So, I just started doing differential equations and I'm trying to understand the basic idea of dN/dt = kN. I understand the whole process of solving a differential equation (the separation of variables, isolating N, finding k etc...) and I'm getting proficient at it. However, the only thing I'm still having difficulty grappling with is why we say that dN/dt = kN to begin with.
The examples used in my textbook:
Radioactive decay. After 3 days, 50 percent of the radioactivity produced by a nuclear explosion has disappeared. How long does it take for 99 percent of this radioactivity to disappear? The rate of change of the mass of our substance is negative, and is proportional at each moment to the mass of the substance at that moment. This statement means that if x = x(t) is the mass of the radioactive substance at time t, then dx/dt = -kx (k > 0).
Population growth. Consider a laboratory culture of bacteria with unlimited food and no enemies. If N = N(t) denotes the number of bacteria present at time t, it is natural to assume that the rate of change of N is proportional to N itself, or dN/dt = kN (k > 0). If the number of bacteria present at the beginning is N_0, and this number doubles after 2 hours (the “doubling time”), how many are there after 6 hours? After t hours?
There's always this idea present in these problems that the rate of change of _____ is proportional at each moment in time to _______ at that moment, or dN/dt = kN. Both mathematically and intuitively, why is this relationship "natural to assume"? What (if any) mathematical property dictates this?
P.S. It might also be worth noting that I don't know anything about population growth outside the context of the ODEs I've been solving.
$\endgroup$1 Answer
$\begingroup$If $N$ represents the population size, then how the population changes will depend on the population size, i.e. the rate of change is a function of $N$\begin{align} \frac{dN}{dt} = f(N). \end{align}In this model, we are also assuming that the rate of change does not explicitly depend on time because whether it is current time or far into the future the only factor that contributes to the change of population should only be determined by its size.
The simplest model is if the rate is proportional to the population, i.e. $f$ is linear.
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