ARTICLE IN PRESS Nonlinear Analysis: Real World Applications (

)

–

Contents lists available at ScienceDirect

Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa

Universal curve, biological time, and dynamically varying scaling exponent in growth law Luciano Medina a , Yisong Yang b,∗ a

Department of Electrical Engineering, Polytechnic Institute of New York University, Brooklyn, NY 11201, United States

b

Department of Mathematics, Polytechnic Institute of New York University, Brooklyn, NY 11201, United States

article

info

Article history: Received 11 September 2008 Accepted 6 November 2008 Keywords: Growth model Universality Dynamical dependence Convergence Transition of stability Period-doubling Chaos

a b s t r a c t A general quantitative model for ontogenetic growth of organisms was derived by West et al. based on fundamental principles for the allocation of metabolic energy in which a universal parameterless growth curve was established in terms of the biomass ratio and a dimensionless biological time. This model was then extended by Guiot et al. to account for the growth of tumors in vivo in which the fractional scaling exponent p becomes a dynamic parameter depending on time t. In this paper, we present a method that may be used effectively to construct a generalized universal growth curve for such a growth law model with an arbitrary dynamically varying fractional exponent. As by-products of this method, we show that the assumption that a biological time variable flows forward in the universal curve allows us to predict the behavior of p with regard to the developmental stages of organisms characterized by body mass which is consistent with the findings of Guiot et al. based on biomedical data, and that, the universal curve is independent of the properties of p as a function of t. We also consider the situation when time increases discretely and the fractional scaling exponent is periodic. We observe the familiar, but superimposed, perioddoubling and transition-to-chaos phenomenon. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Quantitative biology is at the interface of biology and physical sciences and utilizes computational and mathematical methods as major tools to investigate a broad range of bioscience issues arising from the subject areas including biomolecules, cell behavior, genomics, molecular networks, neurons and cognition, population evolution, subcellular processes, protein folding. The study of growth laws is one of the central issues of quantitative biology and may be traced back to the idea of Gompertz [1] that population growth could be modeled by a differential equation which refines the demographic model of Malthus. The Gompertz equation, also known as the Gompertz growth law, has been used in many areas, especially in biology, to model a time series where growth is slowest at the start and end of a period, and led to considerable progress in understanding the growth patterns of various life systems [2]. Besides their power in making qualitative theoretical predictions, growth laws when applied quantitatively to tumors are also valuable in clinical practices such as cancer treatments [3,4]. For example, the well-known Norton–Simon hypothesis [5] states that the rate of tumor volume regression is proportional to the rate of growth, suggesting that tumors given less time to regrow between treatments are more likely to be destroyed [3,5]. Hence mathematical growth models can provide useful insights into tumor biology and lead to more effective tumor treatments. It is in this hope that the study of tumor growth models has remained to be an active area of theoretical biology for many decades [4].

∗

Corresponding author. E-mail address: [email protected] (Y. Yang).

1468-1218/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2008.11.011 Please cite this article in press as: L. Medina, Y. Yang, Universal curve, biological time, and dynamically varying scaling exponent in growth law, Nonlinear Analysis: Real World Applications (2008), doi:10.1016/j.nonrwa.2008.11.011

ARTICLE IN PRESS 2

L. Medina, Y. Yang / Nonlinear Analysis: Real World Applications (

)

–

Recently, West, Brown, and Enquist [6] derived on the basis of fundamental principles for the allocation of metabolic energy a general quantitative model allowing them to predict the parameters governing growth curves from basic cellular properties and to obtain a parameterless universal curve that describes the growth of diverse species. Their work has been used to make quantitative predictions for the growth rates and the timing of life history events of various biological systems and initiated a good amount of research activities and discussions (cf. [7–14] and references therein). Although the growth law equation presented in [6] is not new and a more general equation has already appeared in the earlier phenomenological work of von Bertalanffy [15], West et al. emphasized [6] that what seen is only a structural similarity and the profound difference lies in the fact that their growth law is built over basic principles and that the few parameters governing growth are directly calculable from cellular parameters. In a subsequent study, Guiot, Delsanto, Carpinteri, Pugno, Mansury, and Deisboeck [13] extend the growth model of West et al. [6] so that in the governing equation the fractional scaling exponent p is a dynamically varying parameter depending on time t and show that the model can be used to describe the growth of tumors in vivo. Here we present a method that can be used effectively to construct a generalized universal growth curve for the extended growth model of Guiot et al. in which p is allowed to change arbitrarily. This method leads us to two important conclusions. The first one is the existence of a universal growth curve in the extended model with a dynamically varying scaling exponent and the observation that the curve is truly universal in a broader sense that it is independent of the detailed dynamical properties of p. The second one is that we can use the growth curve to derive a monotonicity condition for p as a function of t with regard to the developmental stages of organisms characterized by body mass which is consistent with the findings of Guiot et al. [13] based on biomedical data, assuming that a ‘‘biological time’’ variable, τ , flows forwardly. We will also study the situation when a discrete time delay is considered. We show that when the fractional scaling exponent is periodic the familiar, but superimposed, period-doubling and transition-to-chaos phenomenon will take place. 2. Growth law Use N to denote the number of cells at time t in the organism and 1N the number of new cells created in the organism during the time interval 1t. The growth law of West et al. [6] is based on the following basic equation which is simply the law of energy conservation, B1t = NBc 1t + Ec 1N ,

