential system for solitary waves and derive conditions--in terms of the model parameters--for the occurrence of the parametric resonance phenomenon. We find that what truly matters for parametric resonance is not the ratio between the strength of the stacking and the inter-strand forces but the ratio between the substrate and the inter-strands. On the basis of these results, the standard objection that longitudinal motion is negligible because of the second order seems to fail, suggesting that all the studies involving the longitudinal degree of freedom in DNA should be reconsidered when the interaction of the RNA polymerase with the DNA macromolecule is not neglected.

A challenge to disease control in modern societies is the spread of pseudo-rational exemption to vaccination, as a consequence of a comparison between the steadily declining risk of infection, and the perceived risk of side effects from the vaccine. Here we consider rational exemption in an SEIR model with information dependent vaccination where individuals use information on the spread of the disease as their information set. Using suitable assumptions, we show the dynamic implications of the interaction between rational exemption and current information. In particular, we show that the vaccinating behavior depending on current information can trigger oscillations, differently from the case of SIR diseases, where an information delay is needed to induce oscillations.

Three behavioral-epidemic models (i.e. epidemic systems including feedbacks that the information about an infectious disease has on its spreading) are introduced. Two relevant feedbacks are explicitly considered: the pseudo-rational exemption to vaccination and the information--related changes in contact patterns by healthy subjects. The global stability analysis of the endemic states is performed by means of the geometric approach to stability, with particular focus on a model of vaccination of adult susceptible subjects. Biological implications of the results are discussed.

We consider an epidemic model for the dynamics of a vaccine-preventable disease, which incorporates the treatment and an imperfect vaccine given to susceptible individuals. We show that in spite of the simple structure of the model, a backward bifurcation may always occur if the treatment rate is above a threshold value. This occurs regardless of the specific form of the force of infection, which is only required to be infinitesimal of the same order of the size of the infectious compartment I, as I→0. This includes many commonly used functionals, as the linear, the monotone saturating Michaelis-Menten and the non-monotone force of infection used to represent the ‘psychological effect’.

SIR and SIS epidemic models with information—related changes in contact patterns are introduced. The global stability analysis of the endemic equilibrium is performed by means of the Li–Muldowney geometric approach. Biological implications of the stability conditions are given.

Using a simple exact solution we point out an anomalous feature associated with the use of the semi-inverse method in a class of shear dependent viscosity non-Newtonian fluids.

In this paper we consider an analytical and numerical study of a reaction-diffusion system for describing the formation of transition front waves in some electrodeposition (ECD) experiments. Towards this aim, a model accounting for the coupling between morphology and composition of one chemical species adsorbed at the surface of the growing cathode is addressed. Through a phase-space analysis we prove the existence of travelling waves, moving with specific wave speed. The numerical approximation of the PDE system is performed by the Method of Lines (MOL) based on high order space semi-discretization by means of the Extended Central Difference Formulae (D2ECDF) introduced in [1]. First of all, to show the advantage of the proposed schemes, we solve the well-known Fisher scalar equation, focusing on the accurate approximation of the wave profile and of its speed. Hence, we provide numerical simulations for the electrochemical reaction-diffusion system and we show that the results obtained are qualitatively in good agreement with experiments for the electrodeposition of Au-Cu alloys.

A compartmental epidemic model, introduced by Gumel and Moghadas [1], is considered. The model incorporates a nonlinear incidence rate and an imperfect preventive vaccine given to susceptible individuals. A bifurcation analysis is performed by applying the bifurcation method introduced in [2], which is based on the use of the center manifold theory. Conditions ensuring the occurrence of backward bifurcation are derived. The obtained results are numerically validated and then discussed from both the mathematical and the epidemiological perspective.

