Closed Form Analytical Solution Neoclassical Model

Equation [9] has a closed-form solution that can be obtained by direct integration in terms of its max


m max m corresponding stationary points. The first stationary point is the trivial one xi s = 0. The nontrivial stationary points are obtained by solving eqn [10] for steady state, that is, (x/x) = 0, leading to the following algebraic equation for these stationary points, x2 s and x3 s:

2r rm

where x2 s

While the negative stationary points are of no interest to cell growth (there is no meaning to negative cell number) their importance here is in the sense that they affect, mathematically, the transient solution. The closed-form solution to eqn [9] can be presented by defining the following parameter:

2rm6zo ^max(rm6 1)

Then, the solution can be expressed in terms of x2 s, x3 s and a in the form ln x\x-x3

where c1 is an integration constant related to the initial conditions by the following relationship:

Equation [20] represents the closed-form solution to eqn [9] for monotonic growth. It is presented in an implicit form and there is no explicit analytical expression for x as a function of t. Nevertheless, one can use the property of the solution being monotonic and evaluate t as a function of x by using eqn [20] for values of xo < x< x2s, when x2 s is the stable positive stationary point, or for values of 0 < x < xo when xi s — 0 is the stable stationary point. Note that when x3s is non-negative (the points on the x-axis represented as a dotted line to the left of xA in Figure 5) it is always globally unstable as can be observed from Figure 5. The monotonic behavior of the solution guarantees that for each value of x there is one and only one value of t. By using this procedure one can vary the values ofx within the range indicated above and obtain the corresponding values of t, producing therefore the numerical values needed for plotting the resulting solution of x as a function of t.

The occurrence of the LIP in time can be evaluated by substituting eqn [12] for xLIP into the closed-form solution eqn [20] in the form

/lip = — ln zo xlip\ /|Xlip -x3S xo J y \ xo x3 s I

where xLIP is evaluated by using eqn [i2].

Since the lag duration A, was defined as the time needed for the solution to reach the value xA = bx3 s for any predetermined value of b > 1, one can use this definition into the closed-form solution eqn [20] to evaluate the lag duration A, in the form

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