A Model of Ontogenetic Growth in Animals Suggests That Lifespan Is a Result of Growth

Abstract

The problem that this study is concerned with is the ontogenetic growth of humans and animals. The aim of this research is to analyze a model of the ontogenetic growth of animals. The target of the analyses is to show a link between growth and longevity. This model has implications for modelling the growth of humans as well. In this study, pigs were considered model animals for humans. Humans and pigs have a number of comparable physiological features as well as a few analogous aspects of growth. The resemblance of the biological functions leads us to think that by modelling the growth of pigs, one can gain a better look into the growth of humans. In this research, we model growth, which is operationalized as weight gain; weight loss was not considered. In this study, a discussion of the translation of the results to the growth and longevity of humans was provided. The lifespan or longevity of animals was not modelled explicitly; predictions were made in accordance with the results of the growth model. The main result of the model is that growth promotes, if not causes, longevity.

1. Introduction

In pigs and humans, the growth and development of these organisms are linked processes. In ontogeny, both processes are reliant on each other and function in a coherent way. However, present knowledge only allows for modelling growth but not development. When modelling the growth of an animal, one can only model certain aspects; growth as a whole process is not within the modelling power. A suitable modelling technique for this task is yet to be discovered. In this study, we offer a possible option.

1.1. Ontogenetic Growth

In this research, the ontogenetic growth of pigs was modelled. Pigs are well-studied animals; these animals have been studied for many decades and for many purposes [1]. Pigs serve as good model animals for humans in several aspects of biology and medicine [2,3,4]. Growth is one of these aspects. One can consider the ontogenetic growth of an animal as a change in its weight and size over time. In this study, we model growth, operationalized as weight gain. Weight loss was not considered. In ontogeny, the growth rate of an animal or human is not constant. The growth rate in ontogeny changes with weight and age and is a species-specific trait. This trait, known as growth rate, has been studied and modelled as a quantitative trait. Since growth is a species-specific trait, its modelling must follow this notion. This means that a model of an animal’s growth must be species-specific. In this study, the model of ontogenetic growth is species-specific to pigs and is used solely for pigs.

Growth is a fundamental feature of living beings. Species are differentiated by size, weight, and shape. Weight and size are essential traits in animal species. For individual animals, these traits are vital for breeding and survival. The rate of change of these traits, the growth rate, is a species-specific feature. The rate of growth is considered to be a genetically determined trait [5,6]. In animals’ ontogeny, growth rates are not constant and vary depending on weight and age. The trajectory of growth is a species-specific and genetically controlled process [5]. It follows that, in animals, the trajectory of growth rates is phenotype-dependent. Animal growth is influenced by many factors. In this study, growth was modelled as an interaction between an organism and its environmental factors; the main and only environmental factor that was considered was the feed or food consumed.

1.2. Models of Animal Growth

The history of modelling animal growth goes back more than one hundred years. During this time span, a few models of animal growth were introduced. All the models were of the same kind; they were interpolation functions. These functions have neither predictive nor explanatory value [7]. They were used for curve fitting to empirical data in both humans and animals [8]. The curve-fitting application does not produce added biological value, nor does it facilitate understanding of animals’ growth. Also, some of the interpolation functions were considered universal; in other words, they were applicable to all animal species. However, this was not the case in [9]. Moreover, these functions were not applied to human weight ranges from birth to their maximum weights; a very limited weight range was usually interpolated. This is due to the fact that the functions are only descriptive [10] and lack the ability to reveal new biological facts. The interpolation functions mentioned above model growth as an epigenetic trait, though growth is a phenotype-determined trait. In addition, from a mathematical point of view, such functions are unreliable. Such functions have been called sloppy or non-identifiable because the parameters in the functions cannot be trusted [11]. Some authors declared that the parameters in the functions have biological meaning. However, the meaning of the parameters was not explained. This summary of the modelling methods suggests that interpolation of growth data is hardly a suitable approach for modelling ontogenetic growth. The well-known approach to modelling processes in biology is systems of differential equations [12]. However, in this field, modelling using dynamic systems is not the conventional method.

