Population Growth Models: Intrinsic Rate of Increase, Finite Rate of Increase, and a Proof of the Rule of 70

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• Exponential Models of Population Growth
  • Continuous
  • Graph Parameters
  • Discrete
  • Relating λ, r, and R
• Proving the Rule of 70
  • Continuous
  • Discrete
• Logistic Models
  • Graph Parameters
• Conclusion

There are many different ways to simulate population growth. Most of these methods have growth rates of some sort, but despite often having similar representations, these growth rates are often completely different. All of the models here are going to disregard immigration and emigration and treat births and deaths as the only forms of population changes. To start, let's look at the most simple form of population growth, an exponential model.

Exponential Models of Population Growth

Continuous

Given a population size N, a simple exponential model for the growth in N is shown below:

$$N_{\mathrm{t}}=N_0(e{)}^{rt}$$

We could also use the differential form: $$\frac{dN}{dt}=rN$$ Where $N_0$ is the initial population, $N_{\mathrm{t}}$ is the population at any time t, r is the instantanous rate of increase, and e is Euler's number (2.718). In this model, we are assuming that the population change is continous (i.e. that births and deaths happen year round) and that there is no immigration or emigration. Additionally, this model is density independent, meaning that growth is unaffected by population density. The growth rate is directly proportional to the population, resulting in a graph similar to below.

Graph Parameters

r:

r value:

$N_0$:

$N_0$ value:

r is known as the instantanous rate of increase and is the per capita birth rate ($b$) minus the per capita death rate ($d$) at any instant. When at its max value, it is known as the intinsic rate of increase. This function only describes a population that undergoes continuous, exponential growth, such as when resources are unlimited at the start of colonization of a new area. When r is negative, the population decreases at an exponential rate, which follows from our definition of r because if r is negative, $b<d$ and more individuals are dying than being born. Likewise, when r is positive, the population increases at an exponential rate because $b>d$.

Discrete

Alternatively, if organisms only reproduce in discrete intervals, we could rewrite the growth equation to $N_t=N_0(1+R{)}^t$, with R (not the same as r) being the geometric rate of increase and being the amount the population changes between discrete intervals.^[1] We could also rewite the discrete equation to $N_t=N_0{\lambda }^t$, with the finite rate of increase (λ) being $\lambda =(1+R)$. The finite rate of increase means that the population increases by a factor of λ for each unit time.

Relating λ, r, and R

Since both population growth equations give us the population at some time t, we can equate the equations for population growth.

$$N_0{e}^{rt}=N_0(1+R{)}^t$$

From this, we can divide both sides by $N_0$ and take the natural logarithm of both sides to give us the following relation:

$$rt=t\ln (1+R)$$ $$r=\ln (1+R)$$ $$r=\ln \left(\lambda \right)$$

However, despite having a simple relationship, we don't really need to use this because populations are usually not both continuous and discrete.

Proving the Rule of 70

For those unfamiliar, the Rule of 70 states that the doubling time of a population (or the price of an asset in economics) can be estimated by $D=\frac{70}{rate}$, with D being the doubling time and rate being the rate at which the population grows in percent (%). We can prove this pretty easily with both the discrete and continuous cases as follows.

Continuous

Firstly, the doubling time is the time for the population to reach two times the original population, so we can substitute $2N_0$ for the left side of the equation.

$$2N_0=N_0{e}^{rt}$$

Solving for t:

$$2=e^{rt}$$ $$\ln 2=rt$$ $$t=\frac{\ln 2}{r}$$

$\ln 2\approx 0.693$, so, when we convert r to percent by multiplying by 100, we must also multiply the numerator by 100, giving us the final approximation:

$$t=\frac{\ln 2}{r}\approx \frac{69.3}{r_{\mathrm{\%}}}\approx \frac{70}{r_{\mathrm{\%}}}$$

with $r_{\mathrm{\%}}$ being the intrinsic rate of increase in percent (%).

Discrete

Using the same process as before, we substitute in $2N_0$.

$$2N_0=N_0(1+R{)}^t$$ $$2=(1+R{)}^t$$

Taking the natural logarithm of both sides:

$$\ln 2=t\ln (1+R)$$

Using the Maclaurin series for $\ln (1+x)$, we find that $\ln (1+x)=0+x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots \approx x$ for small values of x. Therefore,

$$t=\frac{\ln 2}{\ln \lambda }=\frac{\ln 2}{\ln (1+R)}\approx \frac{\ln 2}{R}\approx \frac{69.3}{R_{\mathrm{\%}}}\approx \frac{70}{R_{\mathrm{\%}}}$$

Logistic Models

Logistic models take into account density dependent effects. This results in the growth rate being proportional to both the current population and the difference between the current and the max population possible, the carrying capacity (K). This gives us the following differential model

$$\frac{dN}{dt}=rN\left(\frac{K-N}{K}\right)=rN\left(1-\frac{N}{K}\right)$$

where N is the population, r is the intrinsic rate of increase, and K is the carrying capacity. The integral form of this equation is

$$N_t=N_0\frac{K}{(K-N_0)e^{-rt}+N_0}$$

This model takes into account density dependent and has a clear carrying capacity. However, it is also limited through its lack of a time lag between population and population growth changes (i.e. the growth rate changes as soon as the population does), along with other limitations.

Graph Parameters

r:

r value:

$N_0$:

$N_0$ value:

K:

K value:

The second and third graphs show the population growth rate $\left(\frac{dN}{dt}\right)$ and per capita population growth rate $\left(\frac{dN}{Ndt}\right)$ respectively plotted against population size.

As seen in the second graph, the highest growth rate is at K/2, which is an important property of logistic models. Another hallmark of the logistic model is the linear graph of per capita population growth rate against population size. Additionally, notice how when $N_0>K$, the growth rate is always negative and increases as population size gets closer to the carrying capacity, illustrating the tendency of a logistic graph to tend towards the carrying capacity.

Conclusion

All of these models have limitations and ultimately are only an extremely simplified model of population growth. However, we can learn a lot about population growth from even these simplified models. Thanks for reading!

^[1] Addendum 2/28/24: While r and R are not the same in concept, we can relate the two with $r=\ln (1+R)$. Therefore, using the Maclaurin Series for $\ln (1+x)$, we see that $r=\ln (R+1)\approx R$ for small values of R. Therefore, at the scales of R found in nature, there isn't much of a difference between the different methods of population growth. Only if R is sufficiently large does the difference become noticable (see this graph of the difference between the two values for a demonstration)