A Non-phenomenological Model of Competition and Cooperation to Explain Population Growth Behaviors

Abstract

This paper is an extension of a previous work which proposes a non-phenomenological model of population growth that is based on the interactions among the individuals of a population. In addition to what had already been studied—that the individuals interact competitively—in the present work it is also considered that the individuals interact cooperatively. As a consequence of this new consideration, a richer dynamics is observed. For instance, besides getting the population models already reached from the original version of the model (as the Malthus, Verhulst, Gompertz, Richards, Bertalanffy and power-law growth models), the new formulation also reaches the von Foerster growth model and also a regime of divergence of the population at a finite time. An agent-based model is also presented in order to give support to the analytical results. Moreover, this new approach of the model explains the Allee effect as an emergent behavior of the cooperative and competitive interactions among the individuals. The Allee effect is the characteristic of some populations of increasing the population growth rate in a small-sized population. Whereas the models presented in the literature explain the Allee effect with phenomenological ideas, the model presented here explains this effect by the interactions between the individuals. The model is tested with empirical data to justify its formulation. Another interesting macroscopic emergent behavior from the model proposed is the observation of a regime of population divergence at a finite time. It is interesting that this characteristic is observed in humanity’s global population growth. It is shown that in a regime of cooperation, the model fits very well to the human population growth data from 1000 AD to nowadays.

Appendices

Appendix 1: The Generalized Logarithm and Exponential Function

In this appendix, one presents the generalizations of the logarithmic and exponential functions and some of their properties. The introduction of the functions is shown to be very useful for dealing with the mathematical representation of the population growth model that is presented in this work.

The $\tilde{q}$-logarithm function is defined as

$$\begin{aligned} \ln _{\tilde{q}}(x) = \lim _{\tilde{q}' \rightarrow \tilde{q}}\frac{x^{\tilde{q}'} -1 }{\tilde{q}'} = \int _1^x \frac{dt}{t^{1-\tilde{q}}} \; , \end{aligned}$$

(33)

which is the area of the crooked hyperbole and is controlled by $\tilde{q}$. This equation is a generalization of the natural logarithm function, which is reproduced when $\tilde{q} = 0$. This function was introduced in the context of nonextensive statistical mechanics Tsallis (1988; 1994) and was studied recently by Arruda et al. (2008), Martinez et al. (2008) and Martinez et al. (2009). Some of the properties of this function are as follows: for $\tilde{q} < 0, \ln _{\tilde{q}}(\infty )=-1/\tilde{q}$; for $\tilde{q} > 0, \ln _{\tilde{q}}(0)=-1/\tilde{q}$; for all $\tilde{q}, \ln _{\tilde{q}}(1)=0$; $\ln _{\tilde{q}}(x^{-1}) = - \ln _{-\tilde{q}}(x)$; $\hbox {d} \ln _{\tilde{q}}(x)/\hbox {d}x = x^{\tilde{q}-1}$. Moreover, the $\tilde{q}$-logarithm is a function: convex for $\tilde{q}>1$, linear for $\tilde{q}=1$, and concave for $\tilde{q}<1$.

The inverse of the $\tilde{q}$-logarithm function is the $\tilde{q}$-exponential function, which is by

$$\begin{aligned} \hbox {e}_{\tilde{q}}(x) = \left\{ \begin{array}{ll} \lim _{\tilde{q}^{'} \rightarrow \tilde{q}} (1+ \tilde{q}^{'} x)^{ \frac{1}{ \tilde{q}^{'}} } &{},\quad {\text { if }}\,\tilde{q}x > -1 \\ 0,&{}\quad {\text {otherwise}} \end{array} \right. . \end{aligned}$$

(34)

Some properties of this function are as follows: $\hbox {e}_{\tilde{q}}(0)=1$, for all $\tilde{q}$; $\left[ \hbox {e}_{\tilde{q}}(x) \right] ^a = \hbox {e}_{\tilde{q}/a}(ax)$, where $a$ is a constant; for $a=-1$, one has $1/\hbox {e}_{\tilde{q}}(x) = \hbox {e}_{-\tilde{q}}(-x)$. Moreover, the $\tilde{q}$-exponential is a function: convex for $\tilde{q}<1$; linear for $\tilde{q}=1$; concave for $\tilde{q}>1$.

Appendix 2: A Detailed Calculus of $I_i^{(l)}$

In this appendix, one presents a detailed calculus for the intensity of the interaction felt by a single individual $i$ from the other individuals of the population, which is represented by $I_i^{(l)}$ (see Sect. 2). One follows Mombach et al. (2002) to show that this intensity is independent of the individual; that is, it is the same for all individuals of the population and depends only on the size of the population. More specifically, one shows that $I_i^{(l)} = I^{(l)}(N)$ regardless of $i$.