(1)

where B is the incoming rate of energy flow which is the average resting metabolic rate of the organism at time t, Bc is the metabolic rate of a single cell, Ec is the metabolic energy needed to create a new cell, Bc , Ec , and the mass of a cell, mc , are assumed to be constants which are independent of the total cell number N and time t throughout growth and development. Thus, using m = mc N to denote the mass of the whole organism and taking the continuous limit 1t → 0, it is seen that (1) becomes a differential equation [6], dm dt

 =

mc



Ec

 B−

Bc



Ec

m.

(2)

Of course, the metabolic rate B, or the total amount of nutrients being delivered to the sites per unit time measured by energy, of an organism depends on its body mass m. In order to find a description of such a dependence relation, Banavar, Maritan, and Rinaldo [16] postulate that the fundamental processes of nutrient transfer at the microscopic level are independent of organism size but dependent on the structure of the transportation network. They describe that, in a D-dimensional organism, since the number of nutrient transfer sites scales as LD where L is the mean distance between neighboring sites, B simply scales as LD as well. On the other hand, the total blood volume C for the most efficient networks for which C is minimized scales as LD+1 . Hence B scales as C D/(D+1) . Because the organism mass m scales as C [17], B scales as mD/(D+1) , which leads to the basic metabolic rate — organism mass relation B = B0 mD/(D+1) ,

(3)

where B0 is constant for a given taxon. As a consequence, the Eq. (2) becomes dm dt

= amD/(D+1) − bm,

(4)

where a = B0 mc /Ec and b = Bc /Ec . The Eq. (4) resembles the classical logistic equation and the quantity M = (a/b)D+1 determines the maximum mass, or the carrying capacity. Using the new variable x = 1 − (m/M )1/(D+1) , (4) reduces to x0 = −bx/(D + 1) which gives us the general solution [6,8] m(t ) = M



 1− 1−

 m 1/(D+1)  0

M

e

−bt /(D+1)

D+1

,

(5)

where m0 = m(0) is the initial mass of the organism. It is seen that the model even allows embryonic growth when the value of m0 is theoretically set to zero. Please cite this article in press as: L. Medina, Y. Yang, Universal curve, biological time, and dynamically varying scaling exponent in growth law, Nonlinear Analysis: Real World Applications (2008), doi:10.1016/j.nonrwa.2008.11.011

ARTICLE IN PRESS L. Medina, Y. Yang / Nonlinear Analysis: Real World Applications (

)

–

3

3. Universal curve As noted earlier, historically, the Eq. (4) also appears in the earlier work of von Bertalanffy [15] which relates the first and second terms of its right-hand side to the rates of synthesis and decomposition of organic matter, respectively, and may be viewed as a law of organic matter conservation [11]. In [15], particular attention is given to the value p = D/(D + 1) = 2/3 for the fractional exponent in the growth law so that the first term on the right-hand side of (4) becomes the rate of food intake with the assumption that it is proportional to body surface area S ∝ m2/3 [11,18,19]. In [20], it is maintained that, in endothermic animals, S ∝ mp with p ∼ 0.6–0.8. Introducing the new dimensionless variables  m 1/(D+1) r = , M

τ=

at

(D + 1)M 1/(D+1)

 h m i1/(D+1)  0 − ln 1 − , M

(6)

one obtains the parameterless universal growth curve, r = 1 − e−τ .

