This work is concerned with the exponential turnpike property for optimal control problems of particle systems and their mean-field limit. Under the assumption of the strict dissipativity of the cost function, exponential estimates for both optimal states and optimal control are proven. Moreover, we show that all the results for particle systems can be preserved under the limit in the case of infinitely many particles.

We propose and unify classes of different models for information propagation over graphs. In a first class, propagation is modelled as a wave, which emanates from a set of known nodes at an initial time, to all other unknown nodes at later times with an ordering determined by the arrival time of the information wave front. A second class of models is based on the notion of a travel time along paths between nodes. The time of information propagation from an initial known set of nodes to a node is defined as the minimum of a generalised travel time over subsets of all admissible paths. A final class is given by imposing a local equation of an eikonal form at each unknown node, with boundary conditions at the known nodes. The solution value of the local equation at a node is coupled to those of neighbouring nodes with lower values. We provide precise formulations of the model classes and prove equivalences between them. Finally, we apply the front propagation models on graphs to semi-supervised learning via label propagation and information propagation on trust networks.

Contemporary epidemiological models often involve spatial variation, providing an avenue to investigate the averaged dynamics of individual movements. In this work, we extend a recent model by Vaziry, Kolokolnikov, and Kevrekidis [Royal Society Open Science 9 (10), 2022] that included, in both infected and susceptible population dynamics equations, a cross-diffusion term with the second spatial derivative of the infected population density. Diffusion terms of this type occur, for example, in the Keller–Siegel chemotaxis model. The presented model corresponds to local orderly commute of susceptible and infected individuals and is shown to arise in two dimensions as a limit of a discrete process. The present contribution identifies and studies specific features of the new model’s dynamics, including various types of infection waves and buffer zones protected from the infection. The model with vital dynamics additionally exhibits complex spatio-temporal behaviour that involves the generation of quasiperiodic infection waves and emergence of transient strongly heterogeneous patterns.

Many science phenomena are modelled as interacting particle systems (IPS) coupled on static networks. In reality, network connections are far more dynamic. Connections among individuals receive feedback from nearby individuals and make changes to better adapt to the world. Hence, it is reasonable to model myriad real-world phenomena as co-evolutionary (or adaptive) networks. These networks are used in different areas including telecommunication, neuroscience, computer science, biochemistry, social science, as well as physics, where Kuramoto-type networks have been widely used to model interaction among a set of oscillators. In this paper, we propose a rigorous formulation for limits of a sequence of co-evolutionary Kuramoto oscillators coupled on heterogeneous co-evolutionary networks, which receive both positive and negative feedback from the dynamics of the oscillators on the networks. We show under mild conditions, the mean field limit (MFL) of the co-evolutionary network exists and the sequence of co-evolutionary Kuramoto networks converges to this MFL. Such MFL is described by solutions of a generalised Vlasov equation. We treat the graph limits as signed graph measures, motivated by the recent work in [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261–349]. In comparison to the recently emerging works on MFLs of IPS coupled on non-co-evolutionary networks (i.e., static networks or time-dependent networks independent of the dynamics of the IPS), our work seems the first to rigorously address the MFL of a co-evolutionary network model. The approach is based on our formulation of a generalisation of the co-evolutionary network as a hybrid system of ODEs and measure differential equations parametrised by a vertex variable, together with an analogue of the variation of parameters formula, as well as the generalised Neunzert’s in-cell-particle method developed in [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261–349].

We prove discrete-to-continuum convergence for dynamical optimal transport on $\mathbb{Z}^d$-periodic graphs with cost functional having linear growth at infinity. This result provides an answer to a problem left open by Gladbach, Kopfer, Maas, and Portinale (Calc Var Partial Differential Equations 62(5), 2023), where the convergence behaviour of discrete boundary-value dynamical transport problems is proved under the stronger assumption of superlinear growth. Our result extends the known literature to some important classes of examples, such as scaling limits of $1$-Wasserstein transport problems. Similarly to what happens in the quadratic case, the geometry of the graph plays a crucial role in the structure of the limit cost function, as we discuss in the final part of this work, which includes some visual representations.

