We consider the so-called prion equation with the general incidence term introduced in [Greer et al., 2007], and we investigate the stability of the steady states. The method is based on the reduction technique introduced in [Gabriel, 2012]. The argu
ment combines a recent spectral gap result for the growth-fragmentation equation in weighted $L^1$ spaces and the analysis of a nonlinear system of three ordinary differential equations.
We study a growth maximization problem for a continuous time positive linear system with switches. This is motivated by a problem of mathematical biology (modeling growth-fragmentation processes and the PMCA protocol). We show that the growth rate is
determined by the non-linear eigenvalue of a max-plus analogue of the Ruelle-Perron-Frobenius operator, or equivalently, by the ergodic constant of a Hamilton-Jacobi (HJ) partial differential equation, the solutions or subsolutions of which yield Barabanov and extremal norms, respectively. We exploit contraction properties of order preserving flows, with respect to Hilberts projective metric, to show that the non-linear eigenvector of the operator, or the weak KAM solution of the HJ equation, does exist. Low dimensional examples are presented, showing that the optimal control can lead to a limit cycle.
We consider the self-similar fragmentation equation with a superquadratic fragmentation rate and provide a quantitative estimate of the spectral gap.
We investigate an optimal control problem which arises in the optimization of an amplification technique for misfolded proteins. The improvement of this technique may play a role in the detection of prion diseases. The model consists in a linear syst
em of differential equations with a nonlinear control. The appearance of oscillations in the numerical simulations is understood by using the Perron and Floquet eigenvalue theory for nonnegative irreductible matrices. Then to overcome the unsolvability of the optimal control, we relax the problem. In the two dimensional case we solve explicitely the optimal relaxed control problem when the final time is large enough.
We are concerned with the long-time behavior of the growth-fragmentation equation. We prove fine estimates on the principal eigenfunctions of the growth-fragmentation operator, giving their first-order behavior close to 0 and $+infty$. Using these es
timates we prove a spectral gap result by following the technique in [Caceres, Canizo, Mischler 2011, JMPA], which implies that solutions decay to the equilibrium exponentially fast. The growth and fragmentation coefficients we consider are quite general, essentially only assumed to behave asymptotically like power laws.
Cells grown in culture act as a model system for analyzing the effects of anticancer compounds, which may affect cell behavior in a cell cycle position-dependent manner. Cell synchronization techniques have been generally employed to minimize the var
iation in cell cycle position. However, synchronization techniques are cumbersome and imprecise and the agents used to synchronize the cells potentially have other unknown effects on the cells. An alternative approach is to determine the age structure in the population and account for the cell cycle positional effects post hoc. Here we provide a formalism to use quantifiable age distributions from live cell microscopy experiments to parameterize an age-structured model of cell population response.
