Vegetation patterns are a characteristic feature of semi-deserts occurring on all continents. The Klausmeier-Gray-Scott 2D model for semi-arid ecosystems on a sloped terrain is considered with the addition of a nonlinear cross-diffusion term. Pattern formation driven by cross-diffusion is studied in the resulting system. A weakly nonlinear analysis around the critical value of the cross-diffusion is performed, and the asymptotic expansion is validated by numerical solution of the full system.

A reaction–diffusion system governing the predator–prey interaction with specialist predator and herd behavior for prey is investigated. Linear stability of the interior equilibrium is studied, and conditions guaranteeing the occurrence of Turing instability, induced by cross-diffusion, are found, with a full characterization of the Turing instability region in the parameter space. Numerical simulations on the obtained results are provided.

Intraguild predation, representing a true combination of predation and competition between two species that rely on a common resource, is of foremost importance in many natural communities. We investigate a spatial model of three species interaction, characterized by a Holling type II functional response and linear cross-diffusion. For this model we report necessary and sufficient conditions ensuring the insurgence of Turing instability for the coexistence equilibrium; we also obtain conditions characterizing the different patterns by multiple scale analysis. Numerical experiments confirm the occurrence of different scenarios of Turing instability, also including Turing–Hopf patterns.

A Cournot triopoly is a type of oligopoly market involving three firms that produce and sell homogeneous or similar products without cooperating with one another. In Cournot models, firms' decisions about production levels play a crucial role in determining overall market output. Compared to duopoly models, oligopolies with more than two firms have received relatively less attention in the literature. Nevertheless, triopoly models are more reflective of real-world market conditions, even though analyzing their dynamics remains a complex challenge. A reaction-diffusion system of PDEs generalizing a nonlinear triopoly model describing a master-slave Cournot game is introduced. The effect of diffusion on the stability of Nash equilibrium is investigated. Self-diffusion alone cannot induce Turing pattern formation. In fact, linear stability analysis shows that cross-diffusion is the key mechanism for the formation of spatial patterns. The conditions for the onset of cross-diffusion-driven instability are obtained via linear stability analysis, and the formation of several Turing patterns is investigated through numerical simulations.

A nonlinear crime model is generalized by introducing self- and cross-diffusion terms. The effect of diffusion on the stability of non-negative constant steady states is applied. In particular, the cross-diffusion-driven instability, called Turing instability, is analyzed by linear stability analysis, and several Turing patterns driven by the cross-diffusion are studied through numerical investigations. When the Turing–Hopf conditions are satisfied, the type of instability highlighted in the ODE model persists in the PDE system, still showing an oscillatory behavior.

A reaction-diffusion model, known as the Sel'kov-Schnakenberg model, is considered. The nonlinear stability of the constant steady state is studied by using a special Liapunov functional and a maximum principle for regular solutions.

We present a simple model describing the chemical aggression undergone by calcium carbonate rocks in presence of acid atmosphere. A large literature is available on the deterioration processes of building stones, in particular in connection with problems concerning historical buildings in the field of Cultural Heritage. It is well known that the greatest aggression is caused by sulfur dioxide and nitrate. In this paper we consider the corrosion caused by sulphur dioxide, which, reacting with calcium carbonate, produces gypsum. The model proposed is obtained by considering both the diffusive and convective effects of propagation and assuming that the porous medium is saturated with a compressible fluid having an assigned polytropic constitutive equation for the pressure. The qualitative behavior of the one dimensional solutions in the fastreaction limit is performed.

A classical Lotka-Volterra model with the logistical growth of prey-and-hunting coopera-tion in the functional response of predators to prey was extended by introducing advection terms,which included the velocities of animals. The effect of velocity on the kinetics of the problem wasanalyzed. In order to examine the band behavior of species over time, traveling wave solutions wereintroduced, and conditions for the coexistence of both populations and/or extinction were found.Numerical simulations illustrating the obtained results were performe

A reaction-diffusion system governing the prey-predator interaction with Allee effect on the predators, already introduced by the authors in a previous work is reconsidered with the aim of showing destabilization mechanisms of the biologically meaning equilibrium and detecting some aspects for the eventual oscillatory pattern formation. Extensive numerical simulations, depicting such complex dynamics, are shown. In order to complete the stability analysis of the coexistence equilibrium, a nonlinear stability result is shown.

