Population extinction

Extinction means that no individuals remain in a population. Mathematically, however, the way extinction appears depends strongly on whether population size is represented as a continuous quantity or as a discrete number of individuals.

Core idea. A population can disappear because its average growth is negative, because it crosses a biological threshold, or because random births and deaths happen unfavourably. These are different mechanisms and should not be confused.

What does extinction mean mathematically?

For a population counted as individuals, extinction occurs when

\[N(t)=0.\]

Once the last individual has disappeared, ordinary birth processes cannot restart the population unless immigration or reintroduction is included. Zero is therefore often an absorbing state.

In a deterministic continuous model, however, \(N(t)\) can take non-integer values such as 0.4 or 0.01. This creates an important distinction between mathematical decline toward zero and literal biological extinction.

Deterministic exponential decline

Consider

\[\frac{dN}{dt}=rN,\qquad r\lt0.\]

Its solution is

\[N(t)=N_0e^{rt}.\]

The population becomes smaller and smaller, but for finite \(t\), the mathematical solution remains positive.

Important. In this continuous deterministic model, the population approaches zero asymptotically. It does not literally reach zero at a finite time. A statement such as “the population becomes extinct at time \(t\)” therefore requires an additional extinction threshold or a discrete stochastic model.

Approaching zero is not the same as reaching zero

Deterministic exponential decline approaches zero continuously. The curve can become arbitrarily small without mathematically reaching zero at a finite time.

A practical extinction threshold

In continuous population models, researchers sometimes define a practical threshold \(N_c\). The population is treated as effectively extinct when

\[N(t)\leq N_c.\]

The choice of \(N_c\) depends on the biological interpretation. It is a modelling convention rather than a consequence of the differential equation itself.

Threshold-driven extinction: the Allee effect

A strong Allee effect creates an unstable threshold \(A\). A common model is

\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)\left(\frac{N}{A}-1\right).\]

When \(N\) is between \(A\) and \(K\), the population tends to grow toward \(K\). When \(N\) is below \(A\), its deterministic direction is toward extinction.

Why? At very low density, individuals may have difficulty finding mates, defending themselves, cooperating or reproducing successfully. Small population size can therefore reinforce further decline.

The extinction threshold changes the outcome

With a strong Allee effect, populations starting above the threshold can move toward carrying capacity, while populations starting below it move toward zero.

Deterministic extinction and stochastic extinction are different

A deterministic model gives one trajectory from a given initial condition. If that trajectory grows, the deterministic model predicts persistence.

A stochastic model describes many possible trajectories. Some can grow while others decline to zero even though they all begin from exactly the same initial population.

Why randomness matters especially in small populations

Suppose a population contains 10 individuals. Losing three individuals by chance is a 30% reduction. If the population contains 10,000 individuals, losing three has almost no effect on the total.

Random individual events therefore have proportionally larger effects when populations are small. This is one reason extinction risk can become substantial near zero.

A simple birth–death picture

Suppose each individual gives birth at per-capita rate \(b\) and dies at per-capita rate \(d\). With \(N\) individuals, the total event rates are

\[\text{birth rate}=bN,\qquad \text{death rate}=dN.\]

The average deterministic direction is

\[\frac{dN}{dt}=(b-d)N.\]

If \(b\gt d\), this average direction is upward. But a finite stochastic population can still experience a sequence of deaths before enough births occur and eventually reach zero.

This is the central stochastic idea. Positive expected growth does not mean every population trajectory grows.

Many possible population histories

Illustrative stochastic birth–death trajectories generated from the same initial population. Random event sequences produce different outcomes; some populations can hit the absorbing state at zero while others grow.

Extinction probability

Because different stochastic trajectories have different outcomes, extinction becomes a probability question.

\[q_i=\Pr(\text{eventual extinction}\mid N(0)=i).\]

Here \(q_i\) means the probability that the population eventually reaches zero when it begins with \(i\) individuals.

For a simple continuous-time linear birth–death process with constant per-capita rates \(b\) and \(d\),

\[q_i=\begin{cases}1,&b\leq d,\\\left(\dfrac{d}{b}\right)^i,&b\gt d.\end{cases}\]

Thus, when births do not exceed deaths, eventual extinction occurs with probability 1. When \(b\gt d\), extinction is still possible, but its probability decreases as the initial population becomes larger.

Why starting population size matters

If one individual starts a supercritical birth–death population, an early death immediately causes extinction. Starting with many individuals gives the population more opportunities for births to occur before everyone disappears.

This is why extinction probability commonly falls as initial population size increases.

Probability of extinction versus time to extinction

These answer different questions.

Quantity	Question
extinction probability	How likely is the population eventually to disappear?
extinction probability by time \(T\)	How likely is extinction before a specified time?
time to extinction	How long does it take to reach zero?
mean time to extinction	What is the expected waiting time until extinction?

Absorbing states

In a birth–death Markov chain, state 0 is normally absorbing:

\[P(0\to0)=1.\]

Once extinction occurs, the process remains at zero unless the model explicitly includes immigration, mutation, reintroduction or another mechanism capable of recreating the population.

Environmental stochasticity

Randomness can affect more than individual births and deaths. Weather, food availability, temperature, habitat quality and other environmental conditions can change the demographic rates experienced by the whole population.

A run of poor environmental years can therefore produce extinction even when long-term average conditions appear favourable.

Demographic versus environmental stochasticity

Demographic stochasticity	Environmental stochasticity
randomness in individual births, deaths and transitions	random variation in conditions affecting many individuals
especially important in small populations	can remain important even for large populations
arises because individuals experience discrete random events	arises because model parameters or conditions vary through time

Extinction and harvesting

Harvesting can reduce a population toward a region where demographic randomness becomes important. Constant harvesting can also create a deterministic lower threshold. If an Allee effect is present, harvesting below the Allee threshold can trigger continued decline even after harvesting stops.

Extinction in structured populations

Population size alone may not determine extinction risk. A population containing many individuals but almost no reproductive adults can have a very different future from one of the same size with a healthy age or stage distribution.

Age- and stage-structured models can therefore be extended to calculate extinction risk while accounting for demographic composition.

Extinction and conservation

Conservation questions are often naturally probabilistic: What is the probability of extinction within 50 years? How much must adult survival improve to reduce that probability? How large should a reintroduced population be?

Such questions cannot generally be answered by a single deterministic equilibrium alone.

Extinction and invasive species

The same mathematics can be viewed from the opposite management perspective. For an invasive population, extinction may be the desired outcome. Models can estimate how control, harvesting or reduced reproduction changes the probability and expected time to eradication.

Connection with epidemics

At the beginning of an epidemic, the infectious population can be small. Even when the reproduction potential is high enough for average growth, early random recoveries may eliminate all infectious individuals before a large outbreak develops.

This is the epidemic analogue of stochastic population extinction and is often called early epidemic fade-out.

Which model should be used?

Question	Useful approach
What is the average direction of population change?	deterministic ODE model
Is there a deterministic threshold?	equilibrium and stability analysis
What is the probability of extinction?	stochastic birth–death or Markov model
How long until extinction?	first-passage or time-to-extinction analysis
How does environmental variability affect risk?	stochastic environmental model or simulation

Key idea. Extinction is not simply “population size becoming small.” Deterministic models describe systematic decline and thresholds, while stochastic models allow a finite population to reach exactly zero through random events. For conservation, invasion control and epidemic fade-out, the probability and timing of extinction can be as important as the average population trajectory.