5. Exponential Growth

Abstract

In the previous chapter, we introduced our first population dynamics model, incorporating deaths alone, and demonstrated that this leads to exponential decay. Here, we expand the model to include both births and deaths. Considering these combined processes, we show that populations can either grow or decline exponentially, depending on whether the birth rate exceeds the death rate or vice versa. We delve more deeply into the phenomenon of exponential growth, investigating its remarkable properties. To build intuition, we first consider a thought experiment exploring the explosive potential of exponential growth in a bacterial culture. We then examine some key mathematical underpinnings, showing how exponential functions exhibit geometric behavior when evaluated over arithmetic sequences. Understanding these fundamental characteristics provides critical insight into how even modest growth rates may translate to astonishingly rapid increases over long timescales. In this chapter, our aim is to develop a richer appreciation for both the power and limitations of exponential growth dynamics.

Introduction¶

In the previous chapter, we examined how deaths influence population dynamics. Now, we will expand our study to incorporate the impact of births. It is important to consider that, in the absence of migration, new individuals arise in the population solely through the reproduction of existing individuals, as spontaneous generation is not possible. Therefore, we can modify Eq. (6) to accommodate births as follows:

In this equation, the term $\partial \eta / \partial t$ represents the rate of change of the population density $\eta$ with respect to time $t$. The first term on the right-hand side, $-\partial \eta / \partial \tau$, accounts for the effect of aging on the population. The second term, $\beta(\tau) \eta$, introduces the influence of births, where $\beta(\tau)$ represents the birth rate as a function of age ($\tau$). Finally, the third term, $-\mu(\tau) \eta$, considers the mortality rate $\mu(\tau)$, which represents the rate at which individuals of age $\tau$ die.

Based on the findings presented in the previous chapter, and assuming that the probability density $\rho(\tau, t) = \nu(\tau, t) / N(t)$ remains stationary, we can integrate Eq. (1) to obtain the following expression:

where $\widetilde{\beta}$ and $\widetilde{\mu}$ represent the average birth and death rates, respectively. These parameters are defined as $\widetilde{\beta} = \int_0^\infty \beta(\tau') \rho(\tau') d\tau'$ and $\widetilde{\mu} = \int_0^\infty \mu(\tau') \rho(\tau') d\tau'$. The population size is represented by $N(t) = \int_0^\infty \eta(\tau') d\tau'$. It is worth noting that, if both $\beta$ and $\mu$ are age-independent, we obtain the same result without requiring the stationarity of $\eta$.

By introducing an effective growth rate $\alpha = \widetilde{\beta} - \widetilde{\mu}$, we can reformulate Eq. (2) as follows:

This differential equation has a solution given by:

where $N_0$ represents the initial population size. If the death rate surpasses the birth rate, as studied in the previous chapter, the population would decay exponentially. However, if the opposite scenario occurs, where the birth rate surpasses the death rate, the population would experience exponential growth.

Explosiveness of Exponential Growth¶

To develop an intuitive understanding of exponential growth, let’s consider a thought experiment involving an E. coli bacterial culture initiated from a single bacterium, assuming it can grow exponentially indefinitely. In such a scenario, the time evolution of the culture’s mass is governed by the following equation:

where $m_b$ represents the initial mass of an individual bacterium, and $\alpha$ denotes the growth rate.

Now, let’s explore how long it would take for the culture’s mass to reach the magnitude of planet Earth ($M_E \approx 6 \times 10^{24}$ kg). Given that an individual bacterium weighs approximately 10^-15 kg and an E. coli culture can double its mass every 30 minutes, we can solve for $t$ in Eq. (5):

By substituting $m(t) = M_E$ and $\alpha = \ln(2) / 30 \, \text{min}^{-1}$, we can calculate the required time:

In other words, an exponentially growing bacterial culture starting from a single bacterium would accumulate a mass equal to that of planet Earth in less than 3 days. This thought experiment vividly illustrates the remarkable explosive potential of exponential growth.

Geometric Sequences and the Exponential Function¶

To comprehend the explosive nature of exponential growth, it is valuable to explore a fundamental property of the exponential function. Let us consider the function defined as:

Now, suppose we have a value $x_2$ such that:

In other words:

By solving for $x_2$ in the equation above, we find:

Notably, the value of $x_2$ remains constant regardless of the initial value of $x$. This implies that, irrespective of the starting point, the value of $f(x)$ doubles when a value of $x_2$ is added to the argument. Consequently, if we construct an arithmetic sequence of the form:

the sequence:

will exhibit geometric behavior ($y_{i+1} = 2 y_i$). The remarkable growth potential of geometric sequences has captivated humanity for centuries, giving rise to legends as the one illustrated in Chapter 2.

Discussion¶

While the exponential growth model provides valuable insight into population dynamics, it has important limitations when considered over long time periods. As we have seen, even tiny growth rates predict populations will eventually exceed all available resources as their mass approaches infinity. In reality, finite carrying capacities constrain population expansion. Additionally, exponential growth assumes resources are unlimited and external influences negligible, violating the basic principles of resource competition and natural regulation within closed ecological systems. In a forthcoming chapter, we will introduce a more realistic population model called the logistic model. The logistic equation incorporates density-dependent effects like resource competition and saturation, allowing populations to stabilize at an environmental carrying capacity. Studying the logistic model will provide a more nuanced understanding of how biotic and abiotic factors interact to govern long-term population persistence in nature. This will lay the foundation for exploring more complex dynamic interactions between species in ecological communities.

Exercises¶

1. A geometric progression ($S_n$) is defined as a sequence of non-zero terms where each subsequent term is obtained by multiplying the previous term by a fixed common ratio ($r$):

   $$S_{n+1} = r S_n$$

   (14)

   An exponential function $\left(f(x) = C e^{a x}\right)$ models continuous growth or decay at a constant rate ($a$). Show that the geometric progression provides a discrete analogue of the exponential function by finding the relation between the common ratio $r$ and the exponent parameter $a$ that allows the geometric progression terms $S_n$ to equal the discrete function values $f(n)$: $S_n = f(n)$.

2. In the previous exercise, we showed that a geometric progression provides a discrete analogue of the exponential function when a specific relationship between their parameters exists. Now consider how the concepts of limits and convergence apply to both of these mathematical objects. Specifically find what conditions must be fulfilled for a geometric sequence and exponential function to converge to zero as the index/input-variable approaches infinity? Relate the convergence condition(s) back to the relationship defined between the common ratio $r$ and exponent parameter $a$ in the prior exercise.