In this paper a reaction-diffusion system modelling metal growth processes is considered, to investigate - within the electrodeposition context- the formation of morphological patterns in a finite two-dimensional spatial domain. Nonlinear dynamics of the system is studied from both the analytical and numerical points of view. Phase-space analysis is provided and initiation of spatial patterns induced by diffusion is shown to occur in a suitable region of the parameter space. Investigations aimed at establishing the role of some relevant chemical parameters on stability and selection of solutions are also provided. By the numerical approximation of the equations, simulations are presented which turn out to be in good agreement with experiments for the electrodeposition of Au-Cu and Au-Cu-Cd alloys.

The dynamics of a structured population model including cannibalism is analyzed. Hopf bifurcation threshold for the cannibalistic attack rate is detected. Linear and nonlinear stability analysis through the Lyapunov Direct Method is also provided. The effects of relevant parameters on the stability are discussed. In particular, cannibalism is found to have a stabilizing effect, whereas the prey growth effect is opposite. The result is emphasized by numerical simulations.

The period-doubling route to chaos has occupied a prominent position and it is still object of great interest among the different complex phenomena observed in nonlinear dynamical systems. The reason of such interest is that such route to chaos has been observed in many physical, chemical and ecological models when they change over from simple periodic to complex aperiodic motion. In interlinked social–ecological systems (SESs) there might be an apparent great ability to cope with change and adapt if analysed only in their social dimension. However, such an adaptation may be at the expense of changes in the capacity of ecosystems to sustain the adaptation and it could affect the quality of ecosystem goods and services since it could degrade natural renewable and non-renewable resources and generate traps and breakpoints in the whole SES eventually leading to chaotic behaviour. This paper is rooted in previous results on modelling tourism-based SESs, only recently object of theoretical investigations, focusing on the dynamics of the coexistence between mass-tourists and eco-tourists. Here we describe a finer scale analysis of time-dependent regimes in the ranges of the degradation coefficient (bifurcation parameter), for which the system can exhibit coexistence. This bifurcation parameter is determined by objective changes in the real world in the quality of ecosystem goods and services together with whether and how such changes are perceived by different tourist typologies. Varying the bifurcation parameter, the dynamical system may in fact evolve toward an aperiodical dynamical state in many ways, showing that there could be different scenarios for the transition to chaos. This paper provides a further evidence for the period-doubling route to chaos with reference to tourism-based socio-ecological models, and for a period locking behaviour, where a small variation in the bifurcation parameter can lead to alternating regular and chaotic dynamics. Moreover, for many models undergoing chaos via period-doubling, it has been showed that structural perturbations with real ecological justification, may break and reverse the expected period-doublings, hence inhibiting chaos. This feature may be of a certain relevance also in the context of adaptive management of tourism-based SESs: these period-doubling reversals might in fact be used to control chaos, since they potentially act in way to suppress possibly dangerous fluctuations.

'Rational' exemption to vaccination is due to a pseudo-rational comparison between the low risk of infection, and the perceived risk of side effects from the vaccine. Here we consider rational exemption in an SI model with information dependent vaccination where individuals use information on the disease's spread as their information set. Using suitable assumptions, we show the dynamic implications of the interaction between rational exemption, current and delayed information. In particular, if vaccination decisions are based on delayed informations, we illustrate both global attractivity to an endemic state, and the onset, through Hopf bifurcations, of general Yakubovich oscillations. Moreover, in some relevant cases, we plot the Hopf bifurcation curves and we give a behavioural interpretation of their meaning.

In this paper, we consider a general bilinear three dimensional ODE system, whose structure generalizes many mathematical models of biological interest, including many from epidemics. Our main goal is to find sufficient conditions, expressed in terms of the parameters of the system, ensuring that the geometric approach to global stability analysis, due to [M.Y. Li, J.S. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal. 27 (4) (1996) 1070–1083], may be successfully applied. We completely determine the dynamics of the general system, including thresholds and global stability of the nontrivial equilibrium. The obtained result is applied to several epidemic models. We further show how the role of new parameters on stability of well-established models may be emphasized.