- Email: [email protected]
An open-ended logistic-based growth function: Analytical solutions and the power-law logistic model
e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 531–534
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/ecolmodel
Short communication
An open-ended logistic-based growth function: Analytical solutions and the power-law logistic model John H.M. Thornley a,∗ , John J. Shepherd b , J. France c a b c
Centre for Ecology & Hydrology, Bush Estate, Penicuik, Midlothian EH26 0QB, UK School of Mathematical and Geospatial Sciences, RMIT University, P.O. Box 2476V, Melbourne, Australia Centre for Nutrition Modelling, Department of Animal & Poultry Science, University of Guelph, Guelph, Ont. N1G 2W1, Canada
a r t i c l e
i n f o
a b s t r a c t
Article history:
An open-ended form of the logistic equation was recently proposed, using a model
Received 6 June 2006
comprising two differential equations [Thornley, J.H.M., France, J., 2005. An open-ended
Received in revised form
logistic-based growth function. Ecol. Model. 184, 257–261]. The equations represent the two
11 December 2006
processes of growth and development, and are coupled. In this note, an analytical solu-
Accepted 22 December 2006
tion is developed for constant parameters. The solution can be expressed as a targetted
Published on line 15 February 2007
single-differential-equation model, the -logistic or power-law logistic model, which is a well-known empirical growth equation in ecology and elsewhere. The analysis may facili-
Keywords:
tate mechanistic interpretation and application of the power-law logistic model as well as
Logistic
the original open two-differential-equation model.
-Logistic
© 2007 Elsevier B.V. All rights reserved.
Power-law-logistic Generalized logistic Model Growth Development Variable asymptote Open-ended Analytical solutions
1.
Introduction
Thornley and France (2005) proposed an open-ended form of the logistic growth equation (Eq. (1)), with a two-differentialequation model representing the coupled processes of growth and development (Eqs. (4a) and (4b)). The model is “open” in the sense that dynamic changes in nutrition and environment can influence growth and development, and may affect the position of the asymptote (M∞ , Eq. (8)). Here an analytical solu-
∗
tion is developed (Eq. (12)), applicable when the parameters representing growth conditions are constant. The solution is identical to a well-known modification of the logistic model, known variously as the -logistic model (Gilpin and Ayala, 1973), or the power-law-logistic model (Banks, 1994, p. 105), or a generalized logistic model (von Bertalanffy, 1957; Richards, 1959; Turner et al., 1976). The analysis therefore provides a mechanistic biological interpretation of the parameters of this model (via Eqs. (7), (8) and (11)).
Corresponding author at: 6 Makins Road, Henley-on-Thames, Oxfordshire, RG9 1PP, UK. Tel.: +44 1491 577346. E-mail address: [email protected] (J.H.M. Thornley). 0304-3800/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2006.12.026
532
e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 531–534
Gilpin and Ayala (1973) use the -logistic model in an ecological context, as a variation in the standard LotkaVolterra equations for competition, and apply the results to fruit fly competition data. Some of their parameter values are discussed below in the context of the mechanistic twodifferential-equation model. More generally, power-law logistic models belong to the class of models proposed by von Bertalanffy (1957, 1968) and Richards (1959, 1969), and they have been used for many different purposes. For values of the principal parameter, , assuming ≥ 0, the lower limit (0) gives a Gompertz equation, = 1 gives a logistic equation, and higher values of give logistic-like growth with an increasingly sharper cessation of growth as the asymptote is approached. As a fraction of the asymptote, inflexion can occur over the range from 1/e (Gompertz), through 1/2 (ordinary logistic), to 1 (high values of ). Banks (1994, pp. 105–109) discusses the properties of the model thoroughly. There has been much continuing interest in the logistic equation, as witnessed by the continuing publication of papers which address perceived problems with its interpretation, or modify it for particular purposes (e.g. Hastings et al., 1996; Invernizzi and Terpin, 1997; van Horssen et al., 2002; Fujikawa et al., 2003; Matis et al., 2003; Gabriel et al., 2005). It may be noted that the idea of setting up a logistic equation with a variable carrying capacity determined by a second differential equation was introduced by Cohen (1995) for human populations and by Rodrigues et al. (2005) for ecological systems. Rodrigues et al. (2005) make use of Cohen’s Eq. (6) for carrying capacity. However, Cohen (1995), in his Eq. (6), makes two scientifically unsatisfactory assumptions: first that a rate of change (of carrying capacity) depends directly on another rate of change (of population) and second that an extensive quantity (carrying capacity) depends only on an intensive quantity (proportional rate of population growth). If mechanistic explanations are sought, it is generally preferable to assume that rates of change depend only on state variables, and that extensive quantities depend on at least one other extensive quantity as well as intensive quantities. The original objective of this note was to report on the method for finding analytical solutions to the two-differentialequation model of Thornley and France (2005). Subsequent to a reviewer’s valued comments, a principal objective has been to supply a mechanistic derivation of the power-law logistic model.
