Depleting the signal: Analysis of chemotaxis-consumption models—A survey

We give an overview of analytical results concerned with chemotaxis systems where the signal is absorbed. We recall results on existence and properties of solutions for the prototypical chemotaxis-consumption model and various variants and review more recent findings on its ability to support the emergence of spatial structures

Depleting the signal: Analysis of chemotaxis-consumption models -- A survey

We give an overview of analytical results concerned with chemotaxis systems where the signal is absorbed. We recall results on existence and properties of solutions for the prototypical chemotaxis-consumption model and various variants and review more recent findings on its ability to support the emergence of spatial structures

Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects

We introduce a generalized concept of solutions for reaction–diffusion systems and prove their global existence. The only restriction on the reaction function beyond regularity, quasipositivity and mass control is special in that it merely controls the growth of cross-absorptive terms. The result covers nonlinear diffusion and does not rely on an entropy estimate

Inducing an optical Feshbach resonance via stimulated Raman coupling

We demonstrate a novel method of inducing an optical Feshbach resonance based on a coherent free-bound stimulated Raman transition. In our experiment atoms in a Rb87 Bose-Einstein condensate are exposed to two phase-locked Raman laser beams which couple pairs of colliding atoms to a molecular ground state. By controlling the power and relative detuning of the two laser beams, we can change the atomic scattering length considerably. The dependence of scattering length on these parameters is studied experimentally and modelled theoretically.Comment: 8 pages, 8 figures, submitted to PR

Currency Substitution and The Law of One Price

We study endogenous currency substitution in a decentralized trade environment. Sellers maximize profits from sales of imperfectly substitutable goods by posting prices in either one of two currencies. A unique symmetric equilibrium exists where goods are priced only in the local currency. This occurs if foreign trade is sporadic, there is sufficient but not excessive liquidity, and discounting is low. Excess or scarcity of liquidity, however, induces sellers to extract all surplus from buyers. This destroys the monetary equilibrium and shuts down trade. Equilibria with and without currency substitution coexist on some region of the parameter space, and may be multiple. We prove that purchasing power parity may hold even if foreign trade is costly and the currency's value differs across countries. International circulation of money may expand the extent of the market hence enhance welfare.

Numerical solution of optimal control problems with constant control delays

We investigate a class of optimal control problems that exhibit constant exogenously given delays in the control in the equation of motion of the differential states. Therefore, we formulate an exemplary optimal control problem with one stock and one control variable and review some analytic properties of an optimal solution. However, analytical considerations are quite limited in case of delayed optimal control problems. In order to overcome these limits, we reformulate the problem and apply direct numerical methods to calculate approximate solutions that give a better understanding of this class of optimization problems. In particular, we present two possibilities to reformulate the delayed optimal control problem into an instantaneous optimal control problem and show how these can be solved numerically with a state-of-the-art direct method by applying Bock’s direct multiple shooting algorithm. We further demonstrate the strength of our approach by two economic examples.delayed differential equations, delayed optimal control, numerical optimization, time-to-build

A double critical mass phenomenon in a no-flux-Dirichlet Keller-Segel system

Derived from a biophysical model for the motion of a crawling cell, the system $$(*)~\begin{cases}u_t=\Delta u-\nabla\cdot(u\nabla v)\\0=\Delta v-kv+u\end{cases}$$ is investigated in a finite domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, with $k\geq0$. While a comprehensive literature is available for cases with $(*)$ describing chemotaxis systems and hence being accompanied by homogeneous Neumann-type boundary conditions, the presently considered modeling context, besides yet requiring the flux $\partial_\nu u-u\partial_\nu v$ to vanish on $\partial\Omega$, inherently involves homogeneous Dirichlet conditions for the attractant $v$, which in the current setting corresponds to the cell's cytoskeleton being free of pressure at the boundary. This modification in the boundary setting is shown to go along with a substantial change with respect to the potential to support the emergence of singular structures: It is, inter alia, revealed that in contexts of radial solutions in balls there exist two critical mass levels, distinct from each other whenever $k>0$ or $n\geq3$, that separate ranges within which (i) all solutions are global in time and remain bounded, (ii) both global bounded and exploding solutions exist, or (iii) all nontrivial solutions blow up in finite time. While critical mass phenomena distinguishing between regimes of type (i) and (ii) belong to the well-understood characteristics of $(*)$ when posed under classical no-flux boundary conditions in planar domains, the discovery of a distinct secondary critical mass level related to the occurrence of (iii) seems to have no nearby precedent. In the planar case with the domain being a disk, the analytical results are supplemented with some numerical illustrations, and it is discussed how the findings can be interpreted biophysically for the situation of a cell on a flat substrate