We study an epidemic patch model that describes the disease spread in population with variable latency due to the differences in immunologic tolerance between individuals. We focus on whether the disease can spread in space that leads to the emergence of epidemic wave, that is the travelling wave solution with constant speed. We first establish some properties of the linearized wave profile equations, which are helpful in obtaining the priori estimates of travelling waves and wave speeds. Then, applying the truncation method and limiting arguments, we can obtain threshold propagation dynamics of the epidemic model. Our result gives a complete characterization of the existence, nonexistence and minimal wave speed of travelling waves. To the best of our knowledge, this is the first time to characterize the propagation dynamics of epidemic patch model with variable latency, which contributes to the understanding of the transmission phenomenon of disease.

The article considers systems of interacting particles on networks with adaptively coupled dynamics. Such processes appear frequently in natural processes and applications. Relying on the notion of graph convergence, we prove that for large systems the dynamics can be approximated by the corresponding continuum limit. Well-posedness of the latter is also established.

Adversarial training is a min-max optimization problem that is designed to construct robust classifiers against adversarial perturbations of data. We study three models of adversarial training in the multiclass agnostic-classifier setting. We prove the existence of Borel measurable robust classifiers in each model and provide a unified perspective of the adversarial training problem, expanding the connections with optimal transport initiated by the authors in their previous work [21]. In addition, we develop new connections between adversarial training in the multiclass setting and total variation regularization. As a corollary of our results, we provide an alternative proof of the existence of Borel measurable solutions to the agnostic adversarial training problem in the binary classification setting.

We investigate a recent model proposed in the literature elucidating patterns driven by chemotaxis, similar to viscous fingering phenomena. Notably, this model incorporates a singular advection term arising from a modified formulation of Darcy’s law. It is noteworthy that this type of advection can also be well interpreted as a description of a radial fluid flow source surrounding an aggregation of cells. For the two-dimensional scenario, we establish a precise threshold delineating between blow-up and global solution existence. This threshold is contingent upon the pressure magnitude and the initial total mass of the aggregating cells.

In network science, one of the significant and challenging subjects is the detection of communities. Modularity [1] is a measure of community structure that compares connectivity in the network with the expected connectivity in a graph sampled from a random null model. Its optimisation is a common approach to tackle the community detection problem. We present a new method for modularity maximisation, which is based on the observation that modularity can be expressed in terms of total variation on the graph and signless total variation on the null model. The resulting algorithm is of Merriman–Bence–Osher (MBO) type. Different from earlier methods of this type, the new method can easily accommodate different choices of the null model. Besides theoretical investigations of the method, we include in this paper numerical comparisons with other community detection methods, among which the MBO-type methods of Hu et al. [2] and Boyd et al. [3], and the Leiden algorithm [4].

Coffee berry diseases (CBD) pose significant threats to coffee production worldwide, affecting the livelihoods of millions of farmers and the global coffee market. Fractional calculus provides a powerful framework for describing non-local and memory-dependent phenomena, making it suitable for modelling the long-range interactions inherent in CBD spread. This study aims to formulate and analyse fractional order model for CBD transmission dynamics in the sense of Atangana–Baleanu–Caputo. Fixed point theorems were utilised to test the existence and uniqueness of the model’s solutions using fractional order. The basic reproduction number was calculated utilising the next-generation matrix. The model has locally asymptotically stable equilibrium positions (disease-free and endemic). Furthermore, the Lyapunov function was used to conduct a global stability analysis of the equilibrium locations. A numerical simulation of the CBD model was created using the fractional Adam–Bashforth–Moulton approach to validate the analytical findings. Our findings contribute to the development of more accurate predictive models and inform the design of targeted interventions to mitigate the impact of CBD on coffee production systems.

We study the global well-posedness and uniform boundedness of a two-dimensional reaction–advection–diffusion system with nonlinear advection. This strongly coupled system of nonlinear partial differential equations represents the continuum of a 2D lattice model designed to describe residential burglary, where each location is characterised by a tractability value that varies in both space and time. We show that the model with sublinear advection enhancement is globally well-posed, with a unique solution that is classical and uniformly bounded in time. Our results provide valuable insights into the development of urban crime models with nonlinear advection enhancements, making them suitable for broader applications, including nonlocal or heterogeneous near-repeat victimisation effects.

In this paper, we study the existence of travelling wave solutions and the spreading speed for the solutions of an age-structured epidemic model with nonlocal diffusion. Our proofs make use of the comparison principles both to construct suitable sub/super-solutions and to prove the regularity of travelling wave solutions.