1.3. Ontogenetic Growth and Longevity

Longevity is defined as an individual’s ability to reach a longer lifespan under ideal or prevailing conditions. Like any quantitative trait, longevity is affected by genetic, developmental, epigenetic, and environmental factors and shows a large amount of variation [13]. The maximum longevity in humans and most animal species is not known [14]. The maximum weight of a species is not known as well. Research into the link between growth and longevity has led to controversial results [15,16]. Maxima of both traits are hard to find empirically; however, future research should build on established models. In humans, theoretical longevity is thought to be an important trait [17]. However, no model has been built to estimate the upper limit of this trait. It is thought that both traits—growth and longevity—are interconnected. Moreover, some researchers believe that growth causes aging [18]. A correlation between growth and longevity in animals has been known for years [17]. There is a reported tendency for individual animals from species with larger animals to live longer compared with individual animals from species with smaller animals [15,19]. Reportedly, a possible link between growth and longevity was found [20]. However, the reasons for causation, or a correlation between these traits, should be clarified in a mathematical model. Furthermore, approaches to empirically find maximum longevity, whether in experiments or field observations, are unclear [15]. In this respect, the type of model needed to address these problems is essential for success. In this study, we proposed a possible solution; predictions of pigs’ longevity were derived from a few aspects of the model of animal growth.

2. Methods

In this study, methods of mathematical modelling were used. The methods include the standard techniques, both continuum and hybrid, that are well known in mathematical physics and biophysics. In this study, functional relations between variables were specified in a non-local form as a dynamic system. In this model, a hybrid technique means that both continuum and discrete variables were used. In this model, the growth of animals was modelled as an interaction between individual animals and their environmental factors. In this study, as the main environmental factor was considered feed or food, other factors were treated as optimal. In animals, nutrition or feed intake is the main environmental factor that affects growth and size [6]. The growth was modelled as an increase in weight. Weight gain was operationalized as the conversion of the consumed food or feed into the weight of an animal. The feed conversion efficiency was considered to be essential for the rate of growth. To determine how ontogenetic growth is linked to longevity, we extrapolated from our analyses of the model.

2.1. Variables

In this model, time is treated as a discrete variable. The unit of chronological time was one day. One day is a time span of one cycle of the circadian rhythm that oscillates with periods close to 24 h. During this time, in animals, processes of feed intake, digestion, and absorption go through one cycle [21]. In other words, in animals, one cycle of feed conversion takes one day [22]. This unit of time was chosen as suitable for modelling growth. Furthermore, one day is the standard unit of time to measure longevity. It follows that in both cases, when modelling growth or longevity, the unit of time is the same—one day. In the model, the time or age of animals was counted from their birth.

The following variables were used in the model. If we let M denote an animal’s current live weight, measured in kilograms, then M = {M∈ℝ₊|30 ≤ M ≤ 600}, where an animal’s individual maximum weight M_x = 600 kg. If we let m_o denote an animal’s initial considered weight, measured in kilograms, then m_o ≤ M, where m_o = 30 kg. If we let t denote the chronological discrete current time, measured in days from an animal’s birth, then t = {t∈ℕ|0 ≤ t < ∞}, where ∆t = 1, 2, 3, …, n, with n∈ℕ, and t_o denotes time related to m_o, where t_o = 90 days. If we let K denote the invariant of growth, being nondimensional, then K = {K∈ℝ₊|1≤ K < 11}, where K_o = 1. If we let Z denote the current feed conversion coefficient, being nondimensional, then Z = {Z∈ℝ₊|Z_o ≤ Z < ∞}, where (Z = ∞) → (M = M_x)˅(M = M_xx), where M_xx denotes the species maximum weight. If we let F denote consumed feed, measured in kilograms, then F = {F∈ℝ₊|0 ≤ F < ∞}.

2.2. Methodological Aspects

In animals and humans, weight and weight gain are quantitative traits. In this study, we use the term phenotype, particularly growth phenotype. Phenotype entails a genetic aspect of a trait, and in this sense, it is more informative. However, there is one aspect to consider. A phenotype of a trait of an animal results from the interactions between different genes. But the number of genes involved is not always clear. In the case of quantitative traits, there are speculations about the number of genes involved. In an organism, a phenotype is any distinct and measurable trait; the traits can range from those pertaining to anatomy to biochemistry. In animals, current weight and weight gain are phenotypes. A phenotype is not a constant state; it is a dynamic system. As animals grow or age, their phenotypes change. In this context, one can speak about the trajectory of a phenotype over time. In other words, growth in the juvenile stage is not the same as in adulthood. From a genetics viewpoint, there is one aspect to consider. It follows from the definition of a phenotype that an animal’s phenotype consists of an infinite number of traits that are determined by a finite number of genes [23]. This implies that it is possible that some traits are not genetically determined, also known as the Dirichlet principle. But, in animals, genetics is an essential factor of growth [5,6]. In this study, growth and growth rates were modelled as a genetically determined dynamic system, resulting in phenotypes. This means that in the model, one can find or differentiate between growth phenotypes.