(7)

For 3-dimensional organisms, D = 3, one arrives at the 3/4-exponent growth law deduced in [6,12] and the universal growth curve (7) which beautifully fits the data gathered from a wide range of living systems including mammals, birds, fish, molluscs, and plants. In the study of Banavar, Damuth, Maritan, and Rinaldo [8], it is shown that the universal curve arises from general considerations that are independent of the specific allometric model used and that the data do not distinguish between the exponents p = 3/4 and p = 2/3 for the metabolic rate—mass scaling. In particular, (7) is universally valid for any D > 0 when the dimensionless ‘mass’ ratio r and ‘time’ τ are defined by (6). This issue is also reexamined by Dodds, Rothman, and Weitz [7]. These authors found that p = 3/4 does not yield a significantly better fit of all available data than p = 2/3. 4. Dynamically varying scaling exponent In the newly published work of Guiot et al. [13], it is reported that the universal growth model of West et al. [6] may be used to describe the growth of tumors in vivo provided that the fixed fractional exponent p = D/(D + 1) is allowed to change as a dynamic parameter depending on time t, that is p = p(t ), which reflects changes in the nutrient-supply mechanisms in evolving tumorigenesis [13] (see also [10]). For this purpose, Guiot et al., [13] choose the dimensionless mass parameter m µ= , (8)

M

where M is a certain theoretical mass threshold used here to rescale the original mass m whose importance will be seen soon, and modify the growth law (4) into a dimensionless growth law, dµ

= aµp(t ) − bµ, (9) dt given by a nonautonomous differential equation, where p = p(t ) varies in the range from 2/3 to p = 3/4 or from 2/3 to 1, taking into account of the different developmental stages of the vascular network. It is clear that the positive constants a and b are both of the dimension of time−1 . Based on collected biomedical data for implanted tumors, Guiot et al. [13] found that, after an initial decrease, p starts to grow up, level, then drop down, and that the initial decrease of p is due to the transition from the original (optimal) network to the subsequent pre-angiogenetic structure. They emphasized that the point that p starts increasing after an initial decrease due to the ‘adaptation’ of the implanted tumor to the new environment is the time when the nutrientreplenishment mechanism becomes dominant [13]. Mathematically, this point corresponds to the initial moment t = 0. Of course, without rescaling the mass or using a dimensionless mass parameter, another way to extend (4) to allow a dynamic exponent p is to assume that the coupling parameter a is also a dynamic parameter so that the growth law is modified into dm = a(t )mp(t ) − bm. (10) dt In the following, we concentrate our study on (9), although we shall also consider the more general Eq. (10) briefly. It is foreseeable that the dynamic evolution of the exponent p inevitably introduces new interesting features into the growth model which are not present in the original model with constant p and that these new features provide broader adaptability of the model to more areas of applications, although the lack of analytic solution of the Eqs. (9) and (10) makes it difficult to obtain precise descriptions of the promised new features. 5. Construction of universal growth curve with dynamically varying scaling exponent Here we show that the use of the universal growth curve of the form (7) is not restricted to making data matches. In fact, it can be used to make predictions on the dynamic behavior of the time-dependent scaling exponent p as reported by Please cite this article in press as: L. Medina, Y. Yang, Universal curve, biological time, and dynamically varying scaling exponent in growth law, Nonlinear Analysis: Real World Applications (2008), doi:10.1016/j.nonrwa.2008.11.011

ARTICLE IN PRESS 4

L. Medina, Y. Yang / Nonlinear Analysis: Real World Applications (

)

–

Guiot et al. [10,13] described above. More precisely, we shall show below that, as a consequence of the validity of a generalized universal growth curve of the type (7) driven by a forward-flowing dimensionless time τ , p must increase when the tumor mass m is small but p decreases when m becomes large, compared with a certain threshold mass, M , theoretically designated in (8) as a quantity for rescaling. Such a result may be clueless without looking into the idea of a universal growth curve. This study is also motivated by the work of Gillooly, Charnov, West, Savage, and Brown [9] in which τ is related to the notion of the biological time which depends on the body temperature and body mass, in addition to the development time t, of the organism. With this biological time ticking forward, it may lead to simple and natural consequences for complicated systems, as will be seen here. We consider the situation of a dynamically changing scaling exponent p = p(t ) (0 < p(t ) < 1) over an arbitrary time interval [0, T ] (T > 0). Note that, in the constant scaling exponent case (4) where p = D/(D + 1), the growth phase dm >0 dt dictates the total mass bound

 a D+1

= M. (11) b In the dynamic scaling exponent case (9), a similar (relative) mass bound for µ(t ), that generalizes (11) in view of D = p/(1 − p), reads m(t ) <

µ(t ) <

 a  1−1p(t )

. (12) b This inequality will be observed throughout the interval [0, T ]. Besides, we will also interchangeably use D or p in our growth law equations according to convenience in respective situations. In order to obtain a generalized dimensionless ‘biological’ time τ , we make the partition P: 0 = t0 < t1 < t2 < · · · < tn = T .

