Introduction

Metabolism and growth are two fundamental aspects of living organisms, and thus, it is somewhat natural to try to understand the connections between these two processes. von Bertalanffy (1957) studied ontogenetic growth curves in order to establish connections between metabolism and growth, showing that individual growth curves in living organisms can be reproduced from models of metabolic energy allocation. Although several subsequent growth models (Reiss 1989; West et al. 2001; Ricklefs 2003; Hou et al. 2008) differ significantly on the details of their derivations, they share the same mathematical form of von Bertalanffy (1957), which basically links patterns of assimilation and growth to the (allometric) scaling of the metabolic rate. In particular, West et al. (2001) proposed a general quantitative model based on the allocation of metabolic energy and showed that individual growth curves can be collapsed onto a single universal curve that describes the growth in all the studied species.

Nevertheless, Banavar et al. (2002) later illustrated that the universal growth curve arises from general considerations of energy conservation that are independent of the specific allometric model used by West et al. (2001). In particular, Banavar et al. (2002) showed that the data do not distinguish between specific exponents in the scaling relationship between metabolic rate and mass, with exponents of 2/3 and 3/4 in the metabolic relation fitting equally well. This scaling collapse onto a universal curve, which occurs when some dimensionless quantities are properly defined, is also equivalent to asserting that a single self-similar solution is able to successfully fit all the ontogenetic growth curves. Since the universal curve arises from general considerations, it is desirable to find an independent test of the assumptions behind the models that defines the key dimensionless quantities that can discriminate between the different models for ontogenetic growth.

In general, all ontogenetic growth models critically depend on the metabolic rate, specifically, on the allometric scaling, the exact slope of which is still matter of debate (White Craig et al. 2007). Moreover, allometric descriptions for the metabolic rate like Kleiber (1932) do not fulfill the mathematical property of dimensional homogeneity, that requires to be expressed in a general form that remain true when the size of units is changed, which is a minimal requirement for any meaningful law of nature (Bridgman 1922). Recently, the empirical metabolic rate relation was corrected in order to fulfill dimensional homogeneity (Escala 2019), and a new metabolic rate (B) formula was proposed: \(\mathrm B = \epsilon (T) \, \eta _{O_2} f_H \, M\), where M is the body mass, \(\mathrm f_H\) is a (characteristic) heart frequency, \(\mathrm \eta _{O_2}\) is a specific \(\mathrm {O_2}\) absorption factor for different exercising conditions (basal, maximal, etc.), and \(\mathrm \epsilon (T)= \epsilon _0 \, {e^{\small -E_a/k T}}\) is a temperature correction inspired by the Arrhenius formula (Gillooly et al. 2001), in which \(\mathrm E_a\) is the activation energy and k is the Boltzmann constant.

Compared to Kleiber’s original formulation (Kleiber 1932), \(\mathrm B = B_0 (M/M_0)^{3/4}\), and Rubner’s surface rule of proportionality to 2/3, this new metabolic rate relation has the heart frequency \(\mathrm f_H\) as the controlling variable (a marker of metabolic rate); its advantage is that it is a unique metabolic rate equation for different classes of animals and different exercising conditions that is valid for both basal and maximal metabolic rates, in agreement with empirical data in the literature (Escala 2019). Moreover, using this new metabolic rate relation, departures from 3/4 scaling in the metabolic rate can be explained by variability in the allometric scaling of \(\mathrm f_H\), even for big deviations from 3/4 scaling such as some invertebrates, where heart frequencies scales as \(\mathrm M^{-0.41}\) (in spiders; Carrel and Heathcote 1976) and the metabolic rate scaling is thus predicted by the factor \(\mathrm f_H\) M to be \(\mathrm M^{-0.59}\), being the same slope measured in the metabolic rate of spiders (Anderson 1970, 1974). This also holds for exercise conditions, where the maximal metabolic rate is proportional to \(\mathrm M^{0.85}\) (Savage et al. 2004), which slope can be again predicted by \(\mathrm f_H\) M since heart frequencies \(\mathrm f_H\) are observed to scales as \(\mathrm M^{-0.15}\) (under maximal exercise conditions; Weibel et al. 2004; Weibel and Hoppeler 2005). In addition, Escala (2022) showed that this new metabolic rate relation can be directly linked to the total energy consumed in a lifespan, so it is able to explain the origin of variations in the “rate of living’ theory (Speakman 2005; Ramsey et al. 2000).

In this paper, we explore the implications of this new metabolic relation for ontogenetic growth models, with a focus on independently testing key quantities in our formulation to enable discrimination between previous models. The paper is organized as follows: We start in §2 by applying the results of the new metabolic rate relation (Escala 2019) to an ontogenetic growth equation that shares the same mathematical form as previous models, showing that individual growth curves can also be collapsed onto a single universal curve, but in this case without the aid of any fitting parameter. Section 3 continues by computing the predicted generation time and explaining the origin of several “Life History Invariants,” with satisfactory results. In §4, we study the predicted implications for population growth, showing that they agree with the collected data. Finally, in §5, we discuss the results and final implications of this work.

