Developmental Partial Differential Equations

TL;DR

This paper introduces Developmental Partial Differential Equations (DPDE), coupling PDEs with evolving manifolds, and demonstrates controllability of surface shapes through diffusion-driven growth, with potential biological applications.

Contribution

The paper develops a new theoretical framework for DPDE, combining PDEs with evolving geometries, and proves controllability of surface shapes via diffusion control.

Findings

Established approximate controllability of surface shapes

Provided numerical simulations demonstrating control methods

Applied framework to biological growth models

Abstract

In this paper, we introduce the concept of Developmental Partial Differential Equation (DPDE), which consists of a Partial Differential Equation (PDE) on a time-varying manifold with complete coupling between the PDE and the manifold's evolution. In other words, the manifold's evolution depends on the solution to the PDE, and vice versa the differential operator of the PDE depends on the manifold's geometry. DPDE is used to study a diffusion equation with source on a growing surface whose growth depends on the intensity of the diffused quantity. The surface may, for instance, represent the membrane of an egg chamber and the diffused quantity a protein activating a signaling pathway leading to growth. Our main objective is to show controllability of the surface shape using a fixed source with variable intensity for the diffusion. More specifically, we look for a control driving a symmetric manifold shape to any other symmetric shape in a given time interval. For the diffusion we take directly the Laplace-Beltrami operator of the surface, while the surface growth is assumed to be equal to the value of the diffused quantity. We introduce a theoretical framework, provide approximate controllability and show numerical results. Future applications include a specific model for the oogenesis of Drosophila melanogaster.