(13)

Of course, when the partition is sufficiently fine, the function p(t ) will remain approximately constant in each of the intervals (ti , ti+1 ) (i = 0, 1, 2, . . . , n − 1) and we may well replace p(t ) over each (ti , ti+1 ) by p(ti ). Hence the Eq. (9) is piecewise replaced by dµ = aµpi − bµ, ti < t < ti+1 , dt pi = p(ti ), i = 0, 1, 2, . . . , n − 1;

µ(ti ) = µi ≡ lim µ(t ),

i = 1, 2, . . . , n − 1;

− t →ti

µ0 = µ(0).

(14)

Then, with Mi = (a/b)1/(1−pi ) , we can write down our piecewise solution in the closed form as before,

" µ(t ) = Mi 1 − 1 −



µi

1−pi #

Mi

! 1−1p e − b(1 − pi )(t − ti )

i

,

ti < t < ti+1 ; i = 0, 1, 2, . . . , n − 1. Define the piecewise mass ratio r (t ) =



µ(t )

1−pi

  =

Mi

b a

(15)

µ(t )1−pi ,

ti < t < ti+1 ; i = 0, 1, 2, . . . , n − 1. We formally arrive again at the piecewise defined parameterless universal growth curve r = 1 − e−τ ,

(16)

τi+ < τ < τi−+1 ,

i = 0, 1, 2, . . . , n − 1,

(17)

where the dimensionless (biological) time τ and its partition {τi } are given by



τ = fi (t ) ≡ (1 − pi )b(t − ti ) − ln 1 −

  b a

µ

1−pi i



,

ti < t < ti+1 ,

    b 1 −p τi+ = lim fi (t ) = − ln 1 − µi i , + t → ti

a

τi−+1 = lim fi (t ) − t →ti+1

    b 1 −p = (1 − pi )b(ti+1 − ti ) − ln 1 − µi i , a

i = 0, 1, 2, . . . , n − 1.

(18)

Please cite this article in press as: L. Medina, Y. Yang, Universal curve, biological time, and dynamically varying scaling exponent in growth law, Nonlinear Analysis: Real World Applications (2008), doi:10.1016/j.nonrwa.2008.11.011

ARTICLE IN PRESS L. Medina, Y. Yang / Nonlinear Analysis: Real World Applications (

)

–

5

For any fixed i = 0, 1, 2, . . . , n − 1, the segment given in (17) over (τi+ , τi− +1 ) is a well-defined growth curve. However, in general, these n segments cannot be pieced together to form a globally-defined growth curve because τ may become a multiple valued function of t. In other words, the dimensionless time τ may fail to flow forwardly with respect to the real time t. In fact, in order to make τ a well-defined ‘forward-flowing’ variable, we need to impose the condition

τi− ≤ τi+ ,

i = 1, 2, . . . , n − 1.

(19)

The biological time jump, 1τi , may be expressed as

1τi = τi+ − τi−         b b 1 −p 1−p = − ln 1 − µi i − (1 − pi−1 )b(ti − ti−1 ) + ln 1 − µi−1 i−1 , a

a

(20)

which may be made arbitrarily small provided that p is continuous at ti . Consequently, τ is a continuous time variable change from t wherever p is continuous. In particular, when p is continuous everywhere and the mesh size |P | of the partition (13) goes to zero, where

|P | = max{ti − ti−1 | i = 1, 2, . . . , n},

(21)

we recover the universal growth curve (7) by taking limit in (17) which unambiguously and accurately relates the mass ratio r to the biological time τ . Combining (18) and (19), we see that, if we insist that τ is a forward-flowing variable with respect to the real time t, we must have



(1 − pi−1 )b ≤ −

ln 1 −

b a

1−pi

µi



   1 −p  − ln 1 − ba µi−1 i−1

ti − ti−1

.

(22)

Since the condition (19) concerns the structure of the problem at ti , the left adjacent point ti−1 may be made to be arbitrarily close to ti . Hence, letting ti−1 → ti and replacing ti by t in (22), we deduce by continuity the relation

(1 − p(t ))b ≤ −

d dt

 ln 1 −

  b a

 µ(t )1−p(t ) .

(23)

Using (9) to simplify (23) and applying (12), we obtain p0 (t ) ln µ(t ) ≤ 0,

0 ≤ t ≤ T.

