Transition Points on the Phase Diagram Neoclassical Model

The analysis of the monotonic growth following eqn [9] is undertaken first by plotting the solution on a phase diagram in terms of the specific growth rate (per capita growth rate) (x/x) versus the cell concentration x. There are a variety of regimes, linked to the behavior of the solution, that depend on different ranges of the parameters, mmax, rm, and S, and on the initial conditions expressed by xo and x_o (or alternatively zo). The present article focuses on the analysis of the dimensional form of the equation and in particular on the investigation of the effect of the individual parameters and initial conditions on the solution. The latter provides useful practical information that can be used in comparing the model results with experimental data as well as using it for the inverse problem formulation for parameter estimation. For plotting the solution on a phase diagram, one presents eqn [9] in the form x

Mmax

max max which can be expanded by using a common denominator on its right-hand side. One may separate the parameter domain into two distinct regimes, namely rm < 1/(5 and rm > 1 /(. The solution for rm <1 /( and rm > 1 /( produces a variety of curves on the phase diagram. The regime corresponding to rm <1/(5 was presented in a dimension-less form by Vadasz and Vadasz showing that the curves corresponding to zo > ^max are concave, while the curves corresponding to zo < ^max are convex; however, no LIP and no lag are possible in this parameter regime.

The more interesting growth regime corresponding to rm >1/( is presented on the phase diagram in Figure 5 showing the solutions in terms of the specific growth rate (x/x) versus the cell concentration x. A straight line corresponding to the LGM solution, which occurs when zo = Mmax, divides the phase plane into two regions, namely 2o > Mmax and Zo < Mmax. In the region where Zo > Mmax the curves are concave, while in the region corresponding to zo < ^max the curves are convex. The positive x-axis consists of a continuous distribution of stationary points where (x/x) = 0. The stationary points to the right of the point xA represented by the continuous thick line are stable, while the stationary points to the left of xA represented by the dotted line are unstable as observed in Figure 5 by following the direction of the arrows representing the solution change in the positive time direction. Any point on the phase plane represents a possible initial condition. Once such a point is set (i.e., an initial condition for both xo as well as x_o), the solution follows the corresponding curve that passes through that point in the direction of the arrows toward a stationary point. From eqn [8] it is easy to observe that the value of zo is identical to the value of the specific growth rate at xo = 0, that is, zo = (x/x)x = 0. The value of zo can therefore be established by the point where the curves cross the (x/x) axis.

From Figure 5 it is evident that the region corresponding to zo > ^max (on the right side and above the straight line representing the LGM) is qualitatively similar to that obtained previously, when rm < 1/(5. However, the region corresponding to zo < ^max can be further divided into four important regions, as follows:

Region I:---< zo < ^max, convex curves but rm (

no LIP and no lag.

Region II: 0 < zo <---, LIP exists but no lag.

Region III: -Mmax(rm( l) < zo < 0, both LIP and lag

are possible.

Region IV: zo < - ^max ( m —the solution leads to 4rm(

extinction (below the x-axis and therefore not shown in Figure 5).

The following statements regarding the properties of the solution in each one of these regions were derived in detail by Vadasz and Vadasz. In region I, the curves are convex but there is no possibility of a LIP, as the curves have no x/x x/x

Figure 5 Phase diagram for the solution of monotonie growth corresponding to rm 6 > 1, in terms of the specific growth rate x/x versus the cell concentration x, for rm = 5 x 10"7 (cell/ml)"1, 6 = 107 cell ml"1 and ^max = 3 x 10"4 s_1. Reproduced from, Vadasz P and Vadasz AS (2002) The neoclassical theory of population dynamics in spatially homogeneous environments - Part I: Derivation of universal laws and monotonic growth. PhysicaA 309(3-4): 329-359. Vadasz P and Vadasz AS (2002) The neoclassical theory of population dynamics in spatially homogeneous environments - Part II: Non-monotonic dynamics, overshooting and oscillations. PhysicaA 309(3-4): 360-380, with permission from Elsevier.

