Positive orthant

Let us consider a structure for the multitude of steady states for eqns. (158) or (160) in the positive orthant. For linear systems z = Kz it forms either a ray (in the case of the unique linear law of conservation) emerging from zero, or a cone formed at the linear subspace ker K intersection with the orthant. The structure for the multitude of steady states for the systems involving no intermediate interactions is also rather simple. Let us consider the case of only one linear law of conservation ZmjZ, = c = const, and examine the dependence of steady-state values zf on c. Using eqn. (160), we obtain... 

A proof for this statement is constructed in accordance with the fact that the latter inequality accounts for the sign of the coefficient in the polynomial P(/.) at kn T, which in turn is associated with the index of a steady-state point for the vector field (151) [60]. If this coefficient is positive at any point of the positive orthant R z, zt > 0, i = 1, 2,. . ., n, then the steady-state point is unique. 

Formally, m lies on the intersection of the hyperplane = const with the positive orthant of the N-dimensional components space (Wei, 1962a). 

For exclusively real eigenvalues of fV the time dependence of the average excess production is determined by the choice of initial conditions. As shown in Appendix 5, optimization of (t) is restricted to initial conditions in the positive orthant [yt(0) >0 k = 0,1,.. ., n]. These initial conditions are not difficult to fulfil, and they will apply to many cases in reality. We should keep in mind, nevertheless, that there are other choices of initial conditions, such as the start with a pure master sequence, for which the simple principle does not hold. For one particular type of choice, yi (0) > 1 and y/j(0) < 0 for all /c 1, the average excess production decreases monotonically. 

Analogous to the three-component system, constraint (12) confines the end of the vector a to the (n — l)-dimensional plane passing through the ends of the n unit vectors along the n coordinate axes. A,. Constraint (13) further limits the composition point at the end of the vector a to that part of the plane lying in the positive orthant of the n-dimensional coordinate system. This part of the plane, which forms the (n — l)-dimensional equivalent of an equilateral triangle for three components and a tetrahedron for four components, is called a simplex. The reaction paths in this system will be curves lying within the reaction simplex. 

It follows that all vectors Xy other than Xo must contain elements that are negative amounts as can be seen in Fig. 8. They, therefore, cannot lie in the positive orthant of the A coordinate system and by themselves do not represent realizable compositions. The important point is that, in spite of this, the vectors Xj are directly determinable in terms of realizable initial composition vectors of certain special reaction paths as will be shown in Section II,B,2,d. 

Extents can take on negative values Extent of reaction is positive for products and negative for reactants. Concentrations, by comparison, may never take on negative values. This property provides a convenient set of bounds for all species present in a mixture. If concentration is used over extent for a given reactive system, the associated AR for the system must then also always lie in the positive orthant of concentration space." Additionally, using reaction stoichiometry and mass balance constraints, it is possible to compute a definite bound (called the stoichiometric subspace) that the AR must reside in. Similar bounds are rather less well-defined when extent is employed, and thus greater care must be taken. 

The trajectory of a general autonomous system of differential equations can wander anywhere in the state-space. What kind of restrictions are obtained if one considers the trajectories of a kinetic differential equation It was mentioned earlier (Subsection 4.1.2) that the solutions of a kinetic differential equation remain in the first orthant if they started there. More refined statements regarding positivity and nonnegativity have also been stated as Problem 6 of Subsection 4.1.4. Now let us try to delineate an as narrow as possible set in the state-space for the trajectories. As a next step towards this goal let us write the kinetic differential equation (4.6) in the form... 

---