3. Results

In this study, the model of animal growth was outlined using analytical equations. It was shown that in animals, ontogenetic growth and longevity are related traits. This model introduces a concept that growth, or weight gain, is a result of feed intake and feed conversion to animal body tissues. It was proven that this concept is reasonable for modelling growth. This model has a clear biological meaning as well as a coherent interpretation, and it is provable through experiments.

3.1. A Model of Ontogenetic Growth

In this study, the current weight, M, of an animal was modelled as a function of its age, feed consumed, feed conversion efficiency, and a parameter K. In this model, the parameter K is an invariant of growth. A concise form of the equation is given below.

M = f (t, F, Z, K).

(1)

In more detail, the following is achieved:

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathit{M}}{{\mathit{m}}_{\mathit{o}}}}= \mathbf{2}\mathit{K}-\mathbf{1}+{\displaystyle \frac{\left(\mathit{Z}-\mathbf{2}\mathit{K}\right)\left(\mathit{K}-\mathbf{1}\right)}{\mathit{Z}\mathit{K}}}\\ {\displaystyle \frac{\left(\mathit{Z}-\mathbf{2}\mathit{K}\right)\left(\mathit{K}-\mathbf{1}\right)}{\mathit{Z}\mathit{K}}}={\displaystyle \frac{\mathit{t}-\mathit{K}{\mathit{t}}_{\mathit{o}}}{\mathit{t}}}\end{array}\right.,$$

(2)

$$\mathit{K}={\displaystyle \frac{\mathit{M}\mathit{t}}{{\mathit{m}}_{\mathit{o}}\left(\mathbf{2}\mathit{t}-{\mathit{t}}_{\mathit{o}}\right)}}.$$

(3)

Equations (2) and (3) are the basic relations in the model. From (2) and (3), the following is obtained:

$${\displaystyle \frac{1}{m_o}}·{\displaystyle \frac{∆M}{∆t}}={\displaystyle \frac{K}{t}}·{\displaystyle \frac{Z\left(2K+1\right)-2K}{Z\left(K+1\right)-2K}},$$

(4)

with

$${\displaystyle \frac{1}{m_o}}·{\displaystyle \frac{∆M}{∆t}}={\displaystyle \frac{1}{t}}·{\displaystyle \frac{m_o\left[Z\left(2K+1\right)-2K\right]}{m_o\left[Z\left(2K+1\right)-2K\right]-MZ}},$$

(5)

and

$${\displaystyle \frac{∆K}{∆t}}={\displaystyle \frac{K}{t}}·{\displaystyle \frac{ZK}{Z\left(K+1\right)-2K}},$$

(6)

$${\displaystyle \frac{1}{m_o}}·{\displaystyle \frac{∆M}{∆K}}={\displaystyle \frac{2K+1}{K}}-{\displaystyle \frac{2}{Z}} , Z>0.$$

(7)

Good approximations are derived from

$${\displaystyle \frac{∆F}{∆M}}={\displaystyle \frac{M}{2m_o}}·{\displaystyle \frac{Z\left(4K+1\right)-2K}{Z\left(2K+1\right)-2K}},$$

(8)

$${\displaystyle \frac{∆Z}{∆K}}={\displaystyle \frac{2K\left(3Z-2\right)}{9Z}} , Z>0.$$

(9)

Equations (7)–(9) form a non-local autonomous dynamic system. This system is a main part of the model of animal growth. If we let M_x denote an animal’s individual maximum weight, then under the model conditions, M_x = 600 kg.

Let us find K_x. Under the model conditions, $K_x={K|}_{M=M_x}$. We can consider the limit

$$\underset{M\to M_x}{\lim }\left({\displaystyle \frac{2K-1}{K+1}}\right)=\sqrt{3,} \mathrm{where} {\left(K\to K_x\right)|}_{M\to M_{x }}$$

(10)

From limit (10), it follows that $K_x= 5+3\sqrt{3 }$. In this model, the individual maximum weight of an animal was denoted as M_x; this is the weight of a full-grown animal. The species’ maximum weight was denoted as M_xx. M_xx and M_x differ considerably, where M_xx > M_x.