2.
Logistic equation
The logistic growth equation in differential form is
dM M = rM 1 − dt Mf
;
t = 0, M = M0
(1)
M (kg) is the mass and t (days) is the time. The three parameters are: specific growth rate parameter, r (day−1 ); initial mass, M0 (kg); and final asymptotic mass, Mf (kg). In Eq. (1), final mass parameter Mf is present—the equation contains the information specifying where it is going, even at time t = 0. This can cause problems in the application of the equation, especially
when attempting to use Eq. (1) to model situations where the final asymptote depends on conditions during growth, which is usually the case in plants and animals. Integration of Eq. (1) gives M=
Mf M0 Mf = M0 + (Mf − M0 )e−rt 1 + (Mf /M0 − 1)e−rt
(2)
There is a single inflexion point (i) at mass and time given by Mi =
3.
1 M, 2 f
ti =
1 ln r
M −M 0 f Mf
(3)
Open-ended logistic equations
Thornley and France’s (2005) proposal for opening up the logistic replaced Eq. (1) with two differential equations (there is a minor notational change in the equations as now presented):
dM M = flim G rM 1 − dt Mf
dMf = −flim D D(Mf − M); dt
;
t = 0, M = M0
(4a)
t = 0, Mf = Mf0
(4b)
Final mass Mf (kg) of Eq. (1) becomes a state variable whose initial value (Mf0 ) assumes that development occurs at its standard maximal rate of D, without limitation. D (day−1 ) represents unspecified processes such as development, differentiation, and progress towards maturity. flim G and flim D are dimensionless fractions (0 ≤ flim ≤ 1), which reflect the degree of limitation of growth (G) or development (D). If flim G = flim D = 1, there is no limitation and both processes occur at their standard rates, r and D. If flim G = flim D = 0, there is total limitation of both growth and development—neither process occurs at all; this trivial case is excluded from the following analysis. The dMf /dt equation causes target final mass Mf to move towards current mass M at a rate proportional to D times the degree to which development is limited (flim D ). If flim D = 0 (development is turned off entirely), Mf does not change from its initial value, and, with flim G = 1 also (growth occurs at its full rate), the unmodified logistic (Eq. (1)) is recovered. Although growth and development (or differentiation) cannot rigorously be separated, such separation provides a valuable conceptual approximation in much of biology. A limiting environment can often affect growth and development differently, e.g. in plants and in many poikilotherms, growth may depend on provision of substrates, whereas development is largely a function of temperature. A single cell may grow larger, giving growth, but this process may be impeded by changes in cell wall and other facets of cell architecture. In many organisms, nutrient deprivation can lead to a smaller but nontheless mature adult. Development or differentiation or maturation are processes which, to a first approximation, do not add mass to the cell, organ, or organism, but whose progression makes the adding of mass (growth) more difficult. Thus, development causes achievable or target mass increasingly to approach current mass. Some of these ideas are explored in various contexts by Thornley (1981, pp. 57–63),
533
e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 531–534
Substituting for M∞ in terms of ˇ from Eq. (8) into Eq. (12):
Lainson and Thornley (1982), Thornley and Johnson (2000, pp. 63–72), and Thornley and France (2007, pp. 150–152).
4.
