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In my textbook the exponential growth rate is described as

$$W'= W e^{rt}.$$

Here $W'$ is the final size, $W$ initial size, $r$ growth rate and $t$ time.

I wanted to have a clear mathematical understanding. So far according to what I understand, if the size is growing by 10% per unit of time, then the growth rate in the above function should be

$$r=\ln(1.1).$$

Am I right? Considering infinite resources.

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  • $\begingroup$ So why don't we just write it as W λ^t? I have some faint idea that it's to represent that it's a derivative of itself and that it's change in value depends upon itself but I am not sure how that works exactly. $\endgroup$
    – user145522
    Commented Sep 5 at 5:20
  • $\begingroup$ I've been doing some brainstorming to figure out why we write this function in "e" form, so please tell me if I'm right. It's just to make the calculations of rate a bit more standard (as the rate is changing because it depends on it's initial value, even though the percentage change is same). Without e we would have to find the proportionality factor for every individual percent change to figure out what the rate was. Right?? $\endgroup$
    – user145522
    Commented Sep 5 at 6:46

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My comments disappeared, so I'll explain here:

There are two forms of the exponential growth model, in discrete time or continuous time.

A discrete-time model is most appropriate for a population that grows at discrete times in regular intervals, such as an animal that reproduces during a specific breeding season. We only consider t = 0, 1, 2, etc., and not the times in-between. The discrete-time exponential growth model is derived from N(t + 1) = λ N(t), where N(t + 1) is the population 1 time step after N(t) - that is, the population at any one time step is λ times the population at the previous time step. From this, we get N(t) = N(0) λ^t.

A continuous-time model is appropriate when growth happens continuously, without seasonality, etc. - perhaps a human population - or when time steps are negligible relative to the time period of interest. The continuous-time exponential growth model is dN(t)/dt = r N(t) - that is, the population grows at a rate of r times the population. By solving the differential equation, we get N(t) = N(0) e^(rt). Euler's constant comes up because of its unique property, d[e^x]/dx = e^x, so dN(t)/dt = d[N(0) e^(rt)]/dx = r N(0) e^(rt) = r N(t)

These two models are equivalent when r = ln(λ). (Note that the symbols r and λ are commonly, but far from always used for these values.) So whether we choose to express the model in as N(t) = N(0) λ^t or N(t) = N(0) e^(rt) comes down to whether it is more intuitive to think about the growth rate in discrete time (λ) or continuous time (r).

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Generally you can rearrange the exponential growth equation to get

r = ln(W'/W) / t.

So indeed if time t = 1 and W'/W = 1.1, then your growth rate would be

r = ln(1.1).

For example in microbiology frequently also the doubling time tD is used, thereby W'/W = 2 and growth rate r = ln(2) / tD.

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