First, from the Sect. 2 one has

$$\begin{aligned} I_i^{(l)} = \sum _{j \in r_{ij} \ge r_0} \frac{\left( 1-\delta _{ij}\right) }{|\mathbf {r}_i-\mathbf {r}_j |^{\gamma _l}} + \sum _{j \in r_{ij} < r_0} \left( 1-\delta _{ij}\right) , \end{aligned}$$

(35)

where $\delta _{ij}$, which is the Kronecker’s delta, was introduced to avoid the restriction in the sum. Moreover, $\mathbf {r}_i$ and $\mathbf {r}_j$ represent the position vectors of the individuals $i$ and $j$, respectively, and consequently $r_{ij} = |\mathbf {r}_i - \mathbf {r}_j |$ is the distance between them. Introducing the property

$$\begin{aligned} f(\mathbf {r}_0) = \int _{V_{D}} \hbox {d}^D\mathbf {r} \delta (\mathbf {r} - \mathbf {r}_0) f(\mathbf {r}), \end{aligned}$$

(36)

where $\delta (\cdots )$ is the Dirac’s delta, the expression (35) becomes

$$\begin{aligned} I_i^{(l)}&= \sum _{j=1}^N (1-\delta _{ij}) \Big [ \int _{V_{D} \in r \ge r_0} \hbox {d}^D\mathbf {r} \delta \Big ( \mathbf {r} - (\mathbf {r}_j - \mathbf {r}_i)\Big ) |\mathbf {r}|^{- \gamma _l} \nonumber \\&+ \int _{V_{D} \in r<r_0} \hbox {d}^D\mathbf {r} \delta \Big ( \mathbf {r} - (\mathbf {r}_j - \mathbf {r}_i)\Big ) \Big ] . \end{aligned}$$

(37)

In the last two expressions, was introduced: $D (=1,2, 3)$, which is the Euclidean dimension in which the population is embedded, and $V_{D}$, which is the total (hyper) volume (in $D$ dimensions) that contains the population. The form represented in (37) was obtained by the variable substitution $\mathbf {r}_j - \mathbf {r}_i$ by $\mathbf {r}$, using Dirac’s delta.

Some algebraic manipulation and the introduction of $r\equiv |\mathbf {r}|$ allows to write

$$\begin{aligned} I_i^{(l)}&= \int _{V_{D} \in r \ge r_0}\frac{\hbox {d}^D \mathbf {r} }{\mathbf {r}^{\gamma _l}} \sum _{j \ne i} \delta \Big ( \mathbf {r} - (\mathbf {r}_j - \mathbf {r}_i)\Big ) + \nonumber \\&\int _{V_{D} \in r< r_0} \hbox {d}^D \mathbf {r} \sum _{j \ne i} \delta \Big ( \mathbf {r} - (\mathbf {r}_j - \mathbf {r}_i)\Big ) . \end{aligned}$$

(38)

Note that $dN(\mathbf {r})\equiv \hbox {d}^D \mathbf {r} \sum _{j \ne i} \delta \Big ( \mathbf {r} - (\mathbf {r}_j - \mathbf {r}_i)\Big )$ is the number of individuals which is at the element of (hipper)volume $d^D\mathbf {r}$ at the distance $\mathbf {r}$ from the individual $i$, localized at $\mathbf {r}_i$. In this way, the density of individuals at $\mathbf {r}_i + \mathbf {r}$ (neighbors of $i$), that is $\rho (\mathbf {r}_i~+~\mathbf {r}) =\hbox {d}N(\mathbf {r})/\hbox {d}^D \mathbf {r}$, can be written as

$$\begin{aligned} \rho (\mathbf {r}_i + \mathbf {r}) = \sum _{j \ne i} \delta \Big ( \mathbf {r} - (\mathbf {r}_j -\mathbf {r}_i)\Big ). \end{aligned}$$

(39)

The density of individuals can also be thought of in terms of the scale of the system (in conformity with Falconer 1990). The volume of the system grows in the form $V_{D} \sim L^D$, where $L$ is the typical size of the system. However, the number of individuals grows as the form $N \sim L^{D_{f}}$, where $D_{f}$ is the fractal dimension formed by the spatial structure of the population. By considering $r$, which is the absolute distance from $i$, as a typical distance of the system, one can say that the density of individuals ($V_{D}/N$) has the form

$$\begin{aligned} \rho (\mathbf {r}_i + \mathbf {r}) \equiv \rho (r) = \rho _0 \frac{r^{D_{f}}}{r^D}, \end{aligned}$$

(40)

where $\rho _0$ is a constant which is related to the density of individuals. In fact, if $D=D_{f}$ and the population is homogeneously distributed, then $\rho _0$ is the usual density of individuals.