Ontogenetic growth model

Since different assumptions about energy allocation lead to the same general equation (von Bertalanffy 1957; Reiss 1989; West et al. 2001; Ricklefs 2003; Hou et al. 2008), we will follow the notation in the mass-energy conservation model described in Moses et al. (2008), which revisited the ontogenetic growth model of West et al. (2001), because we will compare our results to theirs, using their parameter estimations. The conservation of energy for the allocation of metabolic energy during growth between the maintenance of existing tissue and the production of new biomass can be expressed as \(\mathrm E_m \,\frac{dm}{dt} = B-B_m \, m\) (Moses et al. 2008), where B is the metabolic rate (in J/s or W), \(\mathrm E_m\) (in J/g) is the energy required to create a unit of biomass, and \(\mathrm B_m\) (in W/g) is the metabolic rate required to maintain an existing unit of biomass.

For the corrected metabolic rate relation, we will restrict ourselves to basal (resting) conditions and neglect temperature variations because ontogenetic growth happens over long periods of time during which such variations might tend to cancel each other out (in wild conditions). Under such conditions, we have the constant factor \(\mathrm \epsilon (T) \, \eta _{O_2} \approx 10^{-4.313} \,mlO_2g^{-1} \approx 10^{-3} \, J/g \equiv E_{2019}\) (converting 1 ltr \(\mathrm O_2\)=20.1 kJ; Schmidt-Nielsen 1984), where \(\mathrm E_{2019}\) is a constant that comes from the best fitted value for the corrected metabolic relation (Eq. 8 of Escala 2019). Therefore, the metabolic rate formula is simply given by \(\mathrm B = E_{2019} \, f_H \, m\) (Escala 2019), and assuming that the heart frequency scales with body mass m as \(\mathrm f_H= f_0 m^{-\alpha }\), (mass-)energy conservation can be expressed as

$$\begin{aligned} \frac{dm}{dt}= a m^{1-\alpha } - bm \, , \end{aligned}$$
(1)

where \(\mathrm a= E_{2019} \, f_0 /E_m\) and \(\mathrm b = B_m/E_m\). The general solution of Eq. 1 is a classical sigmoidal curve, the general form of which was given by von Bertalanffy (1957; Eq. 6). Noting also that for an initial (birth) mass \(m_0\) and final (asymptotic) mass M, the condition \(\mathrm dm/dt= 0\) (at m = M) in Eq. 1 is equivalent to a/b = \(\mathrm M^\alpha\), the solution to Eq. 1 can be written as

$$\begin{aligned} \Big (\frac{m}{M}\Big )^{\alpha } = 1 -\Big [1-\Big (\frac{m_0}{M}\Big )^{\alpha }\Big ] \times \exp \Big ({\small -\frac{at}{{\alpha }^{-1}M^{\alpha }}}\Big ) \, . \end{aligned}$$
(2)

This solution is equivalent to the one found by Banavar et al. (2002), and for the special case of \(\mathrm \alpha = 1/4\), it has the solution given in West et al. (2001) (their Eq. 5). The solution given by Eq. 2 can be rewritten as \(\mathrm r = 1-e^{-\tau }\), which is the same universal growth curve found in West et al. (2001) and Banavar et al. (2002), but in our case, it uses the following variable change:

$$\begin{aligned} \mathrm r \equiv \Big (\frac{m}{M}\Big )^{\alpha } \, \,\, ; \,\,\, \tau \equiv \frac{E_{2019} }{\,E_m} \, \frac{f_H\, t}{\alpha ^{-1}} - ln \Big [1-\Big (\frac{m_0}{M}\Big )^{\alpha }\Big ] \,\, , \end{aligned}$$
(3)

where we also replace \(\mathrm a= E_{2019} \, f_0 /E_m\) in Eq. 2 (in the definition of \(\mathrm \tau\)), and thus, \(\mathrm f_H = f_0 M^{-\alpha }\) is (henceforth) the heart frequency when the animal reaches the final (asymptotic) mass M. For the particular case of \(\mathrm \alpha =1/3\), r also corresponds to the fractional size (\(\mathrm r=(m/M)^{1/3}={ l}/L\)), leading to the classical Bertalanffy growth equation, \(\mathrm { l(\tau )} = L \,(1-e^{-\tau }\)), which has been successfully applied in the fishery industry (Beverton and Holt 1959; Charnov 2008). In this case, we can explicitly express (in terms of physical quantities) the Bertalanffy growth coefficient as \(\mathrm K \equiv \tau /t = \frac{E_{2019} }{\,E_m} \, \frac{f_H}{\alpha ^{-1}}\) (from Eq. 3 for \(\mathrm m_0=0\)).

The advantage of this formulation compared to previous ones is that now the dimensionless time \(\mathrm \tau\) is expressed exclusively in terms of quantities that are well defined, with either clear physical (specific energies \(\mathrm E_{2019} \, and \, E_m\)) or biological (heart frequency \(\mathrm f_H\), body mass M, etc.) meanings, and therefore, the dimensionless variables defined in Eq. 3 are now written in a physically transparent form, without the aid of any fitting parameters. This contrasts, for example, with the parameter a (present in Eqs. 1 and 2 but not in the universal growth curve defined by Eq. 3) that is critical in the procedure of fitting the growth curves in von Bertalanffy (1957), West et al. (2001), Banavar et al. (2002), etc. and in the definition of their dimensionless variables, which has an obscure meaning considering its fractal dimensionality of [mass\(^{\alpha }]\)/[time] and is thus not associable to any physical quantity. This fitting parameter a is not present in the final growth curve solution found in our formulation, \(\mathrm r = 1-e^{-\tau }\), for the variables defined in Eq. 3. It is notable that in this formulation for \(\tau\), the inverse of the heart frequency \(\mathrm f_H^{-1}\) defines the characteristic timescale for ontogenetic growth, \(\mathrm t_{growth} \equiv f_H^{-1} \frac{E_m}{ E_{2019}}\), which is rescaled by the ratio of the specific energies of biomass creation (\(\mathrm E_m)\) and (basal) metabolism (\(\mathrm E_{2019})\).

