High-accuracy positivity-preserving numerical method for Keller-Segel model.

The Keller-Segel model is a time-dependent nonlinear partial differential system, which couples a reaction-diffusion-chemotaxis equation with a reaction-diffusion equation; the former describes cell density, and the latter depicts the concentration of chemoattractants. This model plays a vital role in the simulation of the biological processes. In view of the fact that most of the proposed numerical methods for solving the model are low-accuracy in the temporal direction, we aim to derive a high-precision and stable compact difference scheme by using a finite difference method to solve this model. First, a fourth-order backward difference formula and compact difference operators are respectively employed to discretize the temporal and spatial derivative terms in this model, and a compact difference scheme with the space-time fourth-order accuracy is proposed. To keep the accuracy of its boundary with the same order as the main scheme, a Taylor series expansion formula with the Peano remainder is used to discretize the boundary conditions. Then, based on the new scheme, a multigrid algorithm and a positivity-preserving algorithm which can guarantee the fourth-order accuracy are established. Finally, the accuracy and reliability of the proposed method are verified by diverse numerical experiments. Particularly, the finite-time blow-up, non-negativity, mass conservation and energy dissipation are numerically simulated and analyzed.

The triple-deck stage of marginal separation.

The method of matched asymptotic expansions is applied to the investigation of transitional separation bubbles. The problem-specific Reynolds number is assumed to be large and acts as the primary perturbation parameter. Four subsequent stages can be identified as playing key roles in the characterization of the incipient laminar-turbulent transition process: due to the action of an adverse pressure gradient, a classical laminar boundary layer is forced to separate marginally (I). Taking into account viscous-inviscid interaction then enables the description of localized, predominantly steady, reverse flow regions (II). However, certain conditions (e.g. imposed perturbations) may lead to a finite-time breakdown of the underlying reduced set of equations. The ensuing consideration of even shorter spatio-temporal scales results in the flow being governed by another triple-deck interaction. This model is capable of both resolving the finite-time singularity and reproducing the spike formation (III) that, as known from experimental observations and direct numerical simulations, sets in prior to vortex shedding at the rear of the bubble. Usually, the triple-deck stage again terminates in the form of a finite-time blow-up. The study of this event gives rise to a noninteracting Euler-Prandtl stage (IV) associated with unsteady separation, where the vortex wind-up and shedding process takes place. The focus of the present paper lies on the triple-deck stage III and is twofold: firstly, a comprehensive numerical investigation based on a Chebyshev collocation method is presented. Secondly, a composite asymptotic model for the regularization of the ill-posed Cauchy problem is developed. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s10665-021-10125-3.

A remark on "Biological control through provision of additional food to predators: A theoretical study" [Theor. Popul. Biol. 72 (2007) 111-120].

Biological control, the use of predators and pathogens to control target pests, is a promising alternative to chemical control. It is hypothesized that the introduced predators efficacy can be boosted by providing them with an additional food source. The current literature (Srinivasu, 2007; 2010; 2011) claims that if the additional food is of sufficiently large quantity and quality then pest eradication is possible in finite time. The purpose of the current manuscript is to show that to the contrary, pest eradication is not possible in finite time, for any quantity and quality of additional food. We show that pest eradication will occur only in infinite time, and derive decay rates to the extinction state. We posit a new modeling framework to yield finite time pest extinction. Our results have large scale implications for the effective design of biological control methods involving additional food.

Biological control via "ecological" damping: An approach that attenuates non-target effects.

In this work we develop and analyze a mathematical model of biological control to prevent or attenuate the explosive increase of an invasive species population, that functions as a top predator, in a three-species food chain. We allow for finite time blow-up in the model as a mathematical construct to mimic the explosive increase in population, enabling the species to reach "disastrous", and uncontrollable population levels, in a finite time. We next improve the mathematical model and incorporate controls that are shown to drive down the invasive population growth and, in certain cases, eliminate blow-up. Hence, the population does not reach an uncontrollable level. The controls avoid chemical treatments and/or natural enemy introduction, thus eliminating various non-target effects associated with such classical methods. We refer to these new controls as "ecological damping", as their inclusion dampens the invasive species population growth. Further, we improve prior results on the regularity and Turing instability of the three-species model that were derived in Parshad et al. (2014). Lastly, we confirm the existence of spatiotemporal chaos.

Numerical study of fractional nonlinear Schrödinger equations.

Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation.