Asymptotic analysis and upper semicontinuity with respect to delay term of attractors to binary mixtures of solids

We investigated the asymptotic dynamics of a nonlinear system modeling binary mixture of solids with delay term. Using the recent quasi-stability methods introduced by Chueshov and Lasiecka, we prove the existence, smoothness and finite dimensionality of a global attractor. We also prove the existence of exponential attractors. Moreover, we study the upper semicontinuity of global attractors with respect to small perturbations of the delay terms.

Long Time Behaviour of a Local Perturbation in the Isotropic XY Chain Under Periodic Forcing

We study the isotropic XY quantum spin chain with a time-periodic transverse magnetic field acting on a single site. The asymptotic dynamics is described by a highly resonant Floquet–Schrödinger equation, for which we show the existence of a periodic solution if the forcing frequency is away from a discrete set of resonances. This in turn implies the state of the quantum spin chain to be asymptotically a periodic function synchronised with the forcing, also at arbitrarily low non-resonant frequencies. The behaviour at the resonances remains a challenging open problem.

Asymptotic Dynamics of Young Differential Equations

We provide a unified analytic approach to study the asymptotic dynamics of Young differential equations, using the framework of random dynamical systems and random attractors. Our method helps to generalize recent results (Duc et al. in J Differ Equ 264:1119–1145, 2018, SIAM J Control Optim 57(4):3046–3071, 2019; Garrido-Atienza et al. in Int J Bifurc Chaos 20(9):2761–2782, 2010) on the existence of the global pullback attractors for the generated random dynamical systems. We also prove sufficient conditions for the attractor to be a singleton, thus the pathwise convergence is in both pullback and forward senses.

Plant host shift: The viability and persistence of a polyphagous non-native insect

Drosophila suzukii (Diptera: Drosophilidae) has become a pervasive pest in several countries around the world. Many studies have investigated the preference and attractiveness of potential hosts on this invasive, polyphagous drosophilid. Thus far, no studies have investigated whether a shift of fruit host could affect its ecological viability or spatiotemporal persistence. In this study, we analysed the fecundity and oviposition period jointly with the survival time of D. suzukii subject to a shift in host fruit, using a joint modelling method for longitudinal outcomes and time-until-event outcomes. The number of eggs laid by females was higher in raspberry than in strawberry and when setting adults of F1 generation underwent a first host shift. The joint modelling also suggested that insects reared on raspberry survived longer. We then combined experimental results with a two-patch dispersal model to investigate how host shift in a species that exhibits both passive and density-dependent dispersal may affect its asymptotic dynamics. In line with empirical evidence, we found that a shift in host choice can significantly affect the growth potential and fecundity of a species such as D. suzukii, which ultimately could aid such invasive populations in their ability to persist within a changing environment.

Introduction to matrix population models

Matrix population models (MPMs) are currently used in a range of fields, from basic research in ecology and evolutionary biology, to applied questions in conservation biology, management, and epidemiology. In MPMs individuals are classified into discrete stages, and the model projects the population over discrete time-steps. A rich analytical theory is available for these models, for both the linear deterministic case and for more complex dynamics including stochasticity and density dependence. This chapter provides a non comprehensive introduction to MPMs and some basic results on asymptotic dynamics, life history parameters, and sensitivities and elasticities of the long-term growth rate to projection matrix elements and to underlying parameters. We assume that readers are familiar with basic matrix calculations. Using examples with different kinds of demographic structure, we demonstrate how the general stage-structured model can be applied to each case.

Scattering Map for the Vlasov–Poisson System

We construct (modified) scattering operators for the Vlasov–Poisson system in three dimensions, mapping small asymptotic dynamics as $$t\rightarrow -\infty$$ t → - ∞ to asymptotic dynamics as $$t\rightarrow +\infty$$ t → + ∞ . The main novelty is the construction of modified wave operators, but we also obtain a new simple proof of modified scattering. Our analysis is guided by the Hamiltonian structure of the Vlasov–Poisson system. Via a pseudo-conformal inversion, we recast the question of asymptotic behavior in terms of local in time dynamics of a new equation with singular coefficients which is approximately integrated using a generating function.

On the Asymptotic Behavior of Solutions to the Vlasov–Poisson System

We prove small data modified scattering for the Vlasov–Poisson system in dimension $d=3$, using a method inspired from dispersive analysis. In particular, we identify a simple asymptotic dynamics related to the scattering mass.

Long-time behavior of solutions for a system of N-coupled nonlinear dissipative half-wave equations

In the current paper, we consider a system of N-coupled weakly dissipative fractional nonlinear Schrödinger equations. The well-posedness of the initial value problem is established by a refined analysis based on a limiting argument as well as the study of the asymptotic dynamics of the solutions. This asymptotic behavior is described by the existence of a compact global attractor in the appropriate energy space.

Mathematical Modeling and Dynamics of Oncolytic Virotherapy

Oncolytic virotherapy is a cancer treatment that uses competent replicating viruses to destroy cancer cells. This field progressed from earlier observations of accidental viral infections causing remission in many malignancies to virus drugs targeting and killing cancer cells. In this chapter, we study some basic models of the oncolytic virotherapy and their dynamics. We show how the dynamical system’s theory can capture the behavior of the solutions of those models and provide different approaches to studying the models. We study the thresholds that enable us to classify asymptotic dynamics of the solutions. Fractional-derivative approach tells us about the memory of the derivative and related solutions of the models. We also study the affect of introducing control parameters on the cost of the therapy.