Banavar et al. (2002) showed that the ontogenetic growth curves do not distinguish between exponents of 2/3 and 3/4 (= \(\mathrm 1-\alpha\)) in the mass scaling of the metabolic rate relation, as both are equally good fits to the current data and are consistent with a universal curve \(\mathrm r = 1-e^{-\tau }\) for ontogenetic growth. Therefore, only for consistency, we will assume \(\mathrm \alpha =1/4\), since we will compare our formulation against data using the fitting parameters from West et al. (2001), which assumes this \(\mathrm \alpha\) value.

The universal growth curve \(\mathrm r = 1-e^{-\tau }\), for the \(\mathrm \tau\) defined in Eq. 3 (with \(\mathrm \alpha =1/4\)), is mathematically identical to the solution found in West et al. (2001) (their Eq. 5) when

$$\begin{aligned} \mathrm t_{growth}^{-1} \equiv \frac{E_{2019}}{E_m} \, f_H = \frac{a}{M^{1/4}} \,\, \end{aligned}$$
(4)

is fulfilled, and therefore, it should equally well fit the individual growth curves for the species studied in West et al. (2001) if their measured heart frequencies \(\mathrm f_H\) and estimated specific energies (\(\mathrm E_{2019},\, E_m\)) fulfill the condition given by Eq. 4 for the parameter a and final mass M found in the fitting procedure of the growth curves in West et al. (2001). In order to test whether the condition in Eq. 4 is fulfilled in nature, we collected estimations for the energy required to create a unit of biomass \(\mathrm E_m\) (Moses et al. 2008, taking averages when multiple estimations are available) and measurements of heart frequencies \(\mathrm f_H\) for the species studied in West et al. (2001) in order to supplement them with the fitting parameters a, birth mass \(\mathrm m_0\) and final mass M that best fit the growth curves studied in West et al. (2001). A summary of all the quantities (with the individual references for \(\mathrm f_H\)) are listed in Table 1.

Table 1 Table 1
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Fig. 1
figure 1

The relation between the variables a, \(\mathrm M, \, E_m \, and \, f_H\), as a function of the final body mass M, showing that it does not vary systematically with M, as predicted by Eq. 4. The different species marked with black stars are scattered around the predicted value of \(\mathrm E_{2019}\), which is denoted by the red dashed line. The best-fitted slope is \(\frac{a}{M^{1/4}} \, \frac{E_m}{ f_H} \propto M^{-0.05}\), which is consistent with a zero slope against body mass M. The parameters a and M, taken from West et al. (2001), are determined to best fit the individual growth curves of the different species. In contrast, the parameters \(\mathrm E_{2019}, \, E_m \, and \, f_H\) that define \(\mathrm t_{growth}\) in our formulation are determined independently (see the references in Table 1) from any procedure of fitting to individual growth curves, such as the one in West et al. (2001)

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Figure 1 displays the relation given by Eq. 4 between the variables a, \(\mathrm M, \, E_m \, and \, f_H\) as a function of the final body mass M. In the 7 orders of magnitude in body mass M variations, the relation has a slope consistent with zero (\(\propto M^{-0.05}\), and there is only 0.22 dex in scatter, where 1 dex on a log scale refers to an order of magnitude, comparable to the scatter in the metabolic rate relation found in Escala 2019), confirming that the condition in Eq. 4 is fulfilled. This implies that for the specific dimensionless variables defined in Eq. 3, the universal curve (\(\mathrm r = 1-e^{-\tau }\)) will describe the (different) individual growth curves as well as the models in West et al. (2001) and Banavar et al. (2002). This is found without performing any fitting procedure on individual growth curves; instead, it is derived from the mathematical equivalence between our universal growth curve (\(\mathrm r = 1-e^{-\tau }\), for the \(\tau\) defined in Eq. 3) and West et al. (2001)’s solution when Eq. 4 is fulfilled. In other words, for the same universal solution, our formalism replaces the fitting parameters in West et al. (2001), such as a, which involves obscure units \([g^{1/4}/day]\) and is thus not easily associable to a known physical quantity, by well-defined energies (\(\mathrm E_{2019} \, and \, E_m\)) and frequency (\(\mathrm f_H\)), which are also determined independently rather than by fitting animal growth curves (as in West et al. 2001; Banavar et al. 2002, etc.).