dM − M = −ˇM+1 dt
Analysis
Divide Eq. (4a) by Eq. (4b) to eliminate dt, and integrate (assuming constant parameters). Therefore ln
M M0
=−
f
lim G
r
flim D
D
ln
M f
Mf0
D f 1 + lim D flim G r
ln(M∞ ) =
f
lim D
flim G
D r
ln(M0 ) + ln(Mf0 )
(6)
Asymptote M∞ depends on D/r as well as degree of limitation, and initial value Mf0 (asymptote when there is no limitation). Define =1+
flim D D flim G r
(7)
Note that ≥ 1 for parameters which are greater or equal to zero. Eq. (6) becomes = Mf0 M0−1 = M∞
1 ˇ
(8)
where ˇ is defined for later use. If flim D = 0 (no development occurs), = 1, M∞ = Mf0 . Final asymptote is unchanged for all growth limitations less than complete (0 < flim G ≤ 1). If flim D = 1 and flim G = 0 (normal development and no growth), = ∞, M∞ = M0 ; organism mass remains at its initial value. From Eq. (5), therefore (with Eq. (7)) Mf = Mf0
M −1 0
(9)
M
Substituting for Mf in Eq. (4a) leads to dM M2 rf − r flim G M = − lim G Mf0 dt
M −1 M0
Eq. (13) is of the form of the Bernoulli equation, namely (Jeffrey, 2002, p. 259): dM + P(t)M = Q(t)Mn dt
(n = 1)
(14)
(5)
As t → ∞, the asymptote is obtained by substituting M = Mf = M∞ (say), and solving for M∞ :
(13)
(10)
Integration using standard techniques (Banks, 1994, pp. 106–108; Jeffrey, 2002, p. 259) leads to a modified logistic function: M =
Mf0 M0
M0 + (Mf0 − M0 )e−t
(15)
This can be compared with Eq. (2). If there is normal growth and no development, flim G = 1 and flim D = 0, then = 1 and = r (Eqs. (7) and (11)), the two equations are identical. It can be helpful to write Eq. (15) in terms of the final asymptote (with (Eq. (8)): M =
M∞
1 + ((M∞ /M0 ) − 1)e−t
(16)
This is a simple modification of the ordinary logistic (Eq. (2)). Differentiating Eq. (12) with respect to time, t, and equating to zero gives the inflexion point as a fraction of final mass as Mi = M∞
1 1/
(17)
1+
This fraction is 1/2 for = 1 (the logistic). It approaches unity for increasing (Eq. (7)), so that exponential growth is sustained for longer and inflexion point Mi moves close to the asymptote at M∞ . Note that because asymptote M∞ is lowered towards M0 , Eq. (8), inflexion point Mi decreases absolutely although it is now closer to the asymptote. When = 0, the fraction equals 1/e, and the -logistic can be transformed into a Gompertz growth equation. Time of inflexion can be obtained by (from Eq. (15)) ti =
1 ln
(M∞ /M0 ) − 1
(M∞ /Mi ) − 1
(18)
With Eq. (8) and substituting
5. = r flim G
Discussion
(11)
Eq. (10) becomes
dM = M 1 − dt
M M∞
(12)
The open logistic now takes a targetted form, with asymptote M∞ depending on initial values and other parameters (Eqs. (7) and (8)). It is familiar to ecologists as the -logistic, and to many other biologists concerned with growth. It is also known as the power-law logistic, for obvious reasons. With = 1, Eq. (12) is identical to the standard logistic of Eq. (1).
The ordinary logistic equation (Eq. (2)) is the limiting case of the -logistic equation (Eq. (12)) with zero developmental rate (D = 0 in Eqs. (4b) and (7)). In this case, no process applies which moves the asymptote from its initial value. As a curve-fitting tool, an advantage of the -logistic equation (Eq. (12)) over the ordinary logistic is that for > 1 (Eq. (7)) the inflexion mass can be at higher values than one half the asymptotic mass (Eqs. (3) and (17)), and a sharp knee can be obtained. Moreover, when this occurs, Eqs. (4a), (4b), (7), and (17) permit a biological interpretation of why this might arise. A developmental process which decreases potential growth and lowers asymptotic mass makes greater than unity. This
534
e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 531–534
causes inflexion mass to be at higher values than one half of asymptotic mass, which itself is at lower values. The faster development occurs relative to growth, the greater the value of . This is similar to the mechanistic two-differential-equation form of the Gompertz equation (Thornley and Johnson, 2000, pp. 80–82) where development causes the specific growth rate to decrease rather than asymptotic mass, although in this case, mass at inflexion remains at 1/e of its asymptotic value. Ideally, a mathematical model is framed in such a way that all parameters have zero or positive values (decay is not negative growth; neither is degradation negative synthesis; these are different biological processes). Thus < 1 (Eq. (7)) would not be allowed by the mechanistic formulation of Eqs. (4a) and (4b). Gilpin and Ayala (1973) report values of 0.35 and 0.12 for competing species of Drosophila, which, viewed in terms of Eq. (7), corresponds to a negative value of D (or r). In their context of populations without age structure, the concepts of maturity, development or differentiation seem unlikely to be applicable, leading to the conclusion that their application of the -logistic equation is empirical, or at least, cannot be interpreted mechanistically using Eqs. (4a) and (4b). However, for other biological contexts, the question can be asked: is it meaningful to reverse the arrow of developmental time and allow progress away from maturity to occur (D < 0)? At the cell level, it seems feasible that the cross-linking processes which decrease cell wall elasticity and cell growth might be reversible. At the organ level in plants, it appears reasonable that the developmental processes leading to reproduction and the cessation of vegetative growth could also be reversed. In animals also, it does not seem impossible that some limited reversal of development might be possible. Assume D is negative so that = 0.5. Now, relative to = 1, the asymptote is multiplied by 100 (Eq. (8), with Mf0 = 100 kg and M0 = 1 kg) and inflexion occurs at 0.44 (4/9) of the asymptote (Eq. (17)). If = 0.25, inflexion occurs at 0.4096 (44 /54 ) of the final asymptote. In the limit → 0, the fraction of the final asymptote at which inflexion occurs approaches 1/e, as in the Gompertz growth equation (Thornley and Johnson, 2000, pp. 80–82). The -logistic equation can be written as a Gompertz for this limiting case. Note also that the power-law logistic, known to statisticians as a generalized logistic, is identical to the Richards function when written in the form suggested by Causton et al. (1978) and applied by them to leaf growth (see also Thornley and France, 2007, p. 155). Although Eq. (12) for the -logistic model is in the rate:state (autonomous) format, and it is an equation which doubtless has occurred to many students of the logistic as possessing desirable behaviour (a high inflexion mass relative to the asymptote), it does not seem possible to give Eq. (12) a plausible biological interpretation directly. However, the representation of Eq. (12) via Eqs. (4a) and (4b) provides a view of biological mechanisms which could give rise to such an equation.
