Основні поняття
The core message of this article is to present a framework for efficiently computing the Jacobian matrices of higher-order ecological networks using various tensor decomposition techniques, enabling effective stability analysis of complex ecological systems.
Анотація
The article introduces the higher-order generalized Lotka-Volterra (HOGLV) model to capture higher-order interactions in ecological networks. It then proposes a framework that leverages tensor decomposition methods, including higher-order singular value decomposition (HOSVD), Canonical Polyadic decomposition (CPD), and tensor train decomposition (TTD), to efficiently compute the Jacobian matrices and thus determine the linear stability of the HOGLV model.
The key highlights and insights are:
- The HOGLV model is represented in various tensor decomposition forms to address the exponential growth in the number of model parameters with the maximum order of interactions.
- The computational complexity of computing the Jacobian matrix is analyzed for the full, HOSVD-based, CPD-based, and TTD-based representations of the HOGLV model. The tensor decomposition-based representations significantly outperform the full representation in terms of both memory and computational efficiency.
- The CPD-based representation offers the lowest computational complexity, but may suffer from numerical instability issues for larger system dimensions. In contrast, the TTD-based representation maintains numerical stability while providing a reasonably low computational complexity.
- Numerical examples demonstrate the effectiveness of the proposed framework in computing the Jacobian matrices and analyzing the stability of complex ecological networks with higher-order interactions.
Статистика
The total number of model parameters for the full, HOSVD-based, CPD-based, and TTD-based representations of the HOGLV model across varying system dimensions are provided in Table I.
The computation time for calculating the Jacobian matrix of the TTD-based representation of the HOGLV model for large system dimensions, where the full representation fails due to memory constraints, is shown in Table II.