9,589 research outputs found
Stochastic and deterministic models for age-structured populations with genetically variable traits
Understanding how stochastic and non-linear deterministic processes interact
is a major challenge in population dynamics theory. After a short review, we
introduce a stochastic individual-centered particle model to describe the
evolution in continuous time of a population with (continuous) age and trait
structures. The individuals reproduce asexually, age, interact and die. The
'trait' is an individual heritable property (d-dimensional vector) that may
influence birth and death rates and interactions between individuals, and vary
by mutation. In a large population limit, the random process converges to the
solution of a Gurtin-McCamy type PDE. We show that the random model has a long
time behavior that differs from its deterministic limit. However, the results
on the limiting PDE and large deviation techniques \textit{\`a la}
Freidlin-Wentzell provide estimates of the extinction time and a better
understanding of the long time behavior of the stochastic process. This has
applications to the theory of adaptive dynamics used in evolutionary biology.
We present simulations for two biological problems involving life-history trait
evolution when body size is plastic and individual growth is taken into
account.Comment: This work is a proceeding of the CANUM 2008 conferenc
PDE models of adder mechanisms in cellular proliferation
Cell division is a process that involves many biochemical steps and complex biophysical mechanisms. To simplify the understanding of what triggers cell division, three basic models that subsume more microscopic cellular processes associated with cell division have been proposed. Cells can divide based on the time elapsed since their birth, their size, and/or the volume added since their birth-the timer, sizer, and adder models, respectively. Here, we propose unified adder-sizer models and investigate some of the properties of different adder processes arising in cellular proliferation. Although the adder-sizer model provides a direct way to model cell population structure, we illustrate how it is mathematically related to the well-known model in which cell division depends on age and size. Existence and uniqueness of weak solutions to our 2+1-dimensional PDE model are proved, leading to the convergence of the discretized numerical solutions and allowing us to numerically compute the dynamics of cell population densities. We then generalize our PDE model to incorporate recent experimental findings of a system exhibiting mother-daughter correlations in cellular growth rates. Numerical experiments illustrating possible average cell volume blowup and the dynamical behavior of cell populations with mother-daughter correlated growth rates are carried out. Finally, motivated by new experimental findings, we extend our adder model cases where the controlling variable is the added size between DNA replication initiation points in the cell cycle
Equilibrium points for Optimal Investment with Vintage Capital
The paper concerns the study of equilibrium points, namely the stationary
solutions to the closed loop equation, of an infinite dimensional and infinite
horizon boundary control problem for linear partial differential equations.
Sufficient conditions for existence of equilibrium points in the general case
are given and later applied to the economic problem of optimal investment with
vintage capital. Explicit computation of equilibria for the economic problem in
some relevant examples is also provided. Indeed the challenging issue here is
showing that a theoretical machinery, such as optimal control in infinite
dimension, may be effectively used to compute solutions explicitly and easily,
and that the same computation may be straightforwardly repeated in examples
yielding the same abstract structure. No stability result is instead provided:
the work here contained has to be considered as a first step in the direction
of studying the behavior of optimal controls and trajectories in the long run
Maximum Principle for Linear-Convex Boundary Control Problems applied to Optimal Investment with Vintage Capital
The paper concerns the study of the Pontryagin Maximum Principle for an
infinite dimensional and infinite horizon boundary control problem for linear
partial differential equations. The optimal control model has already been
studied both in finite and infinite horizon with Dynamic Programming methods in
a series of papers by the same author, or by Faggian and Gozzi. Necessary and
sufficient optimality conditions for open loop controls are established.
Moreover the co-state variable is shown to coincide with the spatial gradient
of the value function evaluated along the trajectory of the system, creating a
parallel between Maximum Principle and Dynamic Programming. The abstract model
applies, as recalled in one of the first sections, to optimal investment with
vintage capital
Equilibrium Points for Optimal Investment with Vintage Capital
The paper concerns the study of equilibrium points, namely the stationary solutions to the closed loop equation, of an infinite dimensional and infinite horizon boundary control problem for linear partial differential equations. Sufficient conditions for existence of equilibrium points in the general case are given and later applied to the economic problem of optimal investment with vintage capital. Explicit computation of equilibria for the economic problem in some relevant examples is also provided. Indeed the challenging issue here is showing that a theoretical machinery, such as optimal control in infinite dimension, may be effectively used to compute solutions explicitly and easily, and that the same computation may be straightforwardly repeated in examples yielding the same abstract structure. No stability result is instead provided: the work here contained has to be considered as a first step in the direction of studying the behavior of optimal controls and trajectories in the long run.Linear convex control, Boundary control, Hamilton–Jacobi–Bellman equations, Optimal investment problems, Vintage capital
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