(24)

This simple but elegant inequality immediately allows us to draw the following precise conclusions which are consistent with the findings in [10,13] for the growth of tumors in vivo: During the initial stages of the development of an organism, the small body mass, characterized by µ(t ) < 1 or m(t ) < M , switches on the phase of an ascending (or more precisely, a nondescending) scaling exponent, satisfying p0 (t ) ≥ 0; during later stages, the body mass becomes large, characterized by µ(t ) > 1 or m(t ) > M , and the phase of a descending (or more precisely, a nonascending) scaling exponent prevails, satisfying p0 (t ) ≤ 0. Note that when the body mass of the organism is small or large relative to the threshold mass M for all time, the scaling exponent p(t ) will remain nondescending or nonascending forever. It is also interesting to note that our condition does not rule out, or in fact is compatible with, the constant scaling exponent case, p0 (t ) ≡ 0, as it should be. In general, although the curve segments (17) give rise to a multiple valued curve when p(t ) violates (24), the universal growth curve is unambiguously defined by (7) in the |P | → 0 limit provided that p(t ) is a continuously varying function of t so that its range stays within the interval (0, 1) because (17) may be viewed as a collection of local parametrizations of (7) and the one-to-one correspondence of r and τ defined by (7) implies that any multiple-valuedness at finite |P | disappears when |P | → 0. Consequently, we have obtained a proof of convergence of the piecewise defined curve (17) to the unique universal growth curve (7) as |P | → 0. From the above study, we can conclude with the following theorem. Theorem. For the growth model (9) with a dynamically varying scaling exponent p(t ) which is assumed to be a continuous function of t so that its range is contained in the interval (0, 1) when 0 ≤ t ≤ T , we can use the discrete scheme given by the expressions (13)–(18) to construct a piecewise defined parametrized curve (17) in the (r , τ )-plane, which may be multiple valued in general. In the zero-partition limit |P | → 0, the piecewise defined parametrized curve converges unambiguously to the uniquely defined universal growth curve described explicitly by the Eq. (7). Furthermore, in order to ensure the existence of a universal growth curve of the form (7) which may be obtained from the scheme (13)–(18) in the |P | → 0 limit so that the biological time τ is a forward-flowing variable with respect to the real time t, the time-dependent exponent p(t ) must satisfy the monotonicity condition (24) dictated by the organism mass m(t ) according to either µ(t ) = m(t )/M < 1 or µ(t ) = m(t )/M > 1, respectively, for a certain threshold mass M . Please cite this article in press as: L. Medina, Y. Yang, Universal curve, biological time, and dynamically varying scaling exponent in growth law, Nonlinear Analysis: Real World Applications (2008), doi:10.1016/j.nonrwa.2008.11.011

ARTICLE IN PRESS 6

L. Medina, Y. Yang / Nonlinear Analysis: Real World Applications (

)

–

In the next section, we present several numerical examples of the universal growth curves constructed by the scheme (13)–(18). These examples illustrate that the behavior of these curves is independent of the detailed dynamical properties of the scaling exponent p(t ) and, therefore, is truly universal. We now briefly consider the modified growth law (10). For greater generality, we assume that the factor b is also a dynamic parameter. Hence the governing equation becomes dm

= a(t )mp(t ) − b(t )m. (25) dt For the generalized growth law (25), we may use the same method to construct a piecewise defined universal curve as before by setting µ = m, µi = mi , a = ai = a(ti ), and b = bi = b(ti ) in (14) over each of the intervals ti < t < ti+1 (i = 0, 1, 2, . . . , n − 1). Then, with Mi = (ai /bi )1/(1−pi ) (i = 0, 1, 2, . . . , n − 1) and µ = m in (15), we obtain our piecewise defined approximate solution of (25) as before. The universal curve now may be written as (17) with the dimensionless parameters r and τ defined as in (16) and (18) in which µ, a, b, Mi are all updated suitably as described above. As a consequence of the assumption of a forward-flowing biological time τ as before, the condition (24) can be seen to be revised as 

b(t )

0 ≥

a(t )

b(t ) 0 p (t ) ln m(t ). a( t )

(26)

In the practical special case when b is a constant and the growth law is given by (10), the condition (26) simplifies into a0 (t ) a( t )

+ p0 (t ) ln m(t ) ≤ 0.

(27)

The rest of the discussion/conclusion is parallel to that carried out in details in the foregoing study for (9). In particular, the existence of a universal curve of the form (7) still holds for (10) and (25). 6. Numerical examples Consider (9) with a dynamic scaling exponent p(t ). We partition the time interval [0, T ] according to (13) and approximate (9) by the piecewise defined initial value problems given in (14) so that the solution of (9) is approximated piecewise by the formula (15). Consequently, with the mass ratio r defined in (16) and the biological time τ specified in (18), we arrive at the piecewise defined universal growth curve (17) which approaches a well defined generalized universal growth curve as the mesh size |P | of the partition (13) tends to zero. Here we present several numerical examples with the data specifications a = 0.05,

b = 0.05,

µ(0) = 0.01,

T = 35,

(28)

and an equidistant partition ti − ti−1 = 1t = T /n (i = 1, 2, . . . , n). In all of our computed examples, the results stabilize rapidly. Since the carrying capacity M = 1, the solution of (9) satisfies

µ(t ) < 1,

t > 0.