Figure 5 Phase diagram for the solution of monotonie growth corresponding to rm 6 > 1, in terms of the specific growth rate x/x versus the cell concentration x, for rm = 5 x 10"7 (cell/ml)"1, 6 = 107 cell ml"1 and ^max = 3 x 10"4 s_1. Reproduced from, Vadasz P and Vadasz AS (2002) The neoclassical theory of population dynamics in spatially homogeneous environments - Part I: Derivation of universal laws and monotonic growth. PhysicaA 309(3-4): 329-359. Vadasz P and Vadasz AS (2002) The neoclassical theory of population dynamics in spatially homogeneous environments - Part II: Non-monotonic dynamics, overshooting and oscillations. PhysicaA 309(3-4): 360-380, with permission from Elsevier.

maxima in the non-negative x domain. In region II an LIP exists (the maximum on the phase diagram) but no lag is possible. In region III both an LIP as well as a lag become possible. In region IV (not shown in the figure) the solution leads always to extinction, that is, the trivial stationary point x1s = 0 becomes the only globally stable solution. The reason for the possibility of a lag in region III is the existence of additional positive but unstable stationary points in this region to the left of point xA represented by the dotted line on the x-axis in Figure 5. When the initial conditions are sufficiently close to one of these unstable stationary points the solution spends a relatively long time to escape from its neighborhood. This implies that if 0 < xo ^ ^maxS, that is xo is positive but sufficiently small, and xo < xA, then a lag exists. It will be shown in the next subsection that the value of xA is given by xA = (rmS — 1)/2rm.

The condition for existence of an LIP is that the second time derivative of the logarithm of the cell concentration should vanish at the location of the LIP, implying that [d2(lnx)/d/]LIP = 0 at the LIP. The latter condition is further developed by using the chain rule, leading to [d(x/x)/dx]UP = 0, that is, when (x/x) has a maximum on the (x/x) versus x phase diagram. By applying the latter condition to eqn [10], that is, undertaking the derivative with respect to x of the right-hand side of eqn [10] produces a quadratic algebraic equation for xLIP in the form

Solving eqn [11] for xLIP and accounting only for the non-negative solution (only non-negative values of x are biologically meaningful) reveals the location of the LIP in the form

XLIP

Further derivations reveal the locus of all LIPs on the (x/x) versus x phase diagram that lie on the straight line

Mlip

Mmax 1

-2xlip

2 XLIP

From eqn [13] it can be observed that for rmS ^ 1 and (xUP/S) ^ 1 the maximum specific growth rate ^max and the specific growth rate at the LIP yULIP become equal, that is, ^max = ^LiP x (x/x)LIP. The location of the LIP as evaluated above represents the maxima of the curves on the (x/x) versus x phase diagram and therefore all LIPs in regions II and III lie on the straight line expressed by eqn [13] as indicated on Figure 5.

The first appearance of an LIP occurs when xLIP = 0 (Figure 5) that upon substitution into eqn [13] yields

The last appearance of LIP in the first quadrant of the (x/x) versus x phase diagram (i.e., where x > 0) occurs at xA where (x/x)LIP = 0. By substituting the latter condition into eqn [14] yields xa

The corresponding value of z0,A is evaluated by substituting xA from eqn [15] and (x/x)A = 0 in eqn [8] to produce (see Figure 5)

Z0,A

The special curve representing Pearl's LGM is obtained as a particular case corresponding to zo = ^max. Substituting the latter into eqn [10] yields x/x = ^max [1 — x/S], which describes a straight line on the (x/x) versus x phase diagram as presented in Figure 5. The location of this straight line corresponding to the LGM is invariant to any changes in the value of rm Its location on the phase plane depends only on the values of ^max and S. This can be shown by substituting zo = yUmax into eqn [10] leading to the LGM equation which is independent of rm This is an important property of the LGM indicating that its location on the phase diagram depends on ^max and S only.

While different lag definitions were proposed in the professional literature, for example, by Pirt, and Wangersky and Cunningham, the definition used here is the one that is consistent with Vadasz and Vadasz findings that the lag is essentially related to the existence of unstable stationary points, that is, the continuous collection ofpoints represented by the dotted line on the x-axis to the left of xA in Figure 5. One therefore defines the lag duration A as the amount of time that elapses until the solution reaches a value, which is by a certain percentage above the corresponding unstable stationary point x3s. This implies that xA = bx3s, where b >1 is a constant that specifies by how far is xA from x3 s. The lag duration, A, is therefore defined as the time needed for the solution to reach the value xA for any predetermined value of b. This definition is very similar to the way one defines the time needed for a monotonic solution to reach a steady state. In most of the computations a value of 3.5% above x3s was used as the lag threshold, that is, b = 1.035. The location of xA on the phase diagram can therefore be established for any value of x3s and a corresponding choice for b >1.