Crossing-effect in non-isolated and non-symmetric systems of patches

TL;DR

This paper determines the minimal size of two identical patches in complex systems, providing explicit solutions that unify previous cases and addressing new communication scenarios, with implications for system robustness.

Contribution

It presents an explicit minimal size solution for two-patch systems, unifying various special cases and exploring new communication configurations.

Findings

Explicit minimal size formula for two patches.

Unified framework encompassing previous special cases.

Identification of conditions affecting system robustness.

Abstract

The main result of this article is the determination of the minimal size for the general case of problems with two identical patches. This solution is presented in the explicit form, which allows to recuperate all the cases found in the literature as particular cases, namely, one isolated fragment, one single fragment communicating with its neighborhood, a system with two identical fragments isolated from the matrix but mutually communicating and a system of two identical fragments inserted in a homogeneous matrix. It is also addressed the new problem of a single fragment communicating with the matrix, with different life difficulty of each side. As application, it is found that the internal condition $a_{0}$ can set which system is the worst to life. This prediction confirms and extends the prediction already found in the literature between isolated and non-isolated systems.