(29)

First, the time-dependent scaling exponent p(t ) is made to increase from p(0) = 1/2 to p(35) = 3/4 along three different trial curves which are concave up, linear, and concave down, and given by the expressions p(t ) =

1 2

+

t2 4900

√ ,

1 2

+

t 140

,

1 2

+

35t 140

,

(30)

for 0 ≤ t ≤ 35, respectively. Since p(t ) monotone increases, the corresponding universal growth curves are all smoothly depending on a forward-flowing biological time variable τ . In our numerical realization, we see that these curves coincide within a 2% tolerance error when 1t = 0.5. If smaller tolerance error is desired, we can simply reduce 1t to achieve the same conclusion. In other words, in the situation of our examples, the universal growth curve is independent of the detailed structure of the time-dependent scaling exponent p(t ). (See Fig. 1.) Next, we note that our method works equally well when p(t ) is not monotone. We have carried out several simulations for which p(t ) is allowed to oscillate periodically around a fixed fractional value, say p = 3/4, and we see that when 1t is not sufficiently small we would obtain a relatively fuzzy growth curve. However, when 1t is reduced further, the observed fuzziness quickly disappears and a fairly smooth distinctive curve stands out. To save space, these examples are skipped and we only point out that a universal growth curve exists and can be constructed by our method even when p(t ) is oscillatory. We now illustrate the effectiveness of our method for the case where p(t ) is a randomly changing function. To be specific, we choose p(t ) to be uniformly distributed over 0 < t < 35 with range (0.55, 0.95). In Fig. 2, we plot the piecewise defined universal growth curves for 1t = 1, 1t = 0.5, and 1t = 0.125, respectively. We observe the unambiguous fuzziness of the curves for the first two cases and we see clearly that the biological time τ fails to be a forward-flowing variable. In fact, more precisely, it now becomes multiple valued as expected. In the third case (1t = 0.125), the fuzziness is significantly reduced and a well-defined planar curve starts to appear. Indeed, as 1t is made smaller, the fuzziness may be reduced further so that in the 1t → 0 limit a well-defined universal growth curve again stands out. Please cite this article in press as: L. Medina, Y. Yang, Universal curve, biological time, and dynamically varying scaling exponent in growth law, Nonlinear Analysis: Real World Applications (2008), doi:10.1016/j.nonrwa.2008.11.011

ARTICLE IN PRESS L. Medina, Y. Yang / Nonlinear Analysis: Real World Applications (

)

–

7

Fig. 1. Monotone increasing scaling exponents with different convexity properties give rise to virtually identical universal growth curves.

Fig. 2. Randomly changing exponent and convergence to a well-defined universal growth curve in the limit 1t → 0.

7. Discrete time delay and transition to chaos In this section, we will be given a discrete time step 1t in the fundamental law (1) and look into growth kinetics in discrete time points at i1t for i = 0, 1, 2, . . .. Such a situation is motivated by the idea that growth occurs in distinct cell cycles with characteristic doubling times. We start from a finite 1t and rewrite 1N as N (t + 1t ) − N (t ). Therefore, we can express (1) as N (t + 1t ) = N (t ) +

1t Ec

B(t ) −

Bc Ec

1tN (t ).

(31)

As already mentioned, we now assume a uniform time increment 1t, return to the body mass m = mc N, use the metabolic rate expression (3), and denote by mi the organism mass at time i1t. As a consequence, we can rewrite the discrete growth D/(D+1) law (1) or (31) as mi+1 = α mi − β mi (i = 0, 1, 2, . . .), where α = a1t and β = b1t − 1. The interesting situation is Please cite this article in press as: L. Medina, Y. Yang, Universal curve, biological time, and dynamically varying scaling exponent in growth law, Nonlinear Analysis: Real World Applications (2008), doi:10.1016/j.nonrwa.2008.11.011

ARTICLE IN PRESS 8

L. Medina, Y. Yang / Nonlinear Analysis: Real World Applications (

)

–

Fig. 3. The familiar transition to chaos phenomenon corresponding to D = 2 and D = 3, respectively, are plotted in the same coordinate frame.

when β > 0 or Bc Ec

1t > 1,

(32)

which will be observed throughout this paper. With the renormalized mass variable x = m/(α/β)D+1 , we arrive at the iterative growth law D/(D+1)

xi+1 = β xi

1/(D+1)

(1 − xi

),

D/(D+1)

i = 0, 1, 2, . . . .

