Exponential Growth
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You should be comfortable with basic calculus, specifically differentiating exponential functions and understanding the concept of a derivative as an instantaneous rate of change. Familiarity with the definition of population density as the number of individuals per unit area or volume is also assumed.
Why this matters
Exponential growth describes the theoretical maximum potential of a population when resources are unlimited.
In ecology, understanding exponential growth provides the baseline against which all real-world population dynamics are measured. While no natural population grows exponentially forever due to resource limitations, this model is crucial for understanding rapid population expansions in new environments, the initial phases of epidemics, and the mathematical foundations of the logistic growth model. It highlights the powerful effect of compounding, where the growth rate itself increases as the population size increases, leading to a characteristic J-shaped curve that can overwhelm ecosystems if unchecked.
The Concept of Exponential Growth
Imagine a population of bacteria in a fresh petri dish with abundant nutrients. Initially, there are few individuals, so the total number of new births per hour is small. However, as the population grows, there are more individuals to reproduce. If every individual has the same probability of surviving and reproducing, and if there are no limits on food or space, the population size increases by a constant proportion in each time step. This is the essence of exponential growth. Unlike linear growth, where the population adds a fixed number of individuals per unit time, exponential growth adds a fixed percentage. This means the absolute rate of change, or the number of new individuals added per unit time, is not constant; it is proportional to the current population size. The larger the population, the faster it grows. This creates a feedback loop where growth fuels further growth. In mathematical terms, this is described by a differential equation where the derivative of the population size with respect to time is equal to a constant rate multiplied by the population size itself. The result is a curve that starts shallow but becomes increasingly steep, resembling the letter J. This model assumes an idealized world with no competition, no predation, and no disease, which is a simplification but a necessary one to understand the fundamental reproductive potential of a species, known as the biotic potential.
Mathematical Formulation
To model this process quantitatively, we define \(N(t)\) as the population size at time \(t\). The core assumption of exponential growth is that the per-capita growth rate is constant. Let \(r\) represent the intrinsic rate of increase, which is the difference between the per-capita birth rate and the per-capita death rate. If \(r\) is positive, the population grows; if negative, it declines. The total rate of change of the population, \(dN/dt\), is therefore the product of the current population size and the per-capita rate. This yields the differential equation \(dN/dt = rN\). Solving this equation involves separating variables and integrating, leading to the general solution \(N(t) = N_0 e^{rt}\), where \(N_0\) is the initial population size at \(t=0\). The constant \(e\) is the base of the natural logarithm, approximately 2.718. This equation shows that population size is an exponential function of time. For example, if \(r = 0.1\) per day, the population grows by a factor \(e^{0.1} = 1.105\) each day (slightly more than 10%, because growth compounds continuously): after one day \(N = 1.105 N_0\), after two days \(N = 1.221 N_0\). The compounding effect is evident as the multiplier \(e^{rt}\) grows rapidly with time. This formulation is continuous, meaning it assumes reproduction happens continuously rather than in discrete generations, which is a reasonable approximation for many microorganisms and small mammals with overlapping generations.
Interpreting the Growth Curve
The graph of \(N(t)\) against \(t\) produces a characteristic J-shaped curve. The slope of this curve at any point represents the absolute growth rate, \(dN/dt\). At \(t=0\), the slope is \(rN_0\). As \(t\) increases, the slope increases because \(N\) increases. This means the population is not just growing, but accelerating. The concept of doubling time is particularly useful here. The time \(t_d\) required for the population to double can be derived by setting \(N(t_d) = 2N_0\). Substituting into the exponential equation gives \(2N_0 = N_0 e^{rt_d}\), which simplifies to \(2 = e^{rt_d}\). Taking the natural logarithm of both sides yields \(\ln(2) = rt_d\), so \(t_d = \ln(2)/r \approx 0.693/r\). This relationship shows that the doubling time is inversely proportional to the growth rate. A high \(r\) leads to a short doubling time. For instance, if \(r = 0.693\) per hour, the population doubles every hour. This rapid expansion illustrates why exponential growth is so powerful and potentially destructive in ecological contexts. It also highlights the sensitivity of population dynamics to small changes in \(r\). A slight increase in the birth rate or a decrease in the death rate can significantly reduce the doubling time, leading to much faster population expansion.