Acknowledgements This work is a contribution to work funded by the UK Department of Environment, Food and Rural Affairs (contract EPG
1/1/60). The authors have been much helped by the reviewers’ comments.
references
Banks, R.B., 1994. Growth and Diffusion Phenomena: Mathematical Frameworks and Applications. Spring Verlag, Berlin. Causton, D.R., Elias, C.O., Hadley, P., 1978. Biometrical studies of plant growth. 1. The Richards function and its application in analysing the effects of temperature on leaf growth. Plant Cell Environ. 1, 163–184. Cohen, J.E., 1995. Population growth and Earth’s human carrying capacity. Science 269, 341–346. Fujikawa, H., Kai, A., Morozumi, S., 2003. A new logistic model for bacterial growth. J. Fd. Hyg. Soc. Jpn. 44, 155–160. Gabriel, J.-P., Saucy, F., Bersier, L.-F., 2005. Paradoxes in the logistic equation. Ecol. Model. 185, 147–151. Gilpin, M.E., Ayala, F.J., 1973. Global models of growth and competition. Proc. Natl. Acad. Sci. U.S.A. 70, 3590–3593. Hastings, H.M., Rumignani, D.J., Vassalotti, A.G., Weiss, M.T., Wu, Y.R., 1996. Lack of predictability in model ecosystems based on coupled logistic equations. Ecol. Model. 92, 209–214. Invernizzi, S., Terpin, K., 1997. A generalized logistic model for photosynthetic growth. Ecol. Model. 94, 231–242. Jeffrey, A., 2002. Advanced Engineering Mathematics. Harcourt Academic Press, San Diego. Lainson, R.A., Thornley, J.H.M., 1982. A model for leaf expansion in cucumber. Ann. Bot. 50, 407–425. Matis, J.H., Kiffe, T.T., Renshaw, E., Hassan, J., 2003. A simple saddlepoint approximation for the equilibrium distribution of the stochastic logistic model of population growth. Ecol. Model. 161, 239–248. Richards, F.J., 1959. A flexible growth function for empirical use. J. Exp. Bot. 10, 290–300. Richards, F.J., 1969. The quantitative analysis of growth. In: Steward, F.C. (Ed.), Plant Physiology VA. Academic Press, New York, pp. 3–76. ˜ P., Belbute, J., 2005. Rodrigues, J., Domingos, T., Conceic¸ao, Constraints on dematerialisation and allocation along a sustainable growth path. Ecol. Econ. 54, 382–396. Thornley, J.H.M., 1981. Organogenesis. In: Rose, D.A., Charles-Edwards, D.A. (Eds.), Mathematics and Plant Physiology. Academic Press, London, pp. 49–65. Thornley, J.H.M., Johnson, I.R., 2000. Plant and Crop Modelling. Blackburn Press, Caldwell, NJ (www.blackburnpress.com). Thornley, J.H.M., France, J., 2005. An open-ended logistic-based growth function. Ecol. Model. 184, 257–261. Thornley, J.H.M., France, J., 2007. Mathematical Modelling in Agriculture. CAB International, Wallingford, Oxfordshire, UK. Turner, M.E., Bradley, E.L., Kirk, K.A., Pruitt, K.M., 1976. A theory of growth. Math. Biosci. 29, 367–373. van Horssen, P.W., Pebesma, P.J., Schot, P.P., 2002. Uncertainties in spatially aggregated predictions from a logistic regression model. Ecol. Model. 154, 93–101. von Bertalanffy, L., 1957. Quantitative laws for metabolism and growth. Q. Rev. Biol. 32, 217–231. von Bertalanffy, L., 1968. General System Theory. George Braziller, New York.
