Analysis and Solution of the Autonomous System

An LIP defined by jLIP = (x/x)LIP = max (j) = max (x/x)

xe[xo,6J xe[xo,6J

as the maximum slope of the growth curve on a logarithm of the cell concentration versus time graph (which is approximately in some circumstances, but not identically, equal to the maximum specific growth rate ^max) can be evaluated from eqn [35] by looking for the maximum of the function y. The maximum of y requires that z = y = 0, which upon substitution into eqn [35a] yields

Equation [36] has two solutions consistent with z= y = 0; the first z = y = 0 represents a stationary point, while the second solution represents the locus of the LIP on the phase diagram in the form zLip = jLip=

Note that eqn [37] describes a curve on the phase diagram representing the locus of the LIP, that is, each point on this curve is associated with only one out of a family of growth curves plotted on the phase diagram. It is easy to observe from eqn [37] that the specific growth rate at LIP is approximately equal to the maximum specific growth rate ^max only for values of cell concentration much below the stationary value, such that x ^ 6. Then according to [37] JlipX (x/x)lip « Mmax if x< 6.

Important information can be revealed when searching for the solution at short times. Assuming that the initial cell concentration is much smaller than the stationary value, that is, xo ^ 6 one can always find a time domain tp [0, tm] for which the condition x ^ 6 is satisfied. During this initial time domain (called here 'short time solution') eqn [31] is simplified to d x x dt x x

or in terms of z = y from eqn [35] in the form dz 2

dt dy dt

" Mmaxz z

Equation [39a] can be directly integrated to yield z/(z — y«max) = AoeMm"t, which upon introducing the initial conditions defines the integration constant in the form

From eqn [40] one can observe that as t (actually t becomes very large but still t < tm) z ! ^max (the saturation value of z) and the specific growth rate at the LIP becomes identical with the maximum specific growth rate. Integrating eqn [39b] by using the solution for z from [40] produces the solution for y in the form y = yo + Mmaxt + ln

By substituting now the definition of y = ln x yields the solutions in terms of cell count in the form

From eqns [41] and [42] it may be observed that the cell concentration grows exponentially during this initial growth phase as long as the initial growth rate xo is not zero, nor too small. If, however, x_o is very small, the exponential growth will be delayed and a lag phase will appear, a result that will be better described on a phase diagram in what follows. For now, we need just stating that eqns [41] and [42] represent the initial growth solution during the lag and exponential logarithmic phases (see Figure 2).

The next step is to look for the long-times solution. Vadasz and Vadasz discussed the possibility that the logistic may be a solution to the model. However, an explicit restriction on the initial conditions had to be complied with if this would be the solution for the whole time domain, that is, yo = (x_o/xo) = ^max(1 — xo/5). They also showed that the logistic y = ^max(S - ey)/S is an accurate solution of eqn [32] for any initial conditions as long as it occurs subsequent to a different initial growth so the restriction on the initial conditions is removed. In addition, the logistic equation yields the only stationary point of the original nonautonomous system xs = S. However, since the logistic cannot satisfy any general combination of initial conditions for xo and x_o it can be concluded that this solution applies for long times only, that is, it follows a different solution that also satisfies eqn [32] as well as the initial conditions.

Because of the nonautonomous nature of the original system it is very difficult to draw a phase diagram in terms of y x (x/x) versus x from the original equations. We will adopt instead the approach of using the short-times equation and long-times logistic equation presented in the previous sections to draw a qualitative phase diagram and analyze it. From eqn [39], which is applicable to short times, one obtains by dividing [39a] by [39b] the following result: dz/dy = (^max " z), which can be presented upon separation of variables in the form dz/(umax" z) = dy. Integrating this equation yields ln(yUmax " zo) = "y + Co, leading following substitution of y = ln x to

From the initial conditions at t = 0: x = xo and z = zo, one may evaluate the constant C1 in the form C1 = xo(yUmax — zo), reproducing, upon its substitution into [43], the short-time solution curve to be plotted on the phase diagram:

xo (Mmax

Equation [44] produces a family of hyperbolic curves on the z—x plane that differ by different combinations of initial conditions, (xo, zo). Once the initial conditions (xo, zo) are set, there is a unique hyperbolic curve for any given value of ^max. On the other hand, by using the long-times solution we have for the logistic limit z = Mmax — (Mmax/6)x producing a unique straight line for any given values of ^max and 6. Equation [44] represents on the phase diagram the short-times solution while the straight line logistic represents the long-times solution. They are represented graphically on Figure 15.

