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We investigate a fast-reaction--diffusion system modelling the effect of autotoxicity on plant-growth dynamics, in which the fast-reaction terms are based on the dichotomy between healthy and exposed roots depending on the toxicity. The model was proposed in [Giannino, Iuorio, Soresina, forthcoming] to account for stable stationary spacial patterns considering only biomass and toxicity, and its fast-reaction (cross-diffusion) limit was formally derived and numerically investigated. In this paper, the cross-diffusion limiting system is rigorously obtained as the fast-reaction limit of the reaction-diffusion system with fast-reaction terms by performing a bootstrap argument involving energies. Then, a thorough well-posedness analysis of the cross-diffusion system is presented, including a $L^\infty_{x,t}$-bound, uniqueness, stability, and regularity of weak solutions. This analysis, in turn, becomes crucial to establish the convergence rate for the fast reaction limit, thanks to the key idea of using an inverse Neumann Laplacian operator. Finally, numerical experiments illustrate the analytical findings on the convergence rate.

In this chapter we provide an introduction to fractional dissipative partial differential equations (PDEs) with a focus on trying to understand their dynamics. The class of PDEs we focus on are reaction-diffusion equations but we also provide an outlook on closely related classes of PDEs. To simplify the exposition, we only discuss the cases of fractional time derivatives and fractional space derivatives in the PDE separately. As our main tools, we describe analytical as well as numerical methods, which are generically necessary to study nonlinear dynamics. We start with the analytical study of steady states and local linear stability for fractional time derivatives. Then we extend this view to a global perspective and consider time-fractional PDEs and gradient flows. Next, we continue to steady states, linear stability analysis and bifurcations for space-fractional PDEs. As a final analytical consideration we discuss existence and stability of traveling waves for space-fractional PDEs. In the last parts, we provide numerical discretization schemes for fractional (dissipative) PDEs and we utilize these techniques within numerical continuation in applied examples of fractional reaction-diffusion PDEs. We conclude with a brief summary and outlook on open questions in the field., Comment: book chapter

The fast-reaction limit for reaction--diffusion systems modelling predator--prey interactions is investigated. In the considered model, predators exist in two possible states, namely searching and handling. The switching rate between these two states happens on a much faster time scale than other processes, leading to the consideration of the fast-reaction limit for the corresponding systems. The rigorous convergence of the solution to the fast-reaction system to the ones of the limiting cross-diffusion system has been recently studied in [Conforto, Desvillettes, Soresina, NoDEA, 25(3):24, 2018]. In this paper, we extend these results by proving improved convergence of solutions and slow manifolds. In particular, we prove that the slow manifold converges strongly in all dimensions without additional assumptions, thanks to use of a modified energy function. This consists in a unified approach since it is applicable to both types of fast-reaction systems, namely with the Lotka--Volterra and the Holling-type II terms.

The concept of Turing instability, namely that diffusion can destabilize the uniform steady state, is well known either in the context of partial differential equations (PDEs) or in networks of dynamical systems. Recently reaction-diffusion equations with cross-diffusion terms have been investigated, showing an analogous effect called cross-diffusion induced instability. In this paper, we extend this concept to networks of dynamical systems, showing that the spectrum of the graph Laplacian determines the instability appearance, as well as the spectrum of the Laplace operator in reaction-diffusion equations. We extend to network dynamics a particular network model for competing species, coming from the PDEs context. In particular, the influence of different topology structures on the cross-diffusion induced instability is highlighted, considering regular rings and lattices, and also small-world, Erd\H{o}s-R\'eyni, and Barab\'asi-Albert networks.

In this paper, we study a modification of the mathematical model describing inflammation and demyelination patterns in the brain caused by Multiple Sclerosis proposed in [Lombardo et al. (2017), Journal of Mathematical Biology, 75, 373--417]. In particular, we hypothesize a minimal amount of macrophages to be able to start and sustain the inflammatory response. Thus, the model function for macrophage activation includes an Allee effect. We investigate the emergence of Turing patterns by combining linearised and weakly nonlinear analysis, bifurcation diagrams and numerical simulations, focusing on the comparison with the previous model.

We investigate the one-dimensional defect SCFT defined on the $1/2$ BPS Wilson line/loop in ABJ(M) theory. We show that the supermatrix structure of the defect imposes a covariant supermatrix representation of the supercharges. Exploiting this covariant formulation, we prove the existence of a long multiplet whose highest weight state is a constant supermatrix operator. At weak coupling, we study this operator in perturbation theory and confirm that it acquires a non-trivial anomalous dimension. At strong coupling, we conjecture that this operator is dual to the lowest bound state of fluctuations of the fundamental open string in AdS$_4\times \mathbb{CP}_3$ around the classical $1/2$ BPS solution. Quite unexpectedly, this operator also arises in the cohomological equivalence between bosonic and fermionic Wilson loops. We also discuss some regularization subtleties arising in perturbative calculations on the infinite Wilson line., Comment: 36 pages plus appendices, 5 figures, 5 tables