In this paper, we consider a delayed discrete single population patch model in advective environments. The individuals are subject to both random and directed movements, and there is a net loss of individuals at the downstream end due to the flow into a lake. Choosing time delay as a bifurcation parameter, we show the existence of Hopf bifurcations for the model. In homogeneous non-advective environments, it is well known that the first Hopf bifurcation value is independent of the dispersal rate. In contrast, for homogeneous advective environments, the first Hopf bifurcation value depends on the dispersal rate. Moreover, we show that the first Hopf bifurcation value in advective environments is larger than that in non-advective environments if the dispersal rate is large or small, which suggests that directed movements of the individuals inhibit the occurrence of Hopf bifurcations.

Interacting particle systems (IPSs) are a very important class of dynamical systems, arising in different domains like biology, physics, sociology and engineering. In many applications, these systems can be very large, making their simulation and control, as well as related numerical tasks, very challenging. Kernel methods, a powerful tool in machine learning, offer promising approaches for analyzing and managing IPS. This paper provides a comprehensive study of applying kernel methods to IPS, including the development of numerical schemes and the exploration of mean-field limits. We present novel applications and numerical experiments demonstrating the effectiveness of kernel methods for surrogate modelling and state-dependent feature learning in IPS. Our findings highlight the potential of these methods for advancing the study and control of large-scale IPS.

We investigate the mean-field dynamics of stochastic McKean differential equations with heterogeneous particle interactions described by large network structures. To express a wide range of graphs, from dense to sparse structures, we incorporate the recently developed graph limit theory of graphops into the limiting McKean–Vlasov equations. Global stability of the splay steady state is proven via a generalised entropy method, leading to explicit graph structure-dependent decay rates. We highlight the robustness of the entropy approach by extending the results to the closely related Sakaguchi–Kuramoto model with intrinsic frequency distributions. We also present central examples of random graphs, such as power law graphs and the spherical graphop, and analyse the limitations of the applied methodology.

We introduce a free boundary model to study the effect of vesicle transport onto neurite growth. It consists of systems of drift-diffusion equations describing the evolution of the density of antero- and retrograde vesicles in each neurite coupled to reservoirs located at the soma and the growth cones of the neurites, respectively. The model allows for a change of neurite length as a function of the vesicle concentration in the growth cones. After establishing existence and uniqueness for the time-dependent problem, we briefly comment on possible types of stationary solutions. Finally, we provide numerical studies on biologically relevant scales using a finite volume scheme. We illustrate the capability of the model to reproduce cycles of extension and retraction.

This article studies the dynamical behaviour of classical solutions of a hyperbolic system of balance laws, derived from a chemotaxis model with logarithmic sensitivity, with time-dependent boundary conditions. It is shown that under suitable assumptions on the boundary data, solutions starting in the $H^2$-space exist globally in time and the differences between the solutions and their corresponding boundary data converge to zero as time goes to infinity. There is no smallness restriction on the magnitude of the initial perturbations. Moreover, numerical simulations show that the assumptions on the boundary data are necessary for the above-mentioned results to hold true. In addition, numerical results indicate that the solutions converge asymptotically to time-periodic states if the boundary data are time-periodic.

This paper studies the spatio-temporal dynamics of a diffusive plant-sulphide model with toxicity delay. More specifically, the effects of discrete delay and distributed delay on the dynamics are explored, respectively. The deep analysis of eigenvalues indicates that both diffusion and delay can induce Hopf bifurcations. The normal form theory is used to set up an exact formula that determines the properties of Hopf bifurcation in a diffusive plant-sulphide model. A sufficiently small discrete delay does not affect the stability and a sufficiently large discrete delay destabilizes the system. Nonetheless, a sufficiently small or large distributed delay does not affect the stability. Both delays cause instability by inducing Hopf bifurcation rather than Turing bifurcation.

We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension,

\begin{equation*} u_t = D u_{xx} + u(1-\phi *u), \end{equation*}

$\phi *u$

$\phi (y) \equiv H\left (\frac{1}{4}-y^2\right )$

$u=1$

$D$

$\Delta _1 \approx 0.00297$

$D \ll 1$

$O(1)$

$u=O(1)$

$u$

$D \to 0^+$

$D \geq \Delta _1$

$2 \sqrt{D}$

$0 \lt D \lt \Delta _1$

$u=0$

$D$