We present a simple model describing the chemical aggression undergone by calcium carbonate rocks in presence of acid atmosphere. A large literature is available on the deterioration processes of building stones, in particular in connection with problems concerning historical buildings in the field of Cultural Heritage. It is well known that the greatest aggression is caused by SO2 andNO3. In this paper we consider the corrosion caused by sulphur dioxide, which, reacting with calcium carbonate, produces gypsum. The model proposed is obtained by considering both the diffusive and convective effects of propagation and assuming that the porous medium is saturated with a compressible fluid having an assigned polytropic constitutive equation for the pressure

Analisi, modellizzazione e simulazione delle traiettorie dei visitatori nelle aree museali

We consider a one-dimensional, isentropic, hydrodynamical model for a unipolar semiconductor, with the mobility depending on the electric field. The mobility is related to the momentum relaxation time, and field-dependent mobility models are commonly used to describe the occurrence of saturation velocity, that is, a limit value for the electron mean velocity as the electric field increases. For the steady state system, we prove the existence of smooth solutions in the subsonic case, with a suitable assumption on the mobility function. Furthermore, we prove uniqueness of subsonic solutions for sufficiently small currents.

A prey-predator system with logistic growth of prey and hunting cooperation of predators is studied. The introduction of fractional time derivatives and the related persistent memory strongly characterize the model behavior, as many dynamical systems in the applied sciences are well described by such fractional-order models. Mathematical analysis and numerical simulations are performed to highlight the characteristics of the proposed model. The existence, uniqueness and boundedness of solutions is proved; the stability of the coexistence equilibrium and the occurrence of Hopf bifurcation is investigated. Some numerical approximations of the solution are finally considered; the obtained trajectories confirm the theoretical findings. It is observed that the fractional-order derivative has a stabilizing effect and can be useful to control the coexistence between species.

In this paper, a reaction-diffusion prey-predator system including the fear effect of predator on prey population and group defense has been considered. The conditions for the onset of cross-diffusion-driven instability are obtained by linear stability analysis. The technique of multiple time scales is employed to deduce the amplitude equation near Turing bifurcation threshold by choosing the cross-diffusion coefficient as a bifurcation parameter. The stability analysis of these amplitude equations leads to the identification of various Turing patterns driven by the cross-diffusion, which are also investigated through numerical simulations.

In this paper, a reaction-diffusion prey-predator system including the fear effect of predator on prey population and group defense has been considered. The conditions for the onset of cross-diffusion-driven instability are obtained by linear stability analysis. The technique of multiple time scales is employed to deduce the amplitude equation near Turing bifurcation threshold by choosing the cross-diffusion coefficient as a bifurcation parameter. The stability analysis of these amplitude equations leads to the identification of various Turing patterns driven by the cross-diffusion, which are also investigated through numerical simulations.

A full ODE model for the transmission of cholera is investigated, includ- ing both direct and indirect transmission and a nonlinear growth for pathogens. The direct problem is preliminarily studied and characterized in terms of reproduction number, endemic and disease free equilibria. The inverse problem is then discussed in view of parameter estimation and model identification via a Least Squares Approximation approach. The procedure is applied to real data coming from the recent Yemen cholera outbreak of 2017-2018.

A limitation of current modeling studies in waterborne diseases (one of the leading causesof death worldwide) is that the intrinsic dynamics of the pathogens is poorly addressed, leadingto incomplete, and often, inadequate understanding of the pathogen evolution and its impact ondisease transmission and spread. To overcome these limitations, in this paper, we consider an ODEsmodel with bacterial growth inducing Allee effect. We adopt an adequate functional response tosignificantly express the shape of indirect transmission. The existence and stability of biologicallymeaningful equilibria is investigated through a detailed discussion of both backward and Hopfbifurcations. The sensitivity analysis of the basic reproduction number is performed. Numericalsimulations confirming the obtained results in two different scenarios are shown.

A binary mixture saturating a horizontal porous layer, with large pores and uniformly heated from below, is considered. The instability of a vertical uid motion (throughflow) when the layer is salted by one salt (either from above or from below) is analyzed. Ultimately boundedness of solutions is proved, via the existence of positively invariant and attractive sets (i.e. absorbing sets). The critical Rayleigh numbers at which steady or oscillatory instability occurs, are recovered. Sufficient conditions guaranteeing that a secondary steady motion or a secondary oscillatory motion can be observed after the loss of stability, are found. When the layer is salted from above, a condition guaranteeing the occurrence of "cold" instability is determined. Finally, the influence of the velocity module on the increasing/decreasing of the instability thresholds is investigated.

The self and cross diffusion action on the dynamic of the nonlinear continu- ous duopoly model introduced in [22], is investigated. Under Robin boundary conditions the longtime behavior and the linear and nonlinear stability of the steady states, are studied. The self and cross diffusion parameters guaran- teeing the spreading of the firms outputs, are characterized.

An intraguild predator-prey model with a carrying capacity proportional to the biotic resource, is generalized by introducing a Holling type II functional response. The longtime behaviour of solutions is analyzed and, in particular, absorbing sets in the phase space are determined. The existence of biologically meaningful equilibria (boundary and internal equilibria) has been investigated. Linear and nonlinear stability conditions for biologically meaningful equilibria are performed. Finally, numerical simulations on different regimes of coexistence and extinction of the involved populations have been shown.