In this paper, we consider the Shigesada-Kawasaki-Teramoto (SKT) model, which presents cross-diffusion terms describing competition pressure effects. Even though the reaction part does not present the activator-inhibitor structure, cross-diffusion can destabilise the homogeneous equilibrium. However, in the full cross-diffusion system and weak competition regime, the cross-diffusion terms have an opposite effect and the bifurcation structure of the system modifies increasing the interspecific competition pressure. The major changes in the bifurcation structure, the type of pitchfork bifurcations on the homogeneous branch, as well as the presence of Hopf bifurcation points are here investigated. Through weakly nonlinear analysis, we can predict the type of pitchfork bifurcation. Increasing the additional cross-diffusion coefficients, the first two pitchfork bifurcation points from super-critical become sub-critical, leading to the appearance of a multi-stability region. The interspecific competition pressure also influences the possible appearance of stable time-period spatial patterns appearing through a Hopf bifurcation point.

In this paper, we review several results from singularly perturbed differential equations with multiple small parameters. In addition, we develop a general conceptual framework to compare and contrast the different results by proposing a three-step process. First, one specifies the setting and restrictions of the differential equation problem to be studied and identifies the relevant small parameters. Second, one defines a notion of equivalence via a property/observable for partitioning the parameter space into suitable regions near the singular limit. Third, one studies the possible asymptotic singular limit problems as well as perturbation results to complete the diagrammatic subdivision process. We illustrate this approach for two simple problems from algebra and analysis. Then we proceed to the review of several modern double-limit problems including multiple time scales, stochastic dynamics, spatial patterns, and network coupling. For each example, we illustrate the previously mentioned three-step process and show that already double-limit parametric diagrams provide an excellent unifying theme. After this review, we compare and contrast the common features among the different examples. We conclude with a brief outlook, how our methodology can help to systematize the field better, and how it can be transferred to a wide variety of other classes of differential equations.

We construct the one-dimensional topological sector of $\mathcal N = 6$ ABJ(M) theory and study its relation with the mass-deformed partition function on $S^3$. Supersymmetric localization provides an exact representation of this partition function as a matrix integral, which interpolates between weak and strong coupling regimes. It has been proposed that correlation functions of dimension-one topological operators should be computed through suitable derivatives with respect to the masses, but a precise proof is still lacking. We present non-trivial evidence for this relation by computing the two-point function at twoloop, successfully matching the matrix model expansion at weak coupling and finite ranks. As a by-product we obtain the two-loop explicit expression for the central charge $c_T$ of ABJ(M) theory. Three- and four-point functions up to one-loop confirm the relation as well. Our result points towards the possibility to localize the one-dimensional topological sector of ABJ(M) and may also be useful in the bootstrap program for 3d SCFTs., Comment: 38 pages, 2 figures, 3 tables

Partial differential equations (PDEs) involving fractional Laplace operators have been increasingly used to model non-local diffusion processes and are actively investigated using both analytical and numerical approaches. The purpose of this work is to study the effects of the spectral fractional Laplacian on the bifurcation structure of reaction-diffusion systems on bounded domains. In order to do this we use advanced numerical continuation techniques to compute the solution branches. Since current available continuation packages only support systems involving the standard Laplacian, we first extend the pde2path software to treat fractional PDEs. The new capabilities are then applied to the study of the Allen-Cahn equation, the Swift-Hohenberg equation and the Schnakenberg system (in which the standard Laplacian is each replaced by the spectral fractional Laplacian). Our study reveals some common effects, which contributes to a better understanding of fractional diffusion in generic reaction-diffusion systems. In particular, we investigate the changes in snaking bifurcation diagrams and also study the spatial structure of non-trivial steady states upon variation of the order of the fractional Laplacian. Our results show that the fractional order can induce very significant qualitative and quantitative changes in global bifurcation structures.

In this paper we investigate the bifurcation structure of the triangular SKT model in the weak competition regime and of the corresponding fast-reaction system in 1D and 2D domains via numerical continuation methods. We show that the software pde2path can be adapted to treat cross-diffusion systems, reproducing the already computed bifurcation diagrams on 1D domains. We show the convergence of the bifurcation structure obtained selecting the growth rate as bifurcation parameter. Then, we compute the bifurcation diagram on a 2D rectangular domain providing the shape of the solutions along the branches and linking the results with the Turing instability analysis. In 1D and 2D, we pay particular attention to the fast-reaction limit by always computing sequences of bifurcation diagrams as the time-scale separation parameter tends to zero. We show that the bifurcation diagram undergoes major deformations once the fast-reaction systems limits onto the cross-diffusion singular limit. Furthermore, we find evidence for time-periodic solutions by detecting Hopf bifurcations, we characterize several regions of multi-stability, and improve our understanding of the shape of patterns in 2D for the SKT model.