(33)

1/(D+1)

Note that if we set Fβ (x) = β x (1 − x ), the Eq. (32) becomes xi+1 = Fβ (xi ). We are interested in ‘‘steady’’ states of this growth law among which the simplest ones are the fixed points of Fβ . Of course, x = 0 is always a fixed point. However, no matter how small β > 0 is, such a fixed point is never stable because Fβ (x) > x if x > 0 is sufficiently small. Note that this is yet another feature which is in sharp contrast with the classical logistic model [21,22] for which the fixed point x = 0 is stable as long as β is sufficiently small. Nevertheless, this novel feature does not affect the dynamics of the trajectories away from the local region around x = 0 and the familiar phenomena such as transition from stable steady states to the period-doubling cascade and chaos still take place as in all the Gompertzian-type models [21–23]. For example, for (33), the nontrivial fixed point is given by

 X0 =

β β +1

D+1

,

(34)

which is stable (or unstable) if |Fβ0 (X0 )| < 1 (or |Fβ0 (X0 )| > 1), or equivalently

β < 2D + 1,

stable;

β > 2D + 1,

unstable.

(35)

In other words, below β1 (D) = 2D + 1, the fixed point (34) is the only stable steady state; beyond β1 (D), stable period2 states appear. In general, we use βn (D) to denote the bifurcation point of β at which the 2n−1 -period steady states lose their stability and beyond which the 2n -period stable steady states appear (n = 1, 2, . . .). In Fig. 3, we present our numerical results for the cases D = 2 and D = 3. It is clearly seen that the first bifurcation from fixed-point states to period-2 states appears at β1 (2) = 5 and β1 (3) = 7, respectively, and the transition of stability to higher-period states takes place as the parameter β assumes a series of subsequent higher values β2 (D), β3 (D), . . . , (D = 2, 3), until at a critical value βc (D), beyond which period-3 states appear (therefore states of all integer periods appear), which indicate the occurrence of chaos [24], which are then followed by the onset of infinitely divergent states when the parameter β passes another critical value, βd (D) (say). These results confirm the standard period-doubling and transition-to-chaos picture [21–24]. With an accuracy up to the 4th decimal place, we have β2 (2) = 6.0069, βc (2) = 6.7090, βd (2) = 6.7508, and β2 (3) = 8.4189, βc (3) = 9.3971, βd (3) = 9.4870. It is natural to ask what happens when a dynamically varying dimension number D replaces a constant dimension number (in this situation, the mass variable is understood to be a dimensionless one as in (9)). Here we consider a simple situation where D in (33) varies periodically and we show that we will observe an interesting superimposed period-doubling bifurcation and transition to chaos phenomenon. Please cite this article in press as: L. Medina, Y. Yang, Universal curve, biological time, and dynamically varying scaling exponent in growth law, Nonlinear Analysis: Real World Applications (2008), doi:10.1016/j.nonrwa.2008.11.011

ARTICLE IN PRESS L. Medina, Y. Yang / Nonlinear Analysis: Real World Applications (

)

–

9

Fig. 4. A superimposed period-doubling and transition to chaos phenomenon takes place when D is chosen to oscillate between 2 and 3.

To be specific, we let D oscillate between 2 and 3 periodically with respect to discrete time t = i1t so that D = 2 if i is odd and D = 3 if i is even. In Fig. 4, we present our numerical results. We see that, of course, stable steady states first assume the form of period-2 states. With the same four-decimal place precision, the first bifurcation point that borders the transition from period-2 states to period-4 states is at β1 (2, 3) = 5.9350, followed by higher-order transition points. The chaotic point appears at βc (2, 3) = 6.6625 and the onset of infinitely divergent (escaping) states happens at βd (2, 3) = 6.7504. It is seen without ambiguity that

βa (2, 3) >

1 2

(βa (2) + βa (3)),

a = 1, 2, . . . , c , d.

(36)

