Qi Wang

We study a reaction–advection–diffusion model of a target–offender–guardian system designed to capture interactions between urban crime and policing. Using Crandall–Rabinowitz bifurcation theory and spectral analysis, we establish rigorous conditions for both steady-state and Hopf bifurcations. These results identify critical thresholds of policing intensity at which spatially uniform equilibria lose stability, leading either to persistent heterogeneous hotspots or oscillatory crime–policing cycles. From a criminological perspective, such thresholds represent tipping points in guardian mobility: once crossed, they can lock neighbourhoods into stable clusters of criminal activity or trigger recurrent waves of hotspot formation. Numerical simulations complement the theory, exhibiting stationary patterns, periodic oscillations and chaotic dynamics. By explicitly incorporating law enforcement as a third interacting component, our framework extends classical two-equation models. It offers new tools for analysing non-linear interactions, bifurcations and pattern formation in multi-agent social systems.

Spatial Solow models seek to explain uneven regional economic development by accounting for spatiotemporal heterogeneity in the inputs of economic growth. Central to the Solow or perhaps any long-run economic growth model, the aggregate production function determines how inputs are transformed into outputs, and it shapes the convergence to a stable economic equilibrium in the long run. Identifying this function is therefore essential for understanding the true drivers of economic growth, and applying empirically validated spatial economic models can help develop effective strategies that promote economic progress and ensure sustainable development. We study the identification of spatial Solow models that treat capital mobility and technology diffusion as the primary sources of heterogeneity in a spatial economic market, where wealth flows from richer to poorer regions and technology spreads from more advanced to less innovative areas; the choice of S-shaped production functions enhances the model’s significance to capture poverty traps and economic agglomeration within a unified framework. The recovery of the resulting PDE system is then framed as an inverse problem for which we introduce a nonparametric approach. Numerical studies find that the presence of noise does not diminish computational accuracy or increase computational burden. While missing more data generally leads to larger errors in system recovery and wealth prediction, our approach demonstrates remarkable accuracy even when a majority of spatial data are unavailable. The resilience to data corruption and data loss underscores the practical utility of our method in real-world applications where imperfect measurement is often unavoidable.

We analyze a system of partial differential equations introduced as a model for urban crime with law enforcement. This system is known to obtain spatially localized patterns representing crime hotspots. In this work, we obtain the amplitude equations that describe the hotspot pattern formation in the model using weakly nonlinear analysis techniques. In particular, we find the existence of super- and sub-critical pitchfork bifurcations. Moreover, we propose different suppression strategies and investigate numerically how the suggested approaches effectively eradicate the two types of hotspots. We highlight the challenges in weakly nonlinear analysis and the limitations of the proposed suppression mechanisms, emphasizing the need for further theoretical and practical work on this urban crime model with law enforcement.

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.

This paper proposes and analyzes a fully discrete scheme for nonlinear biharmonic Schrödinger equations. We first write the single equation into a system of problems with second-order spatial derivatives and then discretize the space variable with an ultraweak discontinuous Galerkin scheme and the time variable with the Crank–Nicolson method. The proposed scheme proves to be computationally more efficient compared to the local discontinuous Galerkin method in terms of the number of equations needed to be solved at each single time step, and it is unconditionally stable without imposing any penalty terms. It also achieves optimal error convergence in L² norm both in the solution and in the auxiliary variable with general nonlinear terms. We also prove several physically relevant properties of the discrete schemes, such as the conservation of mass and the Hamiltonian for the nonlinear biharmonic Schrödinger equations. Several numerical studies demonstrate and support our theoretical results.

This study investigates the development of intricate pattern formation in synaptogenesis, which describes the transfer of synapses between neurons via bidirectional active and passive motor-mediated transport in the nervous system. The trafficking model, which consists of three strongly coupled partial differential equations, exhibits remarkable spatiotemporal dynamics that contribute to both static and constantly changing spatial patterns. We perform and detail the steady-state and Hopf bifurcation to present the formation of nonhomogeneous stationary or time-periodic solutions from the homogeneous one. In particular, the steady-state reduces to that of a two-component PDE system, independent of the initial condition and the system parameters. Moreover, we analyze the stability of these nontrivial patterns and our results find a wave mode selection mechanism, which in turn suggests that a large interval length admits time-periodic solutions, while a small interval length most likely drives spatially monotonous nonhomogeneous stationary solutions. Finally, we provide numerical experiments to illustrate and support our theoretical findings.

This paper develops analytical approximations for volatility index (VIX) option pricing under nonaffine generalized autoregressive conditional heteroskedasticity (GARCH) models as advocated by Christoffersen et al. We obtain the approximation formulae for pricing VIX options and then evaluate their performance with three expansions under four empirically well‐tested models. Our numerical experiments find that the weighted expansion generated by the fat‐tailed weighting kernel can significantly reduce approximation error over the Gram‐Charlier expansion; the Taylor expansion of conditional moments can lead to divergence for parameters with certain high persistence in the affine GARCH, nonlinear asymmetric GARCH, and Glosten‐Jagannathan‐Runkle GARCH models, while surviving during high persistence in the exponential GARCH.

This article introduces a class of generative models based on the (G)ARCH-like continuous-time framework to unify econometric and diffusion-based methods for pricing European options. We formulate a partial differential equation (PDE) for the option price when the volatility of the underlying asset is described by a broad class of discrete-time GARCH models. The GARCH-PDE framework combines discrete data from the physical market with a latent stochastic volatility process and it provides flexibility in the pricing process to accommodate any underlying realized volatility dynamics. We reduce solving a two-dimensional degenerate PDE to finding the solutions of a system of one-dimensional PDEs and then obtain closed-form analytical formulas for option pricing under stochastic volatility, which is typically achieved through Monte Carlo simulations. Convergence and error analysis of the analytical formula attest to the option pricing accuracy of the proposed framework using available discrete implied volatility samples without compromising its computational accuracy. Several sets of numerical tests are presented to illustrate our approach and demonstrate its superiority over other empirically well-tested pricing methods.