Moreover, the different species shown in Fig. 1 are scattered around the predicted value of \(\mathrm E_{2019} \, (= 10^{-3} \, Jg^{-1}\); Escala 2019) denoted by the red dashed line, and therefore, Fig. 1 can be considered a third independent estimation of \(\mathrm E_{2019}\) (in addition to those of Escala 2019, 2022), supporting the predictability of the formalism presented and showing a characteristic feature of the physical sciences: fewer (and more universal) constants that have consistent measurements in multiple contexts. Finally, Fig. 1 strongly supports the idea that the inverse of the heart frequency \(\mathrm f_H^{-1}\), rescaled by the ratio of the competing specific energies (biomass creation \(\mathrm E_m\) versus metabolism \(\mathrm E_{2019})\), defines the characteristic timescale for ontogenetic growth \(\mathrm t_{growth} = f_H^{-1} \frac{E_m}{ E_{2019}}\).

Generation time and the origin of some life history invariants

The success of the formulation given by Eq. 3, supported by Fig. 1, motivates us to study its implications, and for that reason, it is interesting to look at the predicted generation time \(\mathrm t_{gen}\), which is the time period required for a young organism to grow to its final size and thus mature to reproductive age. The generation time \(\mathrm t_{gen}\) can be straightforwardly determined from the \(\tau\) defined in Eq. 3, making it possible to arrive at the following relation:

$$\begin{aligned} \mathrm t_{gen} = \Big (\tau ^* + ln \Big [1-\Big (\frac{m_0}{M}\Big )^{\alpha }\Big ]\Big ) \frac{E_m}{ \,E_{2019} } \, \frac{\alpha ^{-1}}{ f_H} \propto f_H ^{-1} \, , \end{aligned}$$
(5)

where \(\tau ^*\) is the approximate dimensionless time value for reaching adult mass (same for all species) in the universal growth curve. Equation 5 gives a generation time that is explicitly dependent on the heart frequency \(\mathrm f_H\), where the allometric scaling of the heart frequency under basal conditions (\(\mathrm f_H \propto M^{-0.25}\); Brody 1945; Calder 1968) gives the well known mass scaling for the generation time (\(\mathrm t_{gen} \propto M^{0.25}\); Bonner 1965). Nevertheless, since most biological rates and times scale as \(\mathrm M^{-1/4}\) and \(\mathrm M^{1/4}\) (Savage et al. 2004; Burger et al. 2021), an interesting possible test is to study allometric variations of \(\mathrm t_{gen}\) in large outliers of 1/4 scaling such as spiders (Anderson 1970, 1974), (White Craig et al. 2007) to test whether they vary as \(\mathrm f_H\), as predicted by Eq. 5. Another interesting possibility is to directly test the predicted inverse correlation between \(\mathrm t_{gen}\) and \(\mathrm f_H\).

One of the advantages of having a generation time that is explicitly dependent on the heart frequency \(\mathrm f_H\) is that it can be directly linked (Escala 2019, 2022) to the total lifespan \(\mathrm t_{life}\) using the empirical relation of the total number of heartbeats (\(\mathrm N_b\)) in a lifetime \(\mathrm t_{life} = N_b/f_H\) found to be valid in mammals (Levine 1997; Cook et. al 2006). Escala (2022) generalized this relation for all types of living organisms using the proportionality between heart and respiration frequencies, \(\mathrm f_H = k \, f_{resp}\) (Schmidt-Nielsen 1984); then, the empirical relation with lifetime can be rewritten so that it is also valid for a total number \(\mathrm N_r \,(=N_b/k)\) of “respiration cycles”: \(\mathrm t_{life} = N_b/f_H = N_b/kf_{resp} = N_r/f_{resp}\). Escala (2022) also used this relation to predict the total lifespan energy consumed in living organisms, with satisfactory results if an approximately constant total number \(\mathrm N_r \sim 10^8\) of respiration cycles per lifetime is required in all living organisms, which further supports the generalization. Combining this relation for a fixed number of respiration cycles per lifetime with Eq. 5, we obtain:

$$\begin{aligned} \, G \equiv \frac{t_{gen}}{ t_{life}} = \Big (\tau ^* + ln \Big [1-\Big (\frac{m_0}{M}\Big )^{\alpha }\Big ]\Big ) \frac{ E_m \alpha ^{-1}}{E_{2019} N_b } \approx \frac{E_m}{k} \, \frac{4\,\tau ^*}{ E_{2019} N_r} \, , \end{aligned}$$
(6)

where we also approximate \(\mathrm m_0<< M\) (in agreement with the species listed in Table 1), neglecting minor changes due to the weakly varying logarithm. Equation 6 directly relates the generation time and lifespan, two quantities that are known to correlate (de Magalhaes et al. 2007) with their respective energies (per unit mass) to create biomass (\(\mathrm E_{m}\)) and sustain lifespan (\(\mathrm N_b E_{2019}\)), by only assuming (mass-)energy conservation (Eq. 1) and the invariant \(\mathrm N_r \, (= 1.62\, 10^8\) respiration cycles per lifetime; Escala 2022) as a critical link between the two timescales.

Equation 6 can be directly compared to the data compiled by Charnov and Berrigan (1990), which summarized published data for the ratio \(\mathrm t_{adult}/t_{gen}\) in different animal groups, where \(\mathrm t_{adult} = t_{life} - t_{gen}\) is the adult lifespan, noting that this definition is related to the G ratio as \(\mathrm t_{gen}/t_{life} = (1 + t_{adult}/t_{gen})^{-1}\). Additionally, we note that in Charnov and Berrigan (1990), lifespans are estimated from the inverse of (instantaneous) mortality rates, being effectively field lifespans and not maximum ones; thus, for comparison with Eq. 6, a correcting factor of 2.5 must be included (McCoy and Gillooly 2008), namely, \(\mathrm G^{min} = t_{gen}/t_{life}^{max} = (2.5(1 + t_{adult}/t_{gen}))^{-1}\).

