Exploring stochasticity and imprecise knowledge based on linear inequality constraints

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.

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

$$\begin{aligned} S\; \text{ convex } \Leftrightarrow \forall \; x_{1},x_{2} \in S, \forall \; \kappa \in (0,1): (1-\kappa )x_{1} + \kappa x_{2} \in S. \end{aligned}$$

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

$$\begin{aligned} \mathcal {P} = \{x\in \mathcal {R}^{n}|P^{x}x \le P^{c}\}. \end{aligned}$$

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.