Let us consider system (2). From (2), one can derive the following equation:

$${\displaystyle \frac{t}{t_o}}={\displaystyle \frac{{ZK}^2}{Z+2K\left(K-1\right)}}$$

(11)

From Equation (11), it follows that the age t_x of an animal that has reached its individual maximum weight M_x is denoted as $t_x={t|}_{K=K_x}$. Let us consider the limit (K → K_x) in (11).

$$\underset{K\to K_x}{\lim }\left({\displaystyle \frac{t}{t_o}}\right)={\displaystyle \frac{K_x^2}{4}} .$$

(12)

In other words,

$${\displaystyle \frac{t_x}{t_o}}={\displaystyle \frac{K_x^2}{4}} .$$

(13)

And one can find that t_x = 6.40 years. At this age, a full-grown animal reaches its weight M_x. From Equation (11) and limit (12), it follows that for Z_x, $Z_x={Z|}_{K=K_{x }}$.

$$Z_x={\displaystyle \frac{{2K}_x\left(K_x-1\right)}{3}}$$

(14)

The model indicates that a full-grown animal of weight M_x attains this weight in 6.40 years, t_x, and needs to consume Z_x kilograms of feed to put on the final kilogram of body weight. When the final weight M_x is reached, growth stops. In this weight, an animal cannot grow; no quantity of feed can increase its weight. As a result, at the point M_x, Z_x grows into infinity, where $Z_x\to \infty$. Let us analyze this relationship. If we consider the limit (Z → ∞) in Equation (7), then

$$\underset{Z\to \infty }{\lim }\left({\displaystyle \frac{1}{m_o}}·{\displaystyle \frac{∆M}{∆K}}\right)={\displaystyle \frac{2K_1+1}{K_1}} , \mathrm{where} {\left(K\to K_1\right)|}_{Z\to \infty }.$$

(15)

In other words, $K_1= {K|}_{\left(M=M_x\right)\wedge \left(Z=\infty \right)}$. From limit (15), one can find K₁ = 10.04975. The biological meaning of K₁ is as follows: In weight M_x, some phenotypes go through a first-order phase transition due to development, Z_x → ∞ → Z_xv, where Z_xv denotes the feed conversion coefficient after the phase transition. The phenotypes with K = K₁ cannot pass through the phase transition; they remain at weight M_x and can reach a lifespan of t₁, where t₁ = 24.90 years.

$${\displaystyle \frac{t_1}{t_o}}=K_1^2 .$$

(16)

The same result can be obtained from Equation (11). If we consider the limit (Z → ∞) in (11)

$$\underset{Z\to \infty }{\lim }\left({\displaystyle \frac{t}{t_o}}\right)=K_1^2 , \mathrm{where} {\left(K\to K_1\right)|}_{Z\to \infty },$$

(17)

then we can subsequently arrive at Equation (16).

3.2. Phase Transition During Growth

When an animal has reached its individual maximum weight, M_x, its growth stops. At this point, the feed conversion coefficient, Z_x, grows into infinity (Z_x → ∞). This development is due to the fact that at point (M_x, Z_x), no quantity of feed can increase an animal’s weight. As a result, a first-order phase transition takes place. This phase transition means that an animal at weight M_x undergoes a change in its growth trajectory. A first-order phase transition is a sudden, catastrophic change of state. In an animal, this kind of change is a sudden transition to a new, emerged growth trajectory. The cause of this change is the feed conversion coefficient Z, which, at point Z_x, becomes unstable. As a result, the following relationship occurs: Z_x → ∞ →Z_xv. Here, Z_xv denotes feed conversion after the phase transition. After the phase transition, the phenotypes can start to grow again. It is now a new, emerging growth trajectory. From this analysis, one can infer that for an animal, it is necessary to experience a first-order phase transition to grow beyond individual maximum weight M_x. This is necessary to reach the species maximum weight M_xx as well.

3.3. Growth and Longevity

From the model, it follows that between the current weight M and parameter K, there is a functional relationship. This fact was used to find maximum longevity. The equation below holds.

$$MK=\int M dK+\int K dM.$$

(18)

Maximum longevity was modelled as t → ∞. We can consider the limit (t → ∞) in Equation (3).

$$\underset{t\to \infty }{\lim }K={\displaystyle \frac{M}{{2m}_o}} .$$

(19)

By substituting (19) into (18), one can find $K_2= {K|}_{\left(M=M_x\right)\wedge \left(t\to \infty \right)},$ given below.

$$K_2={\displaystyle \frac{M_x}{{2m}_o}}+{\displaystyle \frac{m_o}{{2M}_x}} .$$

(20)

Here, K₂ = 10.0250. Under conditions $\left(M=M_x\right)\wedge \left(K=K_2\right)$, t₂ = 49.43 years. This is the maximum longevity of pigs.

