When random matrix theory meets theoretical ecology
Overview
This is an investigation on the P-property of the inverse of a random matrix under positive diagonal deformation. This project is driven by the objective of understanding properties of large dynamic Lotka Volterra systems equilibria for theoretical ecology.
The report presents our study of the phase transition phenomena for the feasibility (whether or not there is species extinction at the equilibrium), Volterra Lyapunov stability (non-negative and globally stable equilibrium) and P-property (unique non-negative equilibrium) in the Lotka Volterra model. The P-property remains an open problem.
Supporting information
From the P-property to regularity of an interval matrix
\[I_N - \Gamma_N \text{ P-matrix} \iff \left[ -2 {\Gamma_N}^{-1}, -2 {\Gamma_N}^{-1} + 2 I_N \right] \text{ is regular}\]Matrices in this interval have the form: \(-2 {\Gamma_N}^{-1} + \Delta\), where \(\Delta\) is a diagonal matrix with entries in \([0,2]\).
The challenge consist in showing that the interval contains no singular matrix.
Jiri Rohn’s algorithm
Consider an exhaustive list of methods to determine the REGularIty or the SINGularity of the interval \([A_c - \Delta, A_c + \Delta]\) where \(A_c = (A - I)^{-1} (A + I)\) and \(\Delta = I\); \(A\) being a matrix of size \((n \times n)\) and \(I\) the identity matrix of size \((n \times n)\).
Conditions for the existence of a singular matrix
- midpoint matrix \(A_c\):
the midpoint matrix \(A_c\) of the interval matrix \([A_c \pm \Delta]\) is singular
- diagonal condition Theorem 2.1 from Oettli and Prager:
\(|A_c x| \leq \Delta |x|\) has a non trivial (i.e. non zero) solution \(x\)
- steepest determinant descent Algorithm 5.1:
investigate determinant bounds of the interval matrix (i.e. the hull of matrices determinant for matrices in the interval)
- two Qz-matrices Theorem 4.3:
the linear programming problem \((\star)\) maximize \(z^T x\) subject to \((A_c - \Delta \cdot diag(z)) x \leq 0\) and \(diag(z) \cdot x \geq 0\), is unbounded for some \(z \in \{ \pm 1 \}^n\)
- main algorithm Theorem 2.2 - find the singular matrix:
loop on \(\{ \pm 1 \}^n\) to identify the possible singular matrix which should have the specific form
- symmetrization Sections 4 and 5 from Rex and Rohn:
both of the following conditions imply the singularity of \([A_c \pm \Delta]\):
- \(\lambda_{\max}({A_c}^T A_c) \leq \lambda_{\min}(\Delta^T \Delta)\)
- \(\Delta^T \Delta - {A_c}^T A_c\) positive definite
- \(\lambda_{\max}({A_c}^T A_c) \leq \lambda_{\min}(\Delta^T \Delta)\)
Conditions for the regularity of the interval
- Beeck’s condition Corollary 3.2 from Beeck:
\(\rho (|{A_c}^{-1}| \Delta)<1\) is regular (for \(A_c\) non singular)
- symmetrization Sections 4 and 5 from Rex and Rohn:
both of the following conditions imply the regularity of \([A_c \pm \Delta]\):
- \(\lambda_{\max}(\Delta^T \Delta) < \lambda_{\min}({A_c}^T A_c)\)
- \({A_c}^T A_c - | \Delta^T \Delta | I\) is positive definite
- \(\lambda_{\max}(\Delta^T \Delta) < \lambda_{\min}({A_c}^T A_c)\)
- two Qz-matrices Theorem 4.3:
the linear programming problem \((\star)\) is bounded for all \(z \in \{ \pm 1 \}^n\)
- main algorithm Theorem 2.2 - all matrices are non singular:
loop on \(\{ \pm 1 \}^n\) to check there is no singular matrix in the whole interval (this is the most expensive track since the algorithm will investigate the values of the sign real spectral radius)