In this paper we consider the Shigesada-Kawasaki-Teramoto (SKT) model to account for stable inhomogeneous steady states exhibiting spatial segregation, which describe a situation of coexistence of two competing species. We provide a deeper understanding on the conditions required on both the cross-diffusion and the reaction coefficients for non-homogeneous steady states to exist, by combining a detailed linearized analysis with advanced numerical bifurcation methods via the continuation software pde2path. We report some numerical experiments suggesting that, when cross-diffusion is taken into account, there exist positive and stable non-homogeneous steady states outside of the range of parameters for which the coexistence homogeneous steady state is positive. Furthermore, we also analyze the case in which self-diffusion terms are considered.

A generalized seasonally-varying predator-prey model with Allee effect in the prey growth is investigated. The analysis is performed only on the basis of some properties determining the shape of the prey growth rate and the trophic interaction functions. General conditions for coexistence are determined, both in the case of weak and strong Allee effect. Finally, a modified Leslie-Gower predator-prey model with Allee effect is investigated. Numerical results illustrate the qualitative behaviors of the system, in particular the presence of periodic orbits.

Pest phenological models describe the cumulative flux of the individuals into each stage of the life cycle of a stage-structured population. Phenological models are widely used tools in pest control decision making. Despite the fact that these models do not provide information on population abundance, they share some advantages with respect to the more sophisticated and complex demographic models. The main advantage is that they do not require data collection to define the initial conditions of model simulation, reducing the effort for field sampling and the high uncertainty affecting sample estimates. Phenological models are often built considering the developmental rate function only. To the aim of adding more realism to phenological models, in this paper we explore the consequences of improving these models taking into consideration three additional elements: the age distribution of individuals which exit from the overwintering phase, the age- and temperature-dependent profile of the fecundity rate function and the consideration of a temperature-dependent mortality rate function. Numerical simulations are performed to investigate the effect of these elements with respect to phenological models considering development rate functions only. To further test the implications of different models formulation, we compare results obtained from different phenological models to the case study of the codling moth (Cydia pomonella) a primary pest of the apple orchard. The results obtained from model comparison are discussed in view of their potential application in pest control decision support.

Simulating the population dynamics of a stage-structured population requires the knowledge of the biodemographic functions characterizing the species, namely development, mortality and fecundity. In general, development and fecundity can satisfactorily be estimated starting from literature data. Unfortunately, this is often not the case of the mortality function, because of the lack of experimental data. To overcome this problem we estimate the mortality rate function from field data on the abundance of the species. The mortality is expressed as a linear combination of cubic splines and the estimation method allows to determine its coefficients taking into account the observations measurement error. Moreover, the variability in the estimate is quantified by means of the confidence bands for both mortality and dynamics. The presented method allows to obtain a more flexible shape for the mortality rate functions compared with previous methods applied to the same pest. The method is applied to the case of Lobesia botrana, the main pest in the European vineyards, with abundance data collected in a location in the North of Italy for five consecutive years. Data collected over three years are used to estimate the mortality and to analyze the variability in the estimate and its effects on the population dynamics. Other two datasets are used to validate the model simulating the dynamics using the estimated mortality.

We consider in this paper a microscopic model (that is, a system of three reaction-diffusion equations) incorporating the dynamics of handling and searching predators, and show that its solutions converge when a small parameter tends to $0$ towards the solutions of a reaction-cross diffusion system of predator-prey type involving a Holling-type II or Beddington-DeAngelis functional response. We also provide a study of the Turing instability domain of the obtained equations and (in the case of the Beddington-DeAngelis functional response) compare it to the same instability domain when the cross diffusion is replaced by a standard diffusion.

We investigate the time-optimal control problem in SIR (Susceptible-Infected-Recovered) epidemic models, focusing on different control policies: vaccination, isolation, culling, and reduction of transmission. Applying the Pontryagin's Minimum Principle (PMP) to the unconstrained control problems (i.e. without costs of control or resource limitations), we prove that, for all the policies investigated, only bang-bang controls with at most one switch are admitted. When a switch occurs, the optimal strategy is to delay the control action some amount of time and then apply the control at the maximum rate for the remainder of the outbreak. This result is in contrast with previous findings on the unconstrained problems of minimizing the total infectious burden over an outbreak, where the optimal strategy is to use the maximal control for the entire epidemic. Then, the critical consequence of our results is that, in a wide range of epidemiological circumstances, it may be impossible to minimize the total infectious burden while minimizing the epidemic duration, and vice versa. Moreover, numerical simulations highlighted additional unexpected results, showing that the optimal control can be delayed also when the control reproduction number is lower than one and that the switching time from no control to maximum control can even occur after the peak of infection has been reached. Our results are especially important for livestock diseases where the minimization of outbreaks duration is a priority due to sanitary restrictions imposed to farms during ongoing epidemics, such as animal movements and export bans.