In this study, longevity was not modelled explicitly; it was derived from a few assumptions based on the model. This means that longevity is a byproduct of the model of growth; this makes the inferences more reliable. In the model, longevity was found as a result of a functional relationship between three variables: current weight M, feed conversion coefficient Z, and parameter K. It follows that the variables contribute both to growth and longevity. In other words, growth and longevity are related traits. When an animal has reached its individual maximum weight M_x, the following equations hold:

$${\displaystyle \frac{M_x}{m_o}}=2K_x-{\displaystyle \frac{4}{K_x}} .$$

(21)

$${\displaystyle \frac{M_x}{m_o}}=2K_1-{\displaystyle \frac{1}{K_1}} .$$

(22)

$${\displaystyle \frac{M_x}{m_o}}=K_2+\sqrt{K_2^{2 }-1} .$$

(23)

One can conclude that parameter K is indispensable to define weight M_x; parameter K is important to specify longevity as well. One can infer that in animals, weight M, parameter K, and longevity are related traits.

4. Discussion

Growth is an essential function of living things. Growth rate and growth dynamics over time are species-specific traits. Is longevity an inborn trait or does it change with weight and age? At what weight can an animal reach its maximum longevity? At what age of an animal can its longevity be safely predicted? In animals, a correlation between growth and longevity has been known for decades [17]. However, is growth a major cause of longevity? These questions are not possible to answer in direct experiments; the answers to these questions are predicted based on a model of animal growth.

4.1. Ontogenetic Growth

Growth of humans and animals is a species-specific trait. Growth rate, growth trajectory, and a number of distinct phenotypes are species-specific traits as well. Together, the trait dynamics in time are considered representative of ontogenetic growth. In this study, a few aspects of the growth were modelled. The results suggest that the ontogenetic growth of animals is best modeled by a dynamic system. And for every single species, the dynamic system is thought to be a unique one. It follows that a general model of animal growth is not viable to build. In line with this, to model growth, each species is considered to be a uniquely growing living being. However, this is yet to be implemented.

The model of pig growth can be considered from different viewpoints. In this study, we discuss two of them. The model of growth can be analyzed as an autonomous dynamic system or as a dynamic system that develops in time. The fact that autonomous dynamic systems (7), (8), and (9) describe growth well leads us to think that time is not a decisive factor in growth. In an organism, growth and development are linked processes. And this aspect is thought to be non-time-dependent; rather, it seems to be animal-dependent, which, in this case, is an adult animal. The time-dependent, non-autonomous, and dynamic systems of growth (2), (3), (5), and (6) describe the same process, though in a different way. In this study, the autonomous dynamic system of growth was analyzed.

During ontogenetic growth, there are two unobservable and unstable states: Z = 1 and Z = ∞. In these two states, at the two points, the growth function is not defined. This means that an animal’s weight function at the point (M_x, Z = ∞) is unstable. The states are unobservable and, for this reason, cannot be explained by experimental technique. However, a mathematical analysis of the states is possible. How animals function under these conditions is important knowledge for both medicine and biology. In ontogenetic growth, the development of a phase transition is of special interest. The phase transition, in this model, is a first-order phase transition, which is a sudden change in the growth trajectory. As a result, new growth trajectories emerge. The emerged trajectories are growth phenotypes that are distinct from previous ones. On the emerging trajectories, animals can continue to grow and potentially attain the maximum weight for their species. In this respect, the model indicates that individual maximum weight M_x is an important trait. The potential to reach this weight can be an inborn quality. And at this weight, animals undergo a phase transition. It follows in animal ontogeny that weight M_x is an important trait. The animals that can reach weight M_x and pass through the phase transition in several aspects are different. Regarding the two factors mentioned, some animals can reach the species maximum weight M_xx, while some animals can reach maximum longevity t₂.

4.2. Longevity as a Result of Growth

In animals, a possible link between growth and longevity was found [20]. However, the physiological details of the link remain unclear. In this section, we discuss the analytical model of pig growth; the functional relation between the variables associated with growth and longevity was analyzed.