Table 2 summarizes the estimations of \(\mathrm G^{min}\) using the data compiled in Charnov and Berrigan (1990) and compares them with the predicted values from Eq. 6, using estimates of \(\mathrm E_m\) for taxonomic groups (average of juvenile estimates; Moses et al. 2008), k values for such groups (Schmidt-Nielsen 1984; Escala 2022) and a dimensionless time \(\tau ^* \sim 5\), the approximate value to reach adult mass in the universal growth curve (i.e., the flat part of the curve in Fig. 2 of West et al. 2001). Overall, we find that the values predicted from Eq. 6 are consistent with the empirically determined ones, as can be seen in Table 2. This is in addition to the consideration that each taxonomic group of birds, mammals, and fishes includes species with up to an order of magnitude difference in \(\mathrm E_m\) values (i.e., cod vs guppy fish in Table 1) and not necessarily the same species studied in Charnov and Berrigan (1990). Therefore, it is even more relevant that the relative \(\mathrm G^{min}\) trends (between birds, mammals, and fish) observed in Charnov and Berrigan (1990) are successfully predicted by the ratio \(\frac{E_m}{k}\).

Table 2 Table 2
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Equation 6 shows that the invariant relation between lifespan and age at maturity that is valid within taxonomic groups, as observed by Charnov and Berrigan (1990), might come directly from the existence of another invariant: the approximately constant total number \(\mathrm N_r \sim 10^8\) of respiration cycles per lifetime. This ratio \(\mathrm t_{adult}/t_{gen}\) is indeed one of the studied dimensionless life-history invariants (Calder 1984; Charnov 1993), which has sometimes been criticized as spurious by being a form of “regressing X on X,” where X is a random number (Nee et al. 2005). In this work, the generation and lifespan timescales are quantities derived independently, giving a quantitative prediction of an approximately invariant ratio within taxonomic groups, including interspecific variations, that comes from the basic energetics of respiration and the creation of new biomass and therefore has a clear physical interpretation and is thus far from spurious.

Traditionally, this relation, expressed in terms of the adult lifespan and age at maturity (which typically marks the end of an animal’s growth), as in Charnov and Berrigan (1990), has been qualitatively explained in terms of life-history evolution theory. Many versions of life-history theories predict that the age of maturity should be positively correlated with the lifespan (Charnov 1993), as these patterns are a reflection of natural selection (Charnov and Berrigan 1991). In this paper, after rewriting the properties of living organisms in a physically transparent form, we predict the value of this life-history invariant in terms of the relevant energetics (\(\mathrm E_m,\, E_{2019}\), k, etc.) and find that the constancy mainly comes from the invariant number \(\mathrm N_r \sim 10^8\) of respiration cycles per lifetime, a generalization of the well-known relation of a constant number of heartbeats in a mammal’s lifetime (Levine 1997; Cook et. al 2006). Nevertheless, we do not study the origin of how the key parameters (\(\mathrm E_m\), k, etc.) vary across species, animal groups, and generations, which should be evolutionary in origin and thus set by natural selection (Charnov 1993; Gardner et al. 2005).

It can be straightforwardly obtained that some of the other life-history invariants (Charnov 1993) come directly from the invariant \(\mathrm G = t_{gen}/t_{life}\) and the universal ontogenetic growth curve. For example, the invariant ratio between the mortality rate (M) and the Bertalanffy growth coefficient (K) (Beverton and Holt 1959; Cushing 1968; Pauly 1980) comes from its relation with \(\tau\) in our formulation, Kt = \(\tau\), which at the generation time corresponds to \(\mathrm Kt_{gen} =\tau ^* \sim 5\) according to the universal growth curve (West et al. 2001; Barnavar et al. 2002). Noting also that the mortality rate M is approximately the inverse of the lifespan 1/\(\mathrm t_{life}\) (in the next section, we will see a more rigorous definition for M), it is straightforward to obtain that M/K = \(\mathrm t_{gen}/( t_{life} \tau ^* )\) = G/5. Another invariant is the ratio between the length at maturity \({l}(t_{gen})\) and the maximum asymptotic length L (Charnov and Berrigan 1991), which also comes from \(\mathrm K t_{gen} = \tau ^*\) by simply recalling the Bertalanffy growth equation (derived in §2): \({l}(t_{gen}) /L = 1-e^{-\tau ^*}\). Finally, the invariant fraction of body mass to reproduction per unit time per life span (Charnov et al. 2001) comes from the definition of the dimensionless mass r (Eq. 3) and the invariant G (Eq. 6), which are related as \(\frac{m(t_{gen})}{M} \frac{t_{life}}{t_{gen}} = (1-e^{-\tau ^*})^{1/\alpha } \,G^{-1}\).