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/ecolmodel
Short communication
An open-ended logistic-based growth function: Analytical solutions and the power-law logistic model John H.M. Thornley a,∗ , John J. Shepherd b , J. France c a b c
Centre for Ecology & Hydrology, Bush Estate, Penicuik, Midlothian EH26 0QB, UK School of Mathematical and Geospatial Sciences, RMIT University, P.O. Box 2476V, Melbourne, Australia Centre for Nutrition Modelling, Department of Animal & Poultry Science, University of Guelph, Guelph, Ont. N1G 2W1, Canada
a r t i c l e
i n f o
a b s t r a c t
Article history:
An open-ended form of the logistic equation was recently proposed, using a model
Received 6 June 2006
comprising two differential equations [Thornley, J.H.M., France, J., 2005. An open-ended
Received in revised form
logistic-based growth function. Ecol. Model. 184, 257–261]. The equations represent the two
11 December 2006
processes of growth and development, and are coupled. In this note, an analytical solu-
Accepted 22 December 2006
tion is developed for constant parameters. The solution can be expressed as a targetted
Published on line 15 February 2007
single-differential-equation model, the -logistic or power-law logistic model, which is a well-known empirical growth equation in ecology and elsewhere. The analysis may facili-
Keywords:
tate mechanistic interpretation and application of the power-law logistic model as well as
Logistic
the original open two-differential-equation model.
-Logistic
© 2007 Elsevier B.V. All rights reserved.
Power-law-logistic Generalized logistic Model Growth Development Variable asymptote Open-ended Analytical solutions
1.
Introduction
Thornley and France (2005) proposed an open-ended form of the logistic growth equation (Eq. (1)), with a two-differentialequation model representing the coupled processes of growth and development (Eqs. (4a) and (4b)). The model is “open” in the sense that dynamic changes in nutrition and environment can influence growth and development, and may affect the position of the asymptote (M∞ , Eq. (8)). Here an analytical solu-
∗
tion is developed (Eq. (12)), applicable when the parameters representing growth conditions are constant. The solution is identical to a well-known modification of the logistic model, known variously as the -logistic model (Gilpin and Ayala, 1973), or the power-law-logistic model (Banks, 1994, p. 105), or a generalized logistic model (von Bertalanffy, 1957; Richards, 1959; Turner et al., 1976). The analysis therefore provides a mechanistic biological interpretation of the parameters of this model (via Eqs. (7), (8) and (11)).
Corresponding author at: 6 Makins Road, Henley-on-Thames, Oxfordshire, RG9 1PP, UK. Tel.: +44 1491 577346. E-mail address: [email protected] (J.H.M. Thornley). 0304-3800/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2006.12.026
532
e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 531–534
Gilpin and Ayala (1973) use the -logistic model in an ecological context, as a variation in the standard LotkaVolterra equations for competition, and apply the results to fruit fly competition data. Some of their parameter values are discussed below in the context of the mechanistic twodifferential-equation model. More generally, power-law logistic models belong to the class of models proposed by von Bertalanffy (1957, 1968) and Richards (1959, 1969), and they have been used for many different purposes. For values of the principal parameter, , assuming ≥ 0, the lower limit (0) gives a Gompertz equation, = 1 gives a logistic equation, and higher values of give logistic-like growth with an increasingly sharper cessation of growth as the asymptote is approached. As a fraction of the asymptote, inflexion can occur over the range from 1/e (Gompertz), through 1/2 (ordinary logistic), to 1 (high values of ). Banks (1994, pp. 105–109) discusses the properties of the model thoroughly. There has been much continuing interest in the logistic equation, as witnessed by the continuing publication of papers which address perceived problems with its interpretation, or modify it for particular purposes (e.g. Hastings et al., 1996; Invernizzi and Terpin, 1997; van Horssen et al., 2002; Fujikawa et al., 2003; Matis et al., 2003; Gabriel et al., 2005). It may be noted that the idea of setting up a logistic equation with a variable carrying capacity determined by a second differential equation was introduced by Cohen (1995) for human populations and by Rodrigues et al. (2005) for ecological systems. Rodrigues et al. (2005) make use of Cohen’s Eq. (6) for carrying capacity. However, Cohen (1995), in his Eq. (6), makes two scientifically unsatisfactory assumptions: first that a rate of change (of carrying capacity) depends directly on another rate of change (of population) and second that an extensive quantity (carrying capacity) depends only on an intensive quantity (proportional rate of population growth). If mechanistic explanations are sought, it is generally preferable to assume that rates of change depend only on state variables, and that extensive quantities depend on at least one other extensive quantity as well as intensive quantities. The original objective of this note was to report on the method for finding analytical solutions to the two-differentialequation model of Thornley and France (2005). Subsequent to a reviewer’s valued comments, a principal objective has been to supply a mechanistic derivation of the power-law logistic model.