There are many controversial reports on the correlation between growth and longevity in animals and humans [17,20]. In mammals, a correlation between weight and maximum lifespan has been known for years [19]. In this regard, one study was considered. In mice, a positive correlation was found between the age at which the mouse reached its maximum weight and its lifespan [24]. In pigs, in their model, a comparable result was found; it is a functional relation between an individual animal’s maximum weight M_x and its longevity. This follows from the model; parameter K describes this relationship. The association is shown in

Table 1. Relationship between variables.

Parameter K	Ko	Kx	K1	K2
Weight M	mo	Mx	Mx	Mx
Time/Age t	to	tx	t1	t2

.

From Table 1, one can see that weight M_x is the suitable weight to predict longevity. The results of the model imply that a link between growth and longevity is physiologically justified. This link is a functional relation between current weight M, feed conversion Z, and age t. In this respect, it can be associated with a deterministic, genetically controlled process or a biological function. In animal ontogeny, it is the growth that contributes to, if not causes, animal longevity. Below, the results are given in an analytical form.

$${\displaystyle \frac{t_x}{t_o}}={\displaystyle \frac{K_x^2}{4}} , \mathrm{where} t_x={t|}_{K=K_x}, \mathrm{M}={\mathrm{M}}_{\mathrm{x}},$$

(24)

$${\displaystyle \frac{t_1}{t_o}}=K_{1 }^2 , \mathrm{where} t_1={t|}_{K=K_1}, \mathrm{M}={\mathrm{M}}_{\mathrm{x}},$$

(25)

$${\displaystyle \frac{t_2}{t_o}}={2K}_2^2-{\displaystyle \frac{1}{2}} , \mathrm{where} t_2={t|}_{K=K_2}, \mathrm{M}={\mathrm{M}}_{\mathrm{x}},$$

(26)

Here, t_x denotes the time at which an animal has reached its individual maximum weight M_x; t₁ denotes obtainable lifespan; t₂ denotes maximum longevity; and t_x = 6.40 years, t₁ = 24.90 years, and t₂ = 49.43 years. The same results can be obtained by solving the following equations:

$$K_x=5+3\sqrt{3}$$

$$K_1=10.04975 .$$

$$K_2=10.0250 .$$

In all cases, M = M_x. It follows from the model that in animals, the individual maximum weight M_x is an important trait. In this weight, the animals can reach their obtainable lifespan t₁ and later attain their maximum longevity t₂. It is possible that a limitation for the long lives of animals is their ability to reach their individual maximum weight M_x. It is likely that the limitation is an inborn quality.

4.3. Implications for Modelling Growth of Humans

Individual humans grow much slower than individual pigs. A human’s growth is species-specific, and this is in line with our modelling strategy. The growth of individual humans must be modelled as a species-specific process. In contrast, the models that use an interpolation method to model the growth of animals do not include human growth data in a set of considered animals. This is understandable; the growth of humans differs from the growth of other animals, even from other mammals, and does not fit into ‘universal’ growth functions. The growth of individual humans is a unique process; it is a unique ontogeny. Individual humans do not grow as animals; the growth of humans differs even from the growth of apes. To model the growth of individual humans, it may be useful to take into account the implications that follow from the model of the growth of pigs.

Due to comparable physiological features in pigs and humans, modelling of the growth of individual humans could be similar to that of pigs. The slow growth of individual humans is not an obstacle to modelling, but this aspect must be taken into consideration. And the straightforward way to achieve this is the use of variable Z, which is the feed or food conversion coefficient. In a model of human growth, its value is much greater than in pigs. In a model of the growth of individual humans, the same variables can be used as for pigs: current weight M, food conversion coefficient Z, and one parameter or parameters. It is feasible that to model the growth of humans, a few parameters are needed. It is feasible that the equations of growth in both species could be alike. It is thought that the growth rate phenotypes differ. With this in mind, one can conceive the form of the equations for the growth of humans.

5. Conclusions

○

In animals, ontogenetic growth and longevity are related traits.

○

The relation between growth and longevity is nonlinear and not smooth.

○

Ontogenetic growth significantly contributes to longevity, if not causes it.

○

The process that contributes both to growth and longevity is a first-order phase transition.

○

The model predicts that maximum theoretical longevity in pigs is 49.43 years.

○

The model predicts that the obtainable lifespan in pigs is 24.90 years.