Using results (40) and (39) in (38), one obtains

$$\begin{aligned} I_i^{(l)} = \rho _0 \int _{V_{D} \in r \ge r_0}d^D \mathbf {r} r^{D_{f} -D -\gamma _l} + \rho _0 \int _{V_{D} \in r < r_0}d^D \mathbf {r} r^{D_{f} -D } . \end{aligned}$$

(41)

Note that the integration argument does not depend on the angular coordinates. Thus, one can write $\hbox {d}^D \mathbf {r}=~r^{D-1}\hbox {d}r \hbox {d}\Omega _{D}$, where $\hbox {d}\Omega _{D}$ is the solid angle, which implies

$$\begin{aligned} I_i^{(l)} = \rho _0\frac{\Omega _{D}}{D_{f}} \int _{0}^{r_0=1} \hbox {d}r r^{D_{f} -1} +\rho _0 \Omega _{D} \int _{r_0=1}^{R_{\hbox {max}}} \hbox {d}r r^{D_{f} -1-\gamma _l}, \end{aligned}$$

(42)

where $\Omega _{D} = \int \hbox {d}\Omega _{D}$. Note that the only term that depends on the Euclidean dimension is the solid angle, and $\Omega _{D}$ assumes the following values according to these tree possibilities: $\Omega _1 = 2$; $\Omega _2 = 2\pi$; $\Omega _3 = 4\pi$. By introducing the constant $\omega _{D} = \rho _0 \Omega _{D}$, which depends only on $D$, one obtains

$$\begin{aligned} I^{(l)} \equiv I_i^{(l)} = \omega _{D} \left( \frac{R_{\hbox {max}}^{D_{f} - \gamma _l} -1 }{D_{f} - \gamma _l}\right) + \frac{\omega _{D}}{D_{f}} \end{aligned}$$

(43)

Thus, $I_i^{(l)}$ does not depend on the label $i$ anymore. As a result, one can say that $I_i^{(l)} = I^{(l)}$ regardless of $i$.

Furthermore, one can introduce the total number of individuals in the relation above by the following thinking. The total number of individuals in the population can be determined by the integral

$$\begin{aligned} N = \int \hbox {d}N(r) =\int _{V_{D}}\hbox {d}^D\mathbf {r} \rho (r). \end{aligned}$$

(44)

Using Eq. (40) and integrating the solid angle, one obtains

$$\begin{aligned} N&= \omega _{D} \int _{0}^{R_{\hbox {max}}} r^{D_{f}-1}\hbox {d}r \end{aligned}$$

(45)

$$\begin{aligned}&= \omega _{D} \int _{0}^{r_0} r^{D_{f}-1} + \omega _{D} \int _{r_0}^{R_{\hbox {max}}} r^{D_{f}-1} \end{aligned}$$

(46)

$$\begin{aligned}&= \omega _{D} \frac{r_0^{D_{f}}}{D_{f}} + \omega _{D} \frac{R_{\hbox {max}}^{D_{f}}}{D_{f}} - \omega _{D} \frac{r_0^{D_{f}}}{D_{f}} . \end{aligned}$$

(47)

Note that the first term on the right in (46) and (47) can be zero (indicating the absence of individuals) or 1 (indicating the presence of a single individual). These values are possible because the ratio of the individual is $r_0$, and hence, there can be at most one individual inside the region that consists of the length between $0$ and $r_0$. Thus, for $r_0=1, \omega _d/D_{f} \sim 1$. $R_{\hbox {max}}$ can be obtained from (47), which is a function of $N$ according to

$$\begin{aligned} R_{\hbox {max}} = \left( \frac{D_{f}}{\omega _{D}} N \right) ^{\frac{1}{D_{f}}}. \end{aligned}$$

(48)

Returning to relation (43), one finds

$$\begin{aligned} I^{(l)} = I^{(l)}(N) = \frac{\omega _{D}}{D_{f}(1- \frac{\gamma _l}{D_{f}})}\left[ \left( \frac{D_{f}}{\omega _{D}}N \right) ^{1- \frac{\gamma _l}{D_{f}}} -1 \right] + \frac{\omega _{D}}{D_{f}}. \end{aligned}$$

(49)

By introducing $\tilde{q}_l=1- \gamma _l/D_{f}$ and the properties of the generalized logarithm (Appendix 1), one obtains

$$\begin{aligned} I^{(l)} = I^{(l)}(N| D, \tilde{q}_l) = \frac{\omega _{D}}{D_{f}} \ln _{\tilde{q}_l} \left( \frac{D_{f}}{\omega _{D}}N \right) + \frac{\omega _{D}}{D_{f}}. \end{aligned}$$

(50)