2.
Logistic equation
The logistic growth equation in differential form is
dM M = rM 1 − dt Mf
;
t = 0, M = M0
(1)
M (kg) is the mass and t (days) is the time. The three parameters are: specific growth rate parameter, r (day−1 ); initial mass, M0 (kg); and final asymptotic mass, Mf (kg). In Eq. (1), final mass parameter Mf is present—the equation contains the information specifying where it is going, even at time t = 0. This can cause problems in the application of the equation, especially
when attempting to use Eq. (1) to model situations where the final asymptote depends on conditions during growth, which is usually the case in plants and animals. Integration of Eq. (1) gives M=
Mf M0 Mf = M0 + (Mf − M0 )e−rt 1 + (Mf /M0 − 1)e−rt
(2)
There is a single inflexion point (i) at mass and time given by Mi =
3.
1 M, 2 f
ti =
1 ln r
M −M 0 f Mf
(3)
Open-ended logistic equations
Thornley and France’s (2005) proposal for opening up the logistic replaced Eq. (1) with two differential equations (there is a minor notational change in the equations as now presented):
dM M = flim G rM 1 − dt Mf
dMf = −flim D D(Mf − M); dt
;
t = 0, M = M0
(4a)
t = 0, Mf = Mf0
(4b)
Final mass Mf (kg) of Eq. (1) becomes a state variable whose initial value (Mf0 ) assumes that development occurs at its standard maximal rate of D, without limitation. D (day−1 ) represents unspecified processes such as development, differentiation, and progress towards maturity. flim G and flim D are dimensionless fractions (0 ≤ flim ≤ 1), which reflect the degree of limitation of growth (G) or development (D). If flim G = flim D = 1, there is no limitation and both processes occur at their standard rates, r and D. If flim G = flim D = 0, there is total limitation of both growth and development—neither process occurs at all; this trivial case is excluded from the following analysis. The dMf /dt equation causes target final mass Mf to move towards current mass M at a rate proportional to D times the degree to which development is limited (flim D ). If flim D = 0 (development is turned off entirely), Mf does not change from its initial value, and, with flim G = 1 also (growth occurs at its full rate), the unmodified logistic (Eq. (1)) is recovered. Although growth and development (or differentiation) cannot rigorously be separated, such separation provides a valuable conceptual approximation in much of biology. A limiting environment can often affect growth and development differently, e.g. in plants and in many poikilotherms, growth may depend on provision of substrates, whereas development is largely a function of temperature. A single cell may grow larger, giving growth, but this process may be impeded by changes in cell wall and other facets of cell architecture. In many organisms, nutrient deprivation can lead to a smaller but nontheless mature adult. Development or differentiation or maturation are processes which, to a first approximation, do not add mass to the cell, organ, or organism, but whose progression makes the adding of mass (growth) more difficult. Thus, development causes achievable or target mass increasingly to approach current mass. Some of these ideas are explored in various contexts by Thornley (1981, pp. 57–63),
533
e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 531–534
Substituting for M∞ in terms of ˇ from Eq. (8) into Eq. (12):
Lainson and Thornley (1982), Thornley and Johnson (2000, pp. 63–72), and Thornley and France (2007, pp. 150–152).
4.
