Population extinction
Extinction occurs when population size reaches zero. In mathematical models, extinction can arise through deterministic decline, threshold effects or random fluctuations.
Deterministic extinction
If a model gives persistent negative growth, the population may approach or reach zero. For example, exponential decline with \(r<0\) gives
\[N(t)=N_0e^{rt}.\]This approaches zero asymptotically in a continuous deterministic model.
Threshold-driven extinction
With a strong Allee effect, populations below the threshold \(A\) are driven toward extinction even when carrying capacity is positive.
Stochastic extinction
In finite populations, random births and deaths can cause extinction even when the average trend predicts growth. Extinction probability and time to extinction therefore become important stochastic quantities.
Biological importance
Extinction analysis is central to conservation, invasive-species control and epidemic fade-out. The appropriate mathematical definition depends on whether the model uses continuous population densities or discrete individual counts.