Evolutionary therapy (ET) applies principles of evolutionary biology to steer tumour dynamics and forestall or delay treatment resistance, typically guided by data-driven mathematical models. Our aim is to assess whether ET protocols, and specifically Zhang et al.’s protocol proposed for metastatic castrate-resistant prostate cancer, can be theoretically effective for fast-growing metastatic cancers such as stage IV non-small-cell lung cancer (NSCLC). Using longitudinal tumour-burden data from NSCLC patients treated with erlotinib, we systematically evaluate 26 two-population differential-equation models based on classical tumour-growth dynamics, with varying assumptions about density- and frequency-dependent interactions, pharmacokinetics, and treatment-induced death. Previous work by Yin et al. on the same dataset employed an exponential model that omitted density- and frequency-dependent interactions; although it provided a good fit to tumour-burden data, its structure would theoretically lead to poorer outcomes under ET protocols. In contrast, our analysis identifies the minimal model structure required to reproduce the resistance-driven regrowth observed in NSCLC, with the Gompertzian model featuring log-kill dynamics and both density- and frequency-dependent interactions providing the best fit. In this model, Zhang et al.’s protocol prolonged median time-to-progression to 42.3 months compared with 24.8 months under maximum tolerated dose. These results indicate that ET is theoretically a viable treatment strategy for NSCLC. This study offers a practical framework for assessing ET feasibility using clinical data and supports future clinical translation of ET in NSCLC.

The dynamics of pedestrian crowds can involve very different scales. While situations involving only a few pedestrians are better described by microscopic models, large crowds can exhibit collective behavior which can be captured by macroscopic equations. Macroscopic models describe crowds as fluids of pedestrians, where individuals cannot be distinguished anymore. This fluid is characterized by some local averages of pedestrian density and velocity. These macroscopic variables are shown to obey conservation equations, which can be solved using the method of characteristics. In contrast with classical fluid equations, the evolution of density and velocity depends on some target or preferred velocity that can be specific to different pedestrian groups. We review the advantages and drawbacks of these conservation laws adapted to the pedestrian case. We also discuss the associated numerical methods, which can be Eulerian or Lagrangian. Particular attention will be devoted to the link between models at microscopic, mesoscopic, and macroscopic scales. Using macroscopic approaches give access to a whole set of methods developed for this kind of partial differential equation, including the study of phase transition, or of travelling waves. Eventually, recent variants that have been proposed in the literature will be outlined.

We study a two-species cross-diffusion model that is inspired by a system of convectiondiffusion equations derived from an agent-based model on a two-dimensional discrete lattice. The latter model has been proposed to simulate gang territorial development through the use of graffiti markings. We find two energy functionals for the system that allow us to prove a weak-stability result and identify equilibrium solutions. We show that under the natural definition of weak solutions, obtained from the weak-stability result, the system does not allow segregated solutions. Moreover, we present a result on the long-term behavior of solutions in the case when the masses of the densities are smaller than a critical value. This result is complemented with numerical experiments.

We develop an agent-based model on a lattice to investigate territorial development motivated by markings such as graffiti, generalizing a previously-published model to account for K groups instead of two groups. We then analyze this model and present two novel variations. Our model assumes that agents’ movement is a biased random walk away from rival groups’ markings. All interactions between agents are indirect, mediated through the markings. We numerically demonstrate that in a system of three groups, the groups segregate in certain parameter regimes. Starting from the discrete model, we formally derive the continuum system of 2K convection– diffusion equations for our model. These equations exhibit cross-diffusion due to the avoidance of the rival groups’ markings. Both through numerical simulations and through a linear stability analysis of the continuum system, we find that many of the same properties hold for the K-group model as for the two-group model. We then introduce two novel variations of the agent-based model, one corresponding to some groups being more timid than others, and the other corresponding to some groups being more threatening than others. These variations present different territorial patterns than those found in the original model. We derive corresponding systems of convection– diffusion equations for each of these variations, finding both numerically and through linear stability analysis that each variation exhibits a phase transition.

We consider a collective behavior model in which individuals try to imitate each others' velocity and have a preferred speed. We show that a phase change phenomenon takes place as diffusion decreases, bringing the system from a "disordered" to an "ordered" state. This effect is related to recently noticed phenomena for the diffusive Vicsek model. We also carry out numerical simulations of the system and give further details on the phase transition.

We consider a hydrodynamic model of swarming behavior derived from the kinetic description of a particle system combining a noisy Cucker-Smale consensus force and self-propulsion. In the large self-propulsive force limit, we provide evidence of a phase transition from disordered to ordered motion which manifests itself as a change of type of the limit model (from hyperbolic to diffusive) at the crossing of a critical noise intensity. In the hyperbolic regime, the resulting model, referred to as the 'Self-Organized Hydrodynamics (SOH)', consists of a system of compressible Euler equations with a speed constraint. We show that the range of SOH models obtained by this limit is restricted. To waive this restriction, we compute the Navier-Stokes diffusive corrections to the hydrodynamic model. Adding these diffusive corrections, the limit of a large propulsive force yields unrestricted SOH models and offers an alternative to the derivation of the SOH using kinetic models with speed constraints.

In this paper we introduce simplified, exact, combinatorial formulas that arise in the vortex interaction model found in [33]. These combinatorial formulas allow for the efficient implementation and development of a new multi-moment vortex method (MMVM) using a Hermite expansion to simulate 2D vorticity. The method naturally allows the particles to deform and become highly anisotropic as they evolve without the added cost of computing the non-local Biot-Savart integral. We present three examples using MMVM. We first focus our attention on the implementation of a single particle, large number of Hermite moments case, in the context of quadrupole perturbations of the Lamb-Oseen vortex. At smaller perturbation values, we show the method captures the shear diffusion mechanism and the rapid relaxation (on Re^1/3 time scale) to an axisymmetric state. We then present two more examples of the full multi-moment vortex method and discuss the results in the context of classic vortex methods. We perform numerical tests of convergence of the single particle method and show that at least in simple cases the method exhibits the exponential convergence typical of spectral methods. Lastly, we numerically investigate the spatial accuracy improvement from the inclusion of higher Hermite moments in the full MMVM.

In this paper, we study simulations of the schooling and swarming behavior of a mathematical model for the motion of pelagic fish. We use a derivative of a discrete model of interacting particles originated by Vicsek and Czirók et al. [A. Czirók, T. Vicsek, Collective behavior of interacting self-propelled particles, Physica A 281 (2000) 17-29; A. Czirók, H. Stanley, T. Vicsek, Spontaneously ordered motion of self-propelled particles, Journal of Physics A: Mathematical General 30 (1997) 1375-1385; T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, O. Shochet, Novel type of phase transition in a system of self-driven particles, Physical Review Letters 75 (6) (1995) 1226-1229; T. Vicsek, A. Czirók, I. Farkas, D. Helbing, Application of statistical mechanics to collective motion in biology, Physica A 274 (1999) 182-189]. Recently, a system of ODEs was derived from this model [B. Birnir, An ODE model of the motion of pelagic fish, Journal of Statistical Physics 128 (1/2) (2007) 535-568], and using these ODEs, we find transitory and long-term behavior of the discrete system. In particular, we numerically find stationary, migratory, and circling behavior in both the discrete and the ODE model and two types of swarming behavior in the discrete model. The migratory solutions are numerically stable and the circling solutions are metastable. We find a stable circulating ring solution of the discrete system where the fish travel in opposite directions within an annulus. We also find the origin of noise-driven swarming when repulsion and attraction are absent and the fish interact solely via orientation.