dM − M = −ˇM+1 dt
Analysis
Divide Eq. (4a) by Eq. (4b) to eliminate dt, and integrate (assuming constant parameters). Therefore ln
M M0
=−
f
lim G
r
flim D
D
ln
M f
Mf0
D f 1 + lim D flim G r
ln(M∞ ) =
f
lim D
flim G
D r
ln(M0 ) + ln(Mf0 )
(6)
Asymptote M∞ depends on D/r as well as degree of limitation, and initial value Mf0 (asymptote when there is no limitation). Define =1+
flim D D flim G r
(7)
Note that ≥ 1 for parameters which are greater or equal to zero. Eq. (6) becomes = Mf0 M0−1 = M∞
1 ˇ
(8)
where ˇ is defined for later use. If flim D = 0 (no development occurs), = 1, M∞ = Mf0 . Final asymptote is unchanged for all growth limitations less than complete (0 < flim G ≤ 1). If flim D = 1 and flim G = 0 (normal development and no growth), = ∞, M∞ = M0 ; organism mass remains at its initial value. From Eq. (5), therefore (with Eq. (7)) Mf = Mf0
M −1 0
(9)
M
Substituting for Mf in Eq. (4a) leads to dM M2 rf − r flim G M = − lim G Mf0 dt
M −1 M0
Eq. (13) is of the form of the Bernoulli equation, namely (Jeffrey, 2002, p. 259): dM + P(t)M = Q(t)Mn dt
(n = 1)
(14)
(5)
As t → ∞, the asymptote is obtained by substituting M = Mf = M∞ (say), and solving for M∞ :
(13)
(10)
Integration using standard techniques (Banks, 1994, pp. 106–108; Jeffrey, 2002, p. 259) leads to a modified logistic function: M =
Mf0 M0
M0 + (Mf0 − M0 )e−t
(15)
This can be compared with Eq. (2). If there is normal growth and no development, flim G = 1 and flim D = 0, then = 1 and = r (Eqs. (7) and (11)), the two equations are identical. It can be helpful to write Eq. (15) in terms of the final asymptote (with (Eq. (8)): M =
M∞
1 + ((M∞ /M0 ) − 1)e−t
(16)
This is a simple modification of the ordinary logistic (Eq. (2)). Differentiating Eq. (12) with respect to time, t, and equating to zero gives the inflexion point as a fraction of final mass as Mi = M∞
1 1/
(17)
1+
This fraction is 1/2 for = 1 (the logistic). It approaches unity for increasing (Eq. (7)), so that exponential growth is sustained for longer and inflexion point Mi moves close to the asymptote at M∞ . Note that because asymptote M∞ is lowered towards M0 , Eq. (8), inflexion point Mi decreases absolutely although it is now closer to the asymptote. When = 0, the fraction equals 1/e, and the -logistic can be transformed into a Gompertz growth equation. Time of inflexion can be obtained by (from Eq. (15)) ti =
1 ln
(M∞ /M0 ) − 1
(M∞ /Mi ) − 1
(18)
With Eq. (8) and substituting
5. = r flim G
Discussion
(11)
Eq. (10) becomes
dM = M 1 − dt
M M∞
(12)
The open logistic now takes a targetted form, with asymptote M∞ depending on initial values and other parameters (Eqs. (7) and (8)). It is familiar to ecologists as the -logistic, and to many other biologists concerned with growth. It is also known as the power-law logistic, for obvious reasons. With = 1, Eq. (12) is identical to the standard logistic of Eq. (1).
The ordinary logistic equation (Eq. (2)) is the limiting case of the -logistic equation (Eq. (12)) with zero developmental rate (D = 0 in Eqs. (4b) and (7)). In this case, no process applies which moves the asymptote from its initial value. As a curve-fitting tool, an advantage of the -logistic equation (Eq. (12)) over the ordinary logistic is that for > 1 (Eq. (7)) the inflexion mass can be at higher values than one half the asymptotic mass (Eqs. (3) and (17)), and a sharp knee can be obtained. Moreover, when this occurs, Eqs. (4a), (4b), (7), and (17) permit a biological interpretation of why this might arise. A developmental process which decreases potential growth and lowers asymptotic mass makes greater than unity. This
534
e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 531–534
causes inflexion mass to be at higher values than one half of asymptotic mass, which itself is at lower values. The faster development occurs relative to growth, the greater the value of . This is similar to the mechanistic two-differential-equation form of the Gompertz equation (Thornley and Johnson, 2000, pp. 80–82) where development causes the specific growth rate to decrease rather than asymptotic mass, although in this case, mass at inflexion remains at 1/e of its asymptotic value. Ideally, a mathematical model is framed in such a way that all parameters have zero or positive values (decay is not negative growth; neither is degradation negative synthesis; these are different biological processes). Thus < 1 (Eq. (7)) would not be allowed by the mechanistic formulation of Eqs. (4a) and (4b). Gilpin and Ayala (1973) report values of 0.35 and 0.12 for competing species of Drosophila, which, viewed in terms of Eq. (7), corresponds to a negative value of D (or r). In their context of populations without age structure, the concepts of maturity, development or differentiation seem unlikely to be applicable, leading to the conclusion that their application of the -logistic equation is empirical, or at least, cannot be interpreted mechanistically using Eqs. (4a) and (4b). However, for other biological contexts, the question can be asked: is it meaningful to reverse the arrow of developmental time and allow progress away from maturity to occur (D < 0)? At the cell level, it seems feasible that the cross-linking processes which decrease cell wall elasticity and cell growth might be reversible. At the organ level in plants, it appears reasonable that the developmental processes leading to reproduction and the cessation of vegetative growth could also be reversed. In animals also, it does not seem impossible that some limited reversal of development might be possible. Assume D is negative so that = 0.5. Now, relative to = 1, the asymptote is multiplied by 100 (Eq. (8), with Mf0 = 100 kg and M0 = 1 kg) and inflexion occurs at 0.44 (4/9) of the asymptote (Eq. (17)). If = 0.25, inflexion occurs at 0.4096 (44 /54 ) of the final asymptote. In the limit → 0, the fraction of the final asymptote at which inflexion occurs approaches 1/e, as in the Gompertz growth equation (Thornley and Johnson, 2000, pp. 80–82). The -logistic equation can be written as a Gompertz for this limiting case. Note also that the power-law logistic, known to statisticians as a generalized logistic, is identical to the Richards function when written in the form suggested by Causton et al. (1978) and applied by them to leaf growth (see also Thornley and France, 2007, p. 155). Although Eq. (12) for the -logistic model is in the rate:state (autonomous) format, and it is an equation which doubtless has occurred to many students of the logistic as possessing desirable behaviour (a high inflexion mass relative to the asymptote), it does not seem possible to give Eq. (12) a plausible biological interpretation directly. However, the representation of Eq. (12) via Eqs. (4a) and (4b) provides a view of biological mechanisms which could give rise to such an equation.