Implications for population growth: the Malthusian parameter and the “Equal Fitness Paradigm”

Malthus (1798) studied the simplest model of population growth, which can be derived by assuming that all individuals are identical and reproduce continuously; therefore, the number of individuals (N) will change with the birth rate B and death rate D as follows:

$$\begin{aligned} \frac{dN}{dt} = B - D = (b_N-d_N )N = r_m N \, , \end{aligned}$$
(7)

where \(\mathrm b_N\) and \(\mathrm d_N\) are the per capita birth and death rates, respectively, and \(\mathrm r_m\) is the Malthusian parameter or intrinsic (maximum) population growth rate. The solution of Eq. 7 has an exponential form given by \(\mathrm N(t) = N_0 \, {e^{r_m t}}\), and therefore, the Malthusian parameter \(\mathrm r_m=b_N-d_N\) has units of inverse time and can be rewritten in the following form: \(\mathrm r_m = \bar{b}/t_{adult} - \bar{d}/t_{life}\). The constants (\(\bar{b}, \bar{d})\) are now dimensionless since we have identified \(\mathrm t_{life}\) as the characteristic timescale for death rates and \(\mathrm t_{adult} = t_{life} - t_{gen}\) as the characteristic timescale for the birth rates. The reason for the latter is that \(\mathrm t_{adult}\) is the reproductive adult lifespan, which is determined by subtracting from the lifespan the time period required to (grow and) mature to reproductive age, \(\mathrm t_{gen}\), which does not fulfill the key assumption in Eq. 7 that the individuals are able to reproduce continuously.

Using Eq. 6, which relates \(\mathrm t_{gen}\) and \(\mathrm t_{life}\), the Malthusian parameter is given by

$$\begin{aligned} \mathrm r_m = \Big (\frac{\bar{b} }{1-G} - \bar{d} \Big ) \, t_{life}^{-1} =\Big (\frac{\bar{b} }{1-G} - \bar{d} \Big ) \, \frac{f_H}{N_b} \propto f_H \, , \end{aligned}$$
(8)

which gives a Malthusian parameter \(\mathrm r_m\) that is again explicitly dependent on the heart frequency \(\mathrm f_H\). The allometric scaling of heart frequency under basal conditions (\(f_H \propto M^{-0.25}\), Brody 1945, Calder 1968) gives the well-known Fenchel (1974) allometry, \(r_m \propto M^{-0.25}\), but again, as most biological rates and times scale as \(\mathrm M^{-1/4}\) and \(\mathrm M^{1/4}\) (Savage et al. 2004; Burger et al. 2021), a more interesting test will be to study allometric variations of \(\mathrm r_m\) in outliers of “1/4 scaling” in terms of the heart frequency \(\mathrm f_H\) to determine whether they vary as \(\mathrm f_H\), as predicted by Eq. 8.

Another prediction of Eq. 8 is an inverse correlation between \(\mathrm r_m \, and \, t_{life}\), which was recently tested by Hatton et al. (2019). Figure 2 displays the data of Hatton et al. (2019) for all types of living organisms, ranging from protists to mammals and birds. Since the present analysis is restricted to organisms with a heart, we denote with the dashed (blue) line the mass scale of \(10^{-3}\) grams, which corresponds to the scale associated with the smallest organisms with hearts, such as fruit and fairy flies, and coincides in Fig. 2 with a transition to a decrease in the scatter (for larger masses); in addition, all individual animal groups (colors in Fig. 2) satisfy this relation for masses larger than \(10^{-3}\) grams. The solid black line corresponds to the best fit to the data, consistent with a clear inverse correlation between \(\mathrm r_m \, and \, t_{life}\), with no residual slope with body mass (\(\mathrm r_m t_{life} = 3.35 \, M^{0.01}\), scatter of 0.5 dex or half an order of magnitude).

Fig. 2
figure 2

Lifetime growth: intrinsic (maximum) population growth rates (\(\mathrm r_m\)) multiplied by the corresponding lifespans (\(\mathrm t_{life}\)) for all types of living organisms, ranging from protists to mammals and birds, compiled by Hatton et al. (2019). The solid black line corresponds to the best fit to the data, with no residual slope with body mass, \(\mathrm r_m t_{life} = 3.35 \, M^{0.01}\), as expected from Eq. 8, with a scatter of 0.5 dex. The dashed blue line denotes the mass scale of \(10^{-3}\) grams, which corresponds to the scale associated with the smallest organisms with hearts, to which the present analysis is restricted

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The relatively low variation (on average, one order of magnitude spread, since the scatter is 0.5 dex) for a database that includes sources of scatter such as mixing lifespans in captivity and in the wild (Hatton et al. 2019; see de Magalhaes and Costa 2009 for more on this issue) and in particular, a negligible slope with body mass as shown in Fig. 2 has been claimed (Hatton et al. 2019) to support the so-called “Equal Fitness Paradigm” (Brown et al. 2018; Burger et al. 2021), which states that “most organisms are more or less equally fit, as evidenced by the persistence of millions of plant, animal and microbe species of widely varying size, form and function in the Earth’s diverse environments" (Brown et al. 2018). In terms of the proposed formalism, Eq. 8 and the low variation of G across animal groups (the factors 2–3 seen in Table 2) also implies a low variation for the parameters \(\bar{b }\, and \, \bar{d}\) across species. These relatively constant values (implied by Fig. 2) for the dimensionless parameters \((\bar{b }, \bar{d})\) associated with the (net) per capita birth and death per characteristic time, in addition to the spectacular diversity of life histories in terms of growth, reproduction, and survivorship over the life cycle, which implies very diverse life history strategies in the very large mass range displayed in Fig. 2, supports the idea that selection mechanisms should operate in terms of equal fit according to Darwinian evolution.

