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Permanence of Stochastic Biological Systems

The interdisciplinary journal of Discontinuity, Nonlinearity, and Complexity, 2019
Area of research related to prey-predator systems is an important topic. The concept of permanence is an important issues related to biological systems. In general permanence is considered as a combination of persistence and boundedness. Following this, this paper reviews few existing definitions of stochastic permanence.
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A Lower Bound for the Permanent of a Doubly Stochastic Matrix

The Annals of Mathematics, 1979
It is shown here that the permanent of an n x n doubly stochastic Let A be an n x n matrix (aij)". The permanent of A is defined by (Al) ~~~~~p(A) = fle Snni=1ai.(i)
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Notes on permanents of doubly stochastic matrices

Linear and Multilinear Algebra, 1983
Using a simple probabilistic interpretation of the permanents of doubly stochastic matrices, we obtain in this paper some results which extend the work in [1]. An upper bound is obtained for the permanent on the doubly stochastic matrices A=(aij) with all of whose entries satisfying the condition .
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An expansion for the permanent of a doubly stochastic matrix

Journal of the Australian Mathematical Society, 1973
The permanent of an n-square matrix A = (aij) is defined by where Sn is the symmetric group of order n. Kn will denote the convex set of all n-square doubly stochastic matrices and K0n its interior. Jn ∈ Kn will be the matrix with all elements equal to 1/n. If M ∈ K0n, then M lies on a line segment passing through Jn and another B ∈ Kn — K0n.
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Permanent of the product of doubly stochastic matrices

Mathematical Proceedings of the Cambridge Philosophical Society, 1966
If A = [aij] is an n × n matrix, the permanent of A is the scalar valued function of A defined bywhere the summation extends over all permutations (i1, i2, …, in) of the integers 1, 2, …, n. If we assume that A is a non-negative matrix (that is, that A has non-negative entries) then we come upon an extremely interesting situation.
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Permanents of special classes of doubly stochastic matrices

Linear and Multilinear Algebra, 1976
We identify the doubly stochastic matrices with at least one zero entry which are closest in the Euclidean norm to Jn , the matrix with each entry equal to 1/n, and we show that at these matrices the permanent function has a relative minimum when restricted to doubly stochastic matrices having zero entries.
Paul J. Knopp, Richard Sinkhorn
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Convexity of the permanent for doubly stochastic matrices

Linear and Multilinear Algebra, 1991
Let Ω n denote the set of all n×n doubly stochastic matrices and let Jn denote the n×n matrix all of whose entries are 1/n. Lih and Wang conjectuted that per[(1−i)Jn +iA≤(1−i)perJn 1i perA for all A∈Ω n and all t∈[0,1/2], and proved their conjecture for n=3.
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Some Conjectures on Permanents of Doubly Stochastic Matrices

Journal of Discrete Mathematical Sciences and Cryptography, 2016
AbstractLet denote the set of all doubly stochastic matrices of order n. Foregger [3] raised a n question whether per per (A) holds for all and , where Jn is the n × n matrix with each entry equal ...
P. Subramanian, K. Somasundaram
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The Stochastic Permanent Break Model and the Fractional Integration Hypothesis

Computational Economics, 2004
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Minimum Permanents of Tridiagonal Doubly Stochastic Matrices

Linear and Multilinear Algebra, 2002
We determine the minimum permanents and minimizing matrices of the tridiagonal doubly stochastic matrices and of certain doubly stochastic matrices with prescribed zero entries.
Seok-Zun Song, Young-Bae Jun
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