Acknowledgements This work is a contribution to work funded by the UK Department of Environment, Food and Rural Affairs (contract EPG
1/1/60). The authors have been much helped by the reviewers’ comments.
references
Banks, R.B., 1994. Growth and Diffusion Phenomena: Mathematical Frameworks and Applications. Spring Verlag, Berlin. Causton, D.R., Elias, C.O., Hadley, P., 1978. Biometrical studies of plant growth. 1. The Richards function and its application in analysing the effects of temperature on leaf growth. Plant Cell Environ. 1, 163–184. Cohen, J.E., 1995. Population growth and Earth’s human carrying capacity. Science 269, 341–346. Fujikawa, H., Kai, A., Morozumi, S., 2003. A new logistic model for bacterial growth. J. Fd. Hyg. Soc. Jpn. 44, 155–160. Gabriel, J.-P., Saucy, F., Bersier, L.-F., 2005. Paradoxes in the logistic equation. Ecol. Model. 185, 147–151. Gilpin, M.E., Ayala, F.J., 1973. Global models of growth and competition. Proc. Natl. Acad. Sci. U.S.A. 70, 3590–3593. Hastings, H.M., Rumignani, D.J., Vassalotti, A.G., Weiss, M.T., Wu, Y.R., 1996. Lack of predictability in model ecosystems based on coupled logistic equations. Ecol. Model. 92, 209–214. Invernizzi, S., Terpin, K., 1997. A generalized logistic model for photosynthetic growth. Ecol. Model. 94, 231–242. Jeffrey, A., 2002. Advanced Engineering Mathematics. Harcourt Academic Press, San Diego. Lainson, R.A., Thornley, J.H.M., 1982. A model for leaf expansion in cucumber. Ann. Bot. 50, 407–425. Matis, J.H., Kiffe, T.T., Renshaw, E., Hassan, J., 2003. A simple saddlepoint approximation for the equilibrium distribution of the stochastic logistic model of population growth. Ecol. Model. 161, 239–248. Richards, F.J., 1959. A flexible growth function for empirical use. J. Exp. Bot. 10, 290–300. Richards, F.J., 1969. The quantitative analysis of growth. In: Steward, F.C. (Ed.), Plant Physiology VA. Academic Press, New York, pp. 3–76. ˜ P., Belbute, J., 2005. Rodrigues, J., Domingos, T., Conceic¸ao, Constraints on dematerialisation and allocation along a sustainable growth path. Ecol. Econ. 54, 382–396. Thornley, J.H.M., 1981. Organogenesis. In: Rose, D.A., Charles-Edwards, D.A. (Eds.), Mathematics and Plant Physiology. Academic Press, London, pp. 49–65. Thornley, J.H.M., Johnson, I.R., 2000. Plant and Crop Modelling. Blackburn Press, Caldwell, NJ (www.blackburnpress.com). Thornley, J.H.M., France, J., 2005. An open-ended logistic-based growth function. Ecol. Model. 184, 257–261. Thornley, J.H.M., France, J., 2007. Mathematical Modelling in Agriculture. CAB International, Wallingford, Oxfordshire, UK. Turner, M.E., Bradley, E.L., Kirk, K.A., Pruitt, K.M., 1976. A theory of growth. Math. Biosci. 29, 367–373. van Horssen, P.W., Pebesma, P.J., Schot, P.P., 2002. Uncertainties in spatially aggregated predictions from a logistic regression model. Ecol. Model. 154, 93–101. von Bertalanffy, L., 1957. Quantitative laws for metabolism and growth. Q. Rev. Biol. 32, 217–231. von Bertalanffy, L., 1968. General System Theory. George Braziller, New York.