Induced kinetic differential equation

Show that the domain of all the solutions of the induced kinetic differential equation of... 

Hint. The induced kinetic differential equation is linear in case (i), while the trajectories in case (ii) remain in a bounded set. 

Construct a reaction which has an induced kinetic differential equation that has solutions defined on the whole positive half-line of real numbers and which still does not belong to any of the reaction classes above. Show that all the co-ordinates of a solution of an induced kinetic differential equation are either strictly positive for all positive times, or everywhere zero. 

Hint. Consider the induced kinetic differential equation as a linear differential equation with variable coefficients. 

Analysing Exercise l(ii) try to characterise a large set of generalised compartmental systems for which the induced kinetic differential equation can be solved in closed form. 

Finally, let us make a remark that may enlighten the significance of relative asymptotic stability. As is known (from Exercise 1 of 4.1.3) the solutions of the induced kinetic differential equation of the reaction... 

Then the induced kinetic differential equation for the corresponding CSTR cannot admit multiple positive steady states, no matter what the feed composition may be and no matter what (positive) values the residence time and rate constants take. 

Then the induced kinetic differential equation admit precisely one positive equilibrium point in each stoichiometric compatibility class. 

Applying the last statement of Subsection 4.5.2, show that the induced kinetic differential equation of the reaction... 

Applying the last statement of Subsection 4.5.2 show that, if the number of atoms is one less than that of the components in an atomic reaction obeying the law of conservation of atomic numbers, and if the stoichio metric matrix is of full rank, then the induced kinetic differential equation of the reaction has no periodic solution. 

Show that the differential equation (4.9) (the induced kinetic differential equation of the Field-Koros-Noyes mechanism model of the Belousov-Zhabotinskii reaction) does have periodic solutions at certain values of the parameters. 

It is an astonishing fact that the converse of the theorem holds as well. If the right-hand side of a differential equation is an (M, M)-polynomial without negative cross-effects then it may be considered as the induced kinetic differential equation of a reaction, or, in other words, if there is no negative cross-effect in the right-hand side then there exists a reaction with the given equation as its deterministic model. 

The right-hand side of the induced kinetic differential equation of reaction (4.21) is the (2, 2)-polynomial 0. Such a case cannot occur with reversible, or even with weakly reversible reactions. Neither can it occur with acyclic reactions. 

On the other hand, it is easy to construct two essentially different conservative, reversible reactions with the same induced kinetic differential equation (see Exercise 4). Such an example shows that it is not true that a kinetic differential equation is induced by a unique reaction even within the class of reversible and conservative reactions. The addition of other, chemically relevant properties may be enough to ensure uniqueness. 

The right-hand side of the induced kinetic differential equation of an acyclic reaction cannot be the polynomial 0. As the reaction is acyclic it should contain at least one point without arrows pointing towards it. If one of these points corresponds to a component then the derivative of this component surely contains a (negative) term that cannot be counterbalanced as a result of other elementary reactions as no other point points toward this point. If all these points correspond to elementary reactions, than the components formed in this elementary reactions have a constant inflow that cannot be counterbalanced. 

Summarising the result of the previous two paragraphs the right-hand side of an induced kinetic differential equation can only be zero, if the reaction is cyclic but not weakly reversible. This class of reactions has again proved to be the most complicated and interesting. 

The induced kinetic differential equations of generalised compartmental systems of the three types are as follows ... 

An easy-to-formulate (but still hard to work through, see Exercise l(ii) of Subsection 4.1.3 and Open Problem 1 of Subsection 4.1.5) generalisation of some of the results collected by Rodiguin Rodiguina (1964) is if the graph of a generalised compartmental system is a tree (i.e. it is acyclic) then the solutions of the induced kinetic differential equation may often happen to be explicitly determined (but not in all cases of tree graph, as one familiar with compartmental systems would expect). 

The core of a system of differential equations of the form (4.23) is the differential equation where the variables having the same index as the zero column vectors of the matrix (a,y) have been deleted. The induced kinetic differential equation of our generalised compartmental system is ... 

The core of this differential equation consists of the first five equations. Clearly, these make up the induced kinetic differential equation of the core of the given generalised compartmental system. This is not by chance, because in general it is true that the induced kinetic differential equation of the core of a generalised compartmental system is the core of the induced kinetic differential equation of the generalised compartmental system. 

There are two problems leading to the notion of continuous components. At first, the number of species or chemical components may reach a huge number in such areas as oil chemistry, polymerisation or biochemistry. In these cases the number of the variables in the induced kinetic differential equation is so large that this system is difficult to treat it may prove more promising to have a continuous manifold of chemical components. 

The meaning of this definition is that no chemical component causes the decrease of another one. Nothing has been supposed about the effect of components on themselves. Our definition is in concord with the definitions used in classical chemical kinetics (cf. Bazsa Beck, 1971). A mechanism is called canonically cross-catalytic, if for all the sets of reaction rate constants the canonic complex chemical reaction corresponding to the induced kinetic differential equation of the complex chemical reaction is cross-catalytic. 

If the induced kinetic differential equation of a complex chemical reaction with a weakly realistic mechanism is a gradient system, then the mechanism itself is canonically cross-catalytic. 

Global methods are aimed at providing information on the change of solutions in a large domain of parameters, i.e. rate constants. As in this case the induced kinetic differential equation should be solved tens of thousands of times in the case of larger systems these methods are only of rather limited use. 

Let us consider the solution of the induced kinetic differential equation at a fixed time point as the function of the rate constants and let us consider the Taylor series of the solution with respect to the parameters around the nominal value of the parameters. The coefficients of this Taylor series are the local sensitivity coefficients. In general only the first order coefficients are considered. 

Hint. The induced kinetic differential equation for x and 3 is a Cauchy-Riemann- (or Erugin-) system, therefore for z = x iy an easily solvable (separable) differential equation can be written down. 

It is not astonishing at all if one considers the deterministic model of the same reaction and verifies that all the solutions of the induced kinetic differential equation are such that they cannot be extended to the whole positive half line, or to use the stochastic terminology, the deterministic model blows up too (cf. Toth, 1986). 

The deterministic model is the (nonautonomous, nonpolynomial) induced kinetic differential equation of the reactions in Fig. 7.13. This model was described in detail by Herodek et al. (1982). Now we give a formal description of a small part of the model. As an example let us consider the time evolution of summer phytoplankton. Our assumption is that it takes part in the elementary reactions No. 2, 6, 13, 26, 34, therefore the equation for is ... 

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