Global dynamics of boundary droplets
Peter W. Bates Jiayin Jin
Discrete & Continuous Dynamical Systems - A 2014, 34(1): 1-17 doi: 10.3934/dcds.2014.34.1
We establish the existence of a global invariant manifold of bubble states for the mass-conserving Allen-Cahn Equation in two space dimensions and give the dynamics for the center of the bubble.
keywords: infinitely dimensional dynamical systems bubble solution Invariant manifold theory global dynamics. mass-conserving Allen-Cahn Equation
Growth regulation and the insulin signaling pathway
Peter W. Bates Yu Liang Alexander W. Shingleton
Networks & Heterogeneous Media 2013, 8(1): 65-78 doi: 10.3934/nhm.2013.8.65
The insulin signaling pathway propagates a signal from receptors in the cell membrane to the nucleus via numerous molecules some of which are transported through the cell in a partially stochastic way. These different molecular species interact and eventually regulate the activity of the transcription factor FOXO, which is partly responsible for inhibiting the growth of organs. It is postulated that FOXO partially governs the plasticity of organ growth with respect to insulin signalling, thereby preserving the full function of essential organs at the expense of growth of less crucial ones during starvation conditions. We present a mathematical model of this reacting and directionally-diffusing network of molecules and examine the predictions resulting from simulations.
keywords: signaling pathway nonlinear system of partial differential equations. Cellular regulatory network
Invariant foliations for random dynamical systems
Ji Li Kening Lu Peter W. Bates
Discrete & Continuous Dynamical Systems - A 2014, 34(9): 3639-3666 doi: 10.3934/dcds.2014.34.3639
We prove the existence of invariant foliations of stable and unstable manifolds of a normally hyperbolic random invariant manifold. The normally hyperbolic random invariant manifold referred to here is that which was shown to exist in a previous paper when a deterministic dynamical system having a normally hyperbolic invariant manifold is subjected to a small random perturbation.
keywords: random normally hyperbolic invariant manifolds Random dynamical systems random invariant foliations.
Singular fold with real noise
Peter W. Bates Ji Li Mingji Zhang
Discrete & Continuous Dynamical Systems - B 2016, 21(7): 2091-2107 doi: 10.3934/dcdsb.2016038
We study the effect of small real noise on the jump behavior near a singular fold point, which is an important step in understanding the burst-spike behavior in many biological models. We show by the theory of center manifolds and random invariant manifolds that if the order of the noise is high enough, trajectories essentially pass the fold point in the manner as though there is no noise.
keywords: random manifolds singular fold blow-up. Random dynamical systems center manifold

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