Abstract
This paper explores the stochastic dynamics of a simple foodweb system using a network model that mimics interacting species in a biosystem. It is shown that the system can be described by a set of ordinary differential equations with real-valued uncertain parameters, which satisfy a set of linear inequality constraints. The constraints restrict the solution space to a bounded convex polytope. We present results from numerical experiments to show how the stochasticity and uncertainty characterizing the system can be captured by sampling the interior of the polytope with a prescribed probability rule, using the Hit-and-Run algorithm. The examples illustrate a parsimonious approach to modeling complex biosystems under vague knowledge.
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Acknowledgments
This work has been partly funded by the IMR Research Program on Marine Processes and Human Influence, ADMAR (RCN Project No. 200497/I30), and KOFA (IMR project 14452-02).
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Appendix: Basic definitions
Appendix: Basic definitions
The following definitions are mostly adapted from Kvasnica (2009).
Definition 1
(Convexity) A set of points S in the n-dimensional space is convex, if the line segment connecting any two points \(x_{1},x_{2} \in S\), belongs completely in S. That is
Definition 2
(Polyhedron) A convex set \(\mathcal {Q}\subseteq \mathcal {R}^{n}\) given as an intersection of a finite number of closed half-spaces \(\mathcal {Q} = \{x\in \mathcal {R}^{n}|Q^{x}x \le Q^{c}\}\), is called a polyhedron.
Definition 3
(Polytope) A bounded polyhedron \(\mathcal {P}\subset \mathcal {R}^{n}\) is called a polytope if
By definition, every polytope represents a convex, bounded and closed set.
Definition 4
(Convex Hull) The convex hull of a set of points is the smallest convex region containing all points in the set.
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Subbey, S., Planque, B. & Lindstrøm, U. Exploring stochasticity and imprecise knowledge based on linear inequality constraints. J. Math. Biol. 73, 575–595 (2016). https://doi.org/10.1007/s00285-015-0959-z
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DOI: https://doi.org/10.1007/s00285-015-0959-z