In the fundamental theorem of natural selection, Fisher (1930) states that the rate of increase in mean fitness (caused by natural selection; Price 1972) is equal to the genetic variance of a species, conceptually linking natural selection with Mendelian genetics. Natural selection can only increase fitness by reducing genetic variance (i.e., selecting away undesirable alleles; Basener and Sanford 2018), and thus, without mutations and given enough time, selection must reduce genetic variance all the way to zero and fitness must reach a maximum, according to Fisher’s theorem and as confirmed through simulations (Basener and Sanford 2018). The “Equal Fitness Paradigm” seems to suggest that a maximum fitness value has been reached by the coexisting species in the current conditions.

Additionally, Demetrius (1974, 1975) found a relation between “evolutionary” entropy (of a population) and reproductive potential with fitness (measured by the Malthusian parameter), in which Fisher’s theorem is also obtained as a corollary for a Hardy-Weinberg equilibrium. In Demetrius’s formalism, the Malthusian parameter is analogous to the Gibbs free energy (Demetrius 1997) and also behaves effectively like a thermodynamic potential: being the maximum (potential) growth that can be performed by a given population, that becomes null when such population reaches abundance equilibrium. In thermodynamic theory, Gibbs free energy is minimized when entropy is maximized, and according to the second law of thermodynamics, maximum entropy is found only in the state of thermodynamic equilibrium, where (the theorem of) equipartition of energy holds. The “Energetic Equivalence Rule” (Damuth 1981), which was recently verified for \(\sim\)3000 species in Hatton et al. (2019), seems to be a version of such an equipartition of energy in living matter and thus is also consistent (under abundance equilibrium) with being in the state in which maximum entropy is reached.

Another relation for \(\mathrm r_m\) that can be derived using the definition of G (Eq. 6) is the relation that predicts an inverse correlation with the generation time \(\mathrm t_{gen}\):

$$\begin{aligned} r_m = \Big ({\frac{G\bar{b }}{1-G}} - {G}\bar{d} \Big ) \, \frac{1}{t_{gen}} \propto t_{gen}^{-1} \, , \end{aligned}$$
(9)

and this is also a relation that can be tested. For this purpose, we use the data in Duncan et al. (2007), which compiles the ages at first reproduction (\(\sim t_{gen}\)) and maximum population growth rates for mammals. Figure 3 displays the data collected in Duncan et al. (2007), which is in overall agreement with Eq. 9, with no relevant slope with body mass. The solid line denotes the best fit to the data (\(\mathrm r_m t_{gen} = 1.38 \, M^{-0.067}\)), which has an average scatter of 0.2 dex. Dillingham et al. (2016) also found equivalent results on this expected inverse relationship using a more sophisticated Bayesian analysis.

Fig. 3
figure 3

Generation times (\(\mathrm t_{gen}\)) times the maximum population growth rates (\(\mathrm r_m\)) for mammals, compiled by Duncan et al. (2007). The solid line denotes \(\mathrm r_m t_{gen} = 1.38 \, M^{-0.067}\), which is the best fit to the data and has an average scatter of 0.2 dex. The best-fitted relation is in overall agreement with the predicted Eq. 9, with no relevant slope with body mass

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The normalizations from the best fit to the data in Figs. 2 and 3 give estimations for \(\mathrm r_m t_{life} \sim 3.35\) (Fig. 2) and \(\mathrm r_m t_{gen} \sim 1.38\) (Fig. 3), which in principle can be combined with Eqs. 8 and 9 to determine some of their constants. Unfortunately, it is not possible to determine the constants \((\bar{b }, \bar{d})\) associated with the dimensionless birth and death rates since Eqs. 8 and 9 are not (linearly) independent equations, but it is still possible to determine that \(\frac{r_m t_{gen}}{r_m t_{life}} = G \sim 0.41\). This is for the average lifespan in the wild because in Hatton et al. (2019), lifespans were normalized to those values, and since maximum lifespans are about 2.5 times the average lifespans in the wild (McCoy and Gillooly 2008), we determine from Figs. 2 and 3 that \(\frac{ t_{gen}}{ t^{max}_{life}} = G^{min} \sim 0.16\),which agrees with the value determined from the average of the predicted G values in Table 2 (\(\mathrm G^{min} \sim 0.17\)).

Summary

In this paper, we explored the implications of the new metabolic relation (Escala 2019) for ontogenetic growth and found that it can be described by a universal ontogenetic growth curve for all studied species without the aid of ad hoc fitting parameters. The same universal growth curve in West et al. (2001) and Banavar et al. (2002) is found when certain dimensionless quantities are properly defined, but in our case, the characteristic growth timescale (\(\mathrm t_{growth}\)) is set by the inverse of the heart frequency \(\mathrm f_H^{-1}\), rescaled by the ratio between the specific energies (per unit of mass) required for metabolism and the creation of new biomass; these are all quantities with clear physical or biological meaning and not obscure fitting parameters. The results for universal growth have the same interpretations as before in terms of conservation of energy, since they come from the same Bertalanffy-type equation, but in our case, the results do not rely on a specific model. Instead, they illustrate the advantages obtained when empirical data are properly described and quantified, reaching this simple description without extra assumptions.

