We study a time-fractional Fisher–KPP equation involving a Riemann–Liouville fractional derivative acting on the diffusion term, as derived by Angstmann and Henry (Entropy, 22:1035, 2020). The model captures memory effects in diffusive population dynamics and serves as a framework for tumor growth modeling. We first establish local well-posedness of weak solutions. The analysis combines a Galerkin approximation with a refined a priori estimate based on a Bihari–Henry–Gronwall inequality, addressing the nonlinear coupling between the fractional diffusion and the reaction term. For small initial data, we further prove global well-posedness and asymptotic stability. A numerical method based on a nonuniform convolution quadrature scheme is then proposed and validated. Simulations demonstrate distinct dynamical behaviors compared to conventional formulations, emphasizing the physical consistency of the present model in describing tumor progression.

In this work, we present a novel stationary Gierer-Meinhardt system incorporating non-local proliferation rates, defined as follows: [Formula presented] This system emerges in various contexts, such as biological morphogenesis, where two interacting chemicals, identified as an activator and an inhibitor, are described, and in ecological systems modeling the interaction between two species, classified as specialists and generalists. The non-local interspecies interactions are represented by the terms J⁎up,J⁎um where the ⁎-symbol denotes the convolution operation in RN with a kernel J∈C1(RN∖0). In the system, we assume that 0<ρ∈C0,γ(RN) with γ∈(0,1), while the parameters satisfy λ,μ,q,m,s>0 and p>1. Under various integrability conditions on the kernel J, we establish the existence and non-existence of classical positive solutions in the function space Cloc2,δ(RN). These results further highlight the influence of the non-local terms, particularly the proliferation rates, in the proposed model. 

Accurate population data is crucial for assessing exposure in disaster risk assessments. In recent years,there has been a signifcant increase in the development of spatially gridded population datasets.Despite these datasets often using similar input data to derive population fgures, notable diferencesarise when comparing them with direct ground-level observations. This study evaluates the precisionand accuracy of food exposure assessments using both known and generated gridded populationdatasets in Sweden. Specifcally focusing on WorldPop and GHSPop, we compare these datasetsagainst ofcial national statistics at a 100 m grid cell resolution to assess their reliability in foodexposure analyses. Our objectives include quantifying the reliability of these datasets and examiningthe impact of data aggregation on estimated food exposure across diferent administrative levels.The analysis reveals signifcant discrepancies in food exposure estimates, underscoring the challengesassociated with relying on generated gridded population data for precise food risk assessments.Our fndings emphasize the importance of careful dataset selection and highlight the potential foroverestimation in food risk analysis. This emphasises the critical need for validations against groundpopulation data to ensure accurate food risk management strategies.

In this paper, we study a stochastic parabolic problem involving anonlocal diffusion operator associated with nonlocal Robin-type boundary conditions. The stochastic dynamics under consideration is driven by a mixtureof a classical Brownian and a fractional Brownian motion with Hurst indexH ∈ (1/2, 1). We first establish local in time existence results and then exploreconditions under which the resulting SPDE exhibits finite-time quenching. Using results on the probability distribution of perpetual integral functionals ofBrownian motion as well as tail estimates for the fractional Brownian motionwe provide analytic estimates for certain quantities of interest, such as upperbounds for quenching times and the corresponding quenching probabilities.The existence of global in time solutions is also investigated and as a consequence a lower estimate of the quenching time is also derived. Our analyticalresults demonstrate the non-trivial impact of the noise on the dynamics ofthe system. The analytic results are complemented with a detailed numericalstudy of the model under Dirichlet boundary conditions. A possible application concerning MEMS technology is considered and the implications of theresults in this context are commented upon.

In this paper, we study a stochastic parabolic problem that emerges in the modeling and control of an electrically actuated MEMS (micro-electro-mechanical system) device. The dynamics under consideration are driven by an one dimensional fractional Brownian motion with Hurst index [Formula: see text]. We derive conditions under which the resulting SPDE has a global in time solution, and we provide analytic estimates for certain statistics of interest, such as quenching times and the corresponding quenching probabilities. Our results demonstrate the non-trivial impact of the fractional noise on the dynamics of the system. Given the significance of MEMS devices in biomedical applications, such as drug delivery and diagnostics, our results provide valuable insights into the reliability of these devices in the presence of positively correlated noise.

The purpose of the current paper is to unveil the key mechanism which is responsible for the occurrence of Turing-type instability for a nonlocal Fisher-KPP model. In particular, we prove that the solution of the considered nonlocal Fisher-KPP equation in the neighborhood of a constant stationary solution is destabilized via a diffusion-driven blow-up. It is also shown that the observed diffusion-driven blow-up is complete, while its blow-up rate is completely classified. Finally, the detected diffusion-driven instability results in the formation of unstable blow-up patterns, which are also identified through the determination of the blow-up profile of the solution.

In the current work, we study a stochastic parabolic problem. The presented problem is motivated by the study of an idealised electrically actuated MEMS (Micro-Electro-Mechanical System) device in the case of random fluctuations of the potential difference, a parameter that actually controls the operation of MEMS device. We first present the construction of the mathematical model, and then, we deduce some local existence results. Next for some particular versions of the model, relevant to various boundary conditions, we derive quenching results as well as estimations of the probability for such singularity to occur. Additional numerical study of the problem in one dimension follows, which also allows the further investigation the problem with respect to its quenching behaviour.

The process of revitalising quiescent cells in order for them to proliferate plays a pivotalrole in the repair of worn-out tissues as well as for tissue homeostasis. This process is also crucialin the growth, development and well-being of higher multi-cellular organisms such as mammals.Deregulation of proliferation-quiescence transition is related to many diseases, such as cancer. Recentstudies have revealed that this proliferation–quiescence process is regulated tightly by the Rb − E2Fbistable switch mechanism. Based on experimental observations, in this study, we formulate amathematical model to examine the effect of the growth factor concentration on the proliferation–quiescence transition in human cells. Working with a non-dimensionalised model, we prove thepositivity, boundedness and uniqueness of solutions. To understand model solution behaviour closeto bifurcation points, we carry out bifurcation analysis, which is further illustrated by the use ofnumerical bifurcation analysis, sensitivity analysis and numerical simulations. Indeed, bifurcationand numerical analysis of the model predicted a transition between bistable and stable states, whichare dependent on the growth factor concentration parameter (GF). The derived predictions confirmexperimental observations.

We consider a parabolic problem with Robin boundary condition which arises when the edge of a micro-electro-mechanical-system (MEMS) device is connected with a flexible nonideal support. Then via a rigorous analysis we investigate the structure of the solution set of the corresponding steady-state problem. We show that a critical value (the pull-in voltage) exists so that the system has exactly two stationary solutions when the applied voltage is lower than this critical value, one stationary solution for applying this critical voltage, and no stationary solution above the critical voltage.