Ecological Implications and Limits
While exponential growth is a fundamental model, it is an idealization. In reality, resources such as food, space, and light are finite. As the population size \(N\) increases, the per-capita availability of these resources decreases. This leads to increased competition, which typically reduces the birth rate and/or increases the death rate. Consequently, the per-capita growth rate \(r\) is not constant in nature; it declines as density increases. This transition from exponential to logistic growth is a central theme in ecology. However, exponential growth is still observed in specific scenarios: when a species colonizes a new, resource-rich environment (such as an island or a new habitat after a disturbance), when a pathogen enters a susceptible host population, or during the early stages of a population's life cycle. Understanding the exponential phase is critical for predicting the initial impact of invasive species or the spread of epidemics. It also serves as the null model for population dynamics; any deviation from exponential growth indicates the presence of density-dependent factors. The parameter \(r\) is often referred to as the biotic potential, representing the maximum reproductive capacity of a species under optimal conditions. Comparing the \(r\) values of different species allows ecologists to classify them as r-strategists (high \(r\), rapid growth, small body size) or K-strategists (low \(r\), slow growth, large body size), which has profound implications for their life history strategies and ecological roles.
The concept
Logistic population growth
Let \(N(t)\) be the size of a single closed, unstructured, well-mixed population, treated as a differentiable function of continuous time, and suppose that the per-capita growth rate declines affinely with density, \(N^{-1}\,dN/dt=r\left(1-N/K\right)\), with the intrinsic rate of increase \(r\gt 0\) and the carrying capacity \(K\gt 0\) both constant, the response to density instantaneous, and no immigration, emigration, demographic stochasticity or Allee effect. Then \(N\) obeys the logistic equation \(dN/dt=rN\left(1-N/K\right)\), whose solution through any \(N_0\gt 0\) exists, is unique, and is \(N(t)=K\left[1+\left(\left(K-N_0\right)/N_0\right)e^{-rt}\right]^{-1}\). The state \(N=K\) is globally asymptotically stable on \(\left(0,\infty\right)\) with linear relaxation rate \(r\), the state \(N=0\) is unstable with rate \(+r\); for \(0\lt N_0\lt K\) the trajectory is strictly increasing, and for \(0\lt N_0\lt K/2\) it is sigmoid, with a single inflection at \(N=K/2\), at which the absolute growth rate attains its maximum \(rK/4\) — a trajectory started at or above \(K/2\) is concave throughout and has no inflection; and the logit \(\ln\left[N/\left(K-N\right)\right]\) is exactly linear in \(t\) with slope \(r\).
A common misconception is that exponential growth implies a constant absolute growth rate. In reality, the per-capita rate is constant, but the absolute rate \(dN/dt\) increases linearly with \(N\). Confusing these two rates leads to incorrect predictions about population size over time. For example, assuming a constant addition of 10 individuals per day when the actual process is exponential will significantly underestimate the population size after several days. This distinction is crucial for accurate modelling and management of populations. C
Worked example
Problem. A population of yeast cells in a nutrient-rich medium has an initial size \(N_0 = 100\) cells. The intrinsic rate of increase is \(r = 0.5\) per hour. Calculate the population size after 4 hours and determine the time required for the population to double.
- Identify the given values: \(N_0 = 100\), \(r = 0.5\ \mathrm{h}^{-1}\), and \(t = 4\ \mathrm{h}\).
- Use the exponential growth equation \(N(t) = N_0 e^{rt}\) to find the population size at \(t=4\).
- Substitute the values: \(N(4) = 100 \times e^{0.5 \times 4}\).
- Calculate the exponent: \(0.5 \times 4 = 2\). Then \(e^2 \approx 7.389\).
- Compute \(N(4) = 100 \times 7.389 = 738.9\). Round to the nearest whole number: 739 cells.
- Calculate the doubling time \(t_d\) using the formula \(t_d = \ln(2)/r\).
- Substitute \(r = 0.5\): \(t_d = 0.693 / 0.5 = 1.386\) hours.
Units check. The rate \(r\) is in \(\mathrm{h}^{-1}\), time \(t\) is in \(\mathrm{h}\), so the exponent \(rt\) is dimensionless, which is correct for the exponential function. The population size is in cells, and the doubling time is in hours, consistent with the input units.
Three quick tests
- If the intrinsic rate of increase \(r\) doubles, what happens to the doubling time \(t_d\)?
Answer
It halves, because \(t_d\) is inversely proportional to \(r\). - Is the absolute growth rate \(dN/dt\) constant in an exponentially growing population?
Answer
No, it increases as the population size increases. - What is the shape of the growth curve for a population undergoing exponential growth?
Answer
J-shaped.
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Lecture notes checked line by line by the editor; every number in the worked example was recomputed.