We also explored the implications of the discussed ontogenetic growth model for the generation time, finding that the predicted \(\mathrm t_{gen}\) can explain the origin of several “Life History Invariants” when is it combined with the invariant number of respiration cycles per lifetime, a relation that comes from the generalization of the well-known relation of the constant number of heartbeats in a mammal’s lifetime (Levine 1997; Cook et al. 2006). In particular, regarding the invariant ratio between the lifespan and age at maturity (\(\mathrm G = \frac{ t_{gen}}{ t_{life}}\)), which has been traditionally explained in terms of life-history evolution theory, in our formalism, the value of this life-history invariant is predicted in terms of the relevant energetics and the invariant number of respiration cycles per lifetime. The predicted G and its variation between taxonomic groups shows consistency with the empirically determined value (Table 2). We also showed that other life-history invariants (Charnov 1993) come directly from the universal ontogenetic growth curve and the invariant G.

We finally studied predictions for population growth, finding that the invariant G implies a Malthusian parameter \(\mathrm r_m\), or intrinsic population growth rate, that is inversely proportional to both \(\mathrm t_{gen} \, and \,t_{life}\). We find that these inverse relations are indeed observed in nature, with no relevant slope with body mass and relatively low scatter (0.5 dex for \(\mathrm r_m t_{life}\) in Fig. 2 and 0.2 dex for \(\mathrm r_m t_{gen}\) in Fig. 3). The ratio \(\mathrm G^{min} = \frac{ t_{gen}}{ t^{max}_{life}}\) that can be estimated from the two best-fitted relations (\(\frac{r_m t_{gen}}{r_m t^{max}_{life}} = G^{min} \sim 0.16\)) is also consistent with our predicted \(\mathrm G^{min} \, (\sim 0.17\); Table 2). We find relatively constant values for the total births and deaths per capita per characteristic time (\(\bar{b }\, and \, \bar{d}\) in Eq. 8), besides the diversity of life histories in living organisms on Earth, which supports the so-called “Equal Fitness Paradigm.”

In our formalism, the allometric scaling relations for several critical timescales and rates (\(\mathrm t_{life}, \, t_{gen} \, and \, r_m\)) are all derived from their proportionality to \(f_H\); therefore, the mass scaling \(f_H \propto M^{-1/4}\) directly explains other well-known allometries (lifespan, generation time, and Fenchel allometries), naturally explaining why most biological times and rates scale as \(\mathrm M^{-1/4}\) and \(\mathrm M^{1/4}\) (Savage et al. 2004; Burger et al. 2021). Additionally, the variations from 1/4 scaling should be explained in terms of the variation in the \(f_H\) mass scaling, since such quantities (\(\mathrm t_{life}, \, t_{gen}, \, and \, r_m\)) should have the same mass scaling as \(f_H\) regardless of whether they are 1/4, being a clear testable prediction. This in fact was the case for the metabolic rate relation (Escala 2019).

Nevertheless, our formalism is empirically motivated and does not explain why \(f_H\) should scale as \(\mathrm M^{-1/4}\) or with another exponent; thus, it is compatible with previous attempts that explained the exponent for the allometric scaling of metabolism and related quantities (West et al. 1997, 1999; Banavar et al. 1999, 2010; Darveau et al. 2002) insofar as the mechanism is based on the anatomy and physiology of the circulatory system. In the context of this work, any attempt to explain the allometric scaling of the heart frequency \(f_H\) will also explain the allometric scaling of metabolism, lifespan, generation time, and the Malthusian parameter, since they are all quantities (either directly or inversely) proportional to \(f_H\) in our formalism, but such explanation could still be one of the previously proposed (West et al. 1997, 1999; Banavar et al. 1999, 2010; Darveau et al. 2002). The difference between these two (compatible) approaches is that the proposed new metabolic formula (and the original one by Kleiber) relates the metabolic rate to macroscopic physiological variables (such as mass, heart frequency), which can be applied to describe a phenomena, such as ontogenetic growth curves. Differently, mechanistic explanations for the “1/4 scaling” of the macroscopic physiological variables (e.g., metabolic rate, heart frequency) are attempts to explain their allometric scaling in terms of a microscopic model (i.e., thru a model of the circulatory system). This is similar to the difference in physics between the (empirically-based and axiomatic) laws of thermodynamics and the statistical mechanics that explain them.

Finally, it is important to emphasize that our predictions were tested without the aid of any free (fitting) parameters. The fitting procedures are shown in Figs. 1, 2, and 3 only for comparison with our predictions, not to determine any free parameters for our formalism. Thus, our formalism follows the criteria suggested by Ginzburg and Jensen (2004) for judging ecological theories, in terms of reducing to a minimum the number of parameters with no empirically determined range. In additionalso directly tested in Fig. 1\(\mathrm (E_{2019})\) and indirectly in terms of the G value empirically determined by Figs. 2 and 3\((E_{2019} \, and \, N_r; \,\) Eq. (6)), further supporting the relations for metabolism and lifespan used in this paper.