From Figure 15 one can distinguish between the short-times hyperbolic curve solution and the straight line representing the logistic. The logistic line intersects the z-axis at z = ^max and the x-axis at the stationary point x= 6. The hyperbolic curves go asymptotically toward z = ^max as x n (in reality, for large values of x) and intersect the x-axis at different values obtained by setting z = 0 in eqn [44] and solving for x leading to the stationary points xs xo l 1

These stationary points depend on the initial conditions. In practice, the procedure follows the following steps (see Figure 15): (1) identify a pair of initial conditions (xo and either xo or zo); (2) for any identified pair of initial conditions there corresponds a unique point on the z—x plane in Figure 15; (3) there is only one hyperbolic curve that passes through this point and therefore only one corresponding stationary point is consistent with any set of initial conditions. The direction of the arrows in Figure 15 indicates the time direction, that is, since above the x-axis (x/x) > 0, it implies that the increase of x in time and the increase in x along the x-axis have the same direction, hence the direction of arrows is as drawn.

o max

max o max

Figure 15 Phase diagram of the solution to the Baranyi and Roberts model drawn from the equations without solving them in the time domain. The hyperbolas represent the short-time solution, while the straight line is the long-time logistic growth. The black markers represent the locus of the LIP, while the white markers represent the unstable stationary points. The arrows represent the positive time direction. Reproduced from Vadasz P and Vadasz AS (2006) Biological implications from an autonomous version of Baranyi & Roberts growth model. International Journal of Food Microbiology 114: 357-365, with permission from Elsevier.

A numerical integration technique is required for solving [46] as this equation is not separable, highlighting again the convenience of using the short- and long-times approximations. Nevertheless, this numerical integration produces the matching solution between the short- and long-times solutions that was drawn in Figure 15 and represented by the dotted curves. The black markers in Figure 15 represent the accurate location (locus) of the LIP following eqn [37]. Note that the phase diagrams were plotted by using the short- and long-term equations, not their solution in the time domain. Only the equations are needed to plot the phase diagrams and the information produced is complementary to any other information that can be extracted from the growth curves once the solution in the time domain is evaluated too.

Figure 15 Phase diagram of the solution to the Baranyi and Roberts model drawn from the equations without solving them in the time domain. The hyperbolas represent the short-time solution, while the straight line is the long-time logistic growth. The black markers represent the locus of the LIP, while the white markers represent the unstable stationary points. The arrows represent the positive time direction. Reproduced from Vadasz P and Vadasz AS (2006) Biological implications from an autonomous version of Baranyi & Roberts growth model. International Journal of Food Microbiology 114: 357-365, with permission from Elsevier.

The latter also establishes the fact that the collection of stationary points given by eqn [45] are unstable, while the stationary point at x = 6 is stable, as established by the direction of the arrows, that is, the solution moves away from the unstable stationary points and towards the stable one. A common property of unstable stationary points is that any solution that starts in their neighborhood spends a considerable amount of time in escaping that neighborhood (the value of the specific growth rate X/x is very small there), actually explaining the lag process. With the help of the graphical description presented in Figure 15, it is now clear what the conditions for obtaining a lag phase should be. First, xo ^ 6 in order for the approximation leading to the hyperbolic curves to be applicable; second, (Xo/xo) ~ 0 (in reality x_o/xo needs to be very small, but not identically zero, in order to be close to one of the unstable stationary points). The intersection between the hyperbolic curves with the logistic line does not have a smooth derivative, that is, dz/dx is not continuous there. This implies that there is a solution matching between the short-times solution and the long-times logistic solution. This matching solution can be found only by using the complete equation for the accurate solution, that is, eqn [35]. Dividing eqn [35a] by [35b] yields dz dy

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