Radioactive Decay ($dN/dt = - \lambda N$)

The equation $\frac{dN}{dt} = -\lambda N$ holds if

  1. the rate of the decay of any atom at any point of the time is constant and independent of any external factors, like the number of atoms left
  2. there are no other effects influencing the number of atoms other than the radioactive decay

The second one can be easily broken, for example by having the atoms to be produced in another process (as is the case with $\,^{14}C$). The first one is usully true, as the atoms aree usually far enough from each other to not affect their decays. However, in some cases, when they are more concentrated, it is possible for them to affect one another, and something like a chain reaction may happen (even in natural situation, like in the uranium deposits in Oklo.