In fact, βa (2, 3) is more and more ‘‘tuned’’ towards βa (2) as a assumes higher and higher values, which implies the eventual dominance of low dimension number, D = 2, for long-time behavior of trajectories. 8. Summary In this paper, we present a study of the universal growth curve associated with the dimensionless growth law of Guiot et al. [13] in which the fractional scaling exponent p is a dynamically varying function, which generalizes the growth model of West et al. [6]. We obtain an approximation method that allows us to construct the universal curve effectively which depends on a generalized biological time τ . We can show that when τ is a forward-flowing variable with respect to the real time t, the fractional exponent p must depend on t as a nondecreasing or nonincreasing function according to whether the organism mass m is below or above a certain threshold mass M . Our numerical examples show that the universal curve exists, in the generalized growth law in which p is taken to be a function of t, regardless whether p varies continuously or randomly with respect to t and is independent of the detailed structure of p as a function of t. In other words, the universal curve is universal in the broader sense that it stays invariant with respect to the variation of its dynamic scaling exponent. Acknowledgments The research of the authors were supported in part by Othmer Institute for Interdisciplinary Studies at Polytechnic University. The research of the second author was also supported in part by National Science Foundation under Grant DMS–0406446. It is a pleasure to thank Joel Rogers for several valuable comments on this work. References [1] B. Gompertz, Philos. Trans. R. Soc. 115 (1825) 513. [2] J.D. Murray, Mathematical Biology, Springer, Berlin, 1993; S. Brody, Bioenergetics and Growth, Hafner Press, Darien, 1964; S.C. Stearns, The Evolution of Life Histories, Oxford U. Press, Oxford, 1992; M.J. Reiss, The Allometry of Growth and Reproduction, Cambridge U. Press, Cambridge, 1989. [3] C. Schmidt, J. Natl. Cancer Inst. 96 (2004) 1492. [4] S.E. Clare, F. Nakhlis, J.C. Panetta, Breast Cancer Res. 2 (2000) 430. [5] L. Norton, R. Simon, Nature 264 (1976) 542. [6] G.B. West, J.H. Brown, B.J. Enquist, Nature 413 (2001) 628. Please cite this article in press as: L. Medina, Y. Yang, Universal curve, biological time, and dynamically varying scaling exponent in growth law, Nonlinear Analysis: Real World Applications (2008), doi:10.1016/j.nonrwa.2008.11.011

ARTICLE IN PRESS 10 [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

[18] [19] [20] [21] [22] [23] [24]

L. Medina, Y. Yang / Nonlinear Analysis: Real World Applications (

)

–

P.S. Dodds, D.H. Rothman, J.S. Weitz, J. Theoret. Biol. 209 (2001) 9. J.R. Banavar, J. Damuth, A. Maritan, A. Rinaldo, Nature 420 (2002) 626. J.F. Gillooly, E.L. Charnov, G.B. West, V.M. Savage, J.H. Brown, Nature 417 (2002) 70. C. Guiot, P.G. Degiorgis, P.P. Delsanto, P. Gabriele, T.S. Deisboeck, J. Theoret. Biol. 225 (2003) 147. A.M. Makarieva, V.G. Gorshkov, B.-L. Li, Ecol. Model. 176 (2004) 15. G.B. West, J.H. Brown, Phys. Today 57 (2004) 36. C. Guiot, P.P. Delsanto, A. Carpinteri, N. Pugno, Y. Mansury, T.S. Deisboeck, J. Theoret. Biol. 240 (2006) 459. P. Castorina, P.P. Delsanto, C. Guiot, Phys. Rev. Lett. 96 (2006) 188701. L. von Bertalanffy, Q. Rev. Biol. 32 (1957) 217. J.R. Banavar, A. Maritan, A. Rinaldo, Nature 399 (1999) 130. T.A. McMahon, J.T. Bonner, On Size and Life, Scientific American Library, New York, 1983; J.T. Bonner, The Evolution of Complexity by Means of Natural Selection, Princeton U. Press, Princeton, 1983; R.H. Peters, The Ecological Implications of Body Size, Cambridge U. Press, Cambridge, 1983; N.K. Schmidt, Why is Animal Size so Important? Cambridge U. Press, Cambridge, 1984; W.A. Calder III, Size, Function and Life History, Harvard U. Press, Cambridge, Massachusetts, 1984; J.H. Brown, Macroecology, U. Chicago Press, Chicago, 1995. S.A.L.M. Kooijman, Dynamic Energy and Mass Budgets in Biological Systems, Cambridge U. Press, Cambridge, 2000. R.M. Nisbet, E.B. Muller, K. Lika, S.A.L.M. Kooijman, J. Animal Ecol. 69 (2000) 913. A.C. Economos, J. Theoret. Biol. 94 (1982) 25. R.M. May, Simple mathematical models with very complicated dynamics, Nature 261 (1976) 459. R.V. Jensen, Classical chaos, Amer. Scientist 75 (1987) 168. P. Cvitanovic, Universality in Chaos, 2nd ed., Bristol and New York, 1989. T.-Y. Li, J.A. Yorke, Amer. Math. Monthly 82 (1975) 985.

Please cite this article in press as: L. Medina, Y. Yang, Universal curve, biological time, and dynamically varying scaling exponent in growth law, Nonlinear Analysis: Real World Applications (2008), doi:10.1016/j.nonrwa.2008.11.011