Question

Consider a population of organisms living in some bounded environment, say the Earth. Let \(X_{n}\) be the number of organisms alive at time \(n\) and observe that \(\\{0\\}\) is an absorbing state, \(X_{n}=0\) implies \(X_{n+m}=0\) for all \(m\). It is reasonable to suppose that for every \(N\) there exists \(\delta>0\) satisfying $$ \operatorname{Pr}\left[X_{n+1}=0 \mid X_{1}, \ldots, X_{n}\right] \geq \delta, \quad \text { if } \quad X_{n} \leq N $$ \(n=1,2, \ldots\) Let \(\&\) be the event of eventual extinction $$ \varepsilon=\left\\{X_{k}=0 \text { for some } k=1,2, \ldots\right\\} $$ Show that with probability one, either \(\&\) occurs or else \(X_{n} \rightarrow \infty\) as \(n \rightarrow \infty\). Since the latter cannot occur in a bounded environment, eventual extinction is certain.

Short answer

Using mathematical induction on the given inequality, we demonstrated that with probability one, either the event of eventual extinction \(\varepsilon\) occurs, or the population size \(X_n\) tends to infinity as the time goes to infinity. However, since the environment is bounded, the population size going to infinity is impossible. Therefore, we conclude that eventual extinction is certain in a bounded environment.

Step-by-step solution

First, let's look at the given inequality: $$ \operatorname{Pr}\left[X_{n+1}=0 \mid X_{1}, \ldots, X_{n}\right] \geq \delta, \quad \text { if } \quad X_{n} \leq N $$ This inequality tells us that the conditional probability of the population size decreasing to zero given the population size at time \(n\) is less than or equal to \(N\), is always greater than the positive number \(\delta\). #Step 2: Analyze the event of eventual extinction#

Let \(\varepsilon\) be the event of eventual extinction, $$ \varepsilon=\left\{X_{k}=0 \text { for some } k=1,2, \ldots\right\} $$ We want to show that with probability one, either \(\varepsilon\) occurs or the population size goes to infinity as the time goes to infinity. #Step 3: Use mathematical induction to prove the statement#

Let's use mathematical induction on \(n\) to prove that either \(\varepsilon\) occurs or \(X_n \rightarrow \infty\) as \(n \rightarrow \infty\). We start with the base case for \(n=1\): Base case: If \(X_n \leq N\) for \(n=1\), the probability of extinction is at least \(\delta\). Then, from the given inequality, we have $$ \operatorname{Pr}\left[X_{2}=0 \mid X_{1}\right] \geq \delta $$ Inductive step: Assume the statement is true for \(n=k\), i.e., $$ \operatorname{Pr}\left[X_{k+1}=0 \mid X_{1}, \ldots, X_{k}\right] \geq \delta $$ when \(X_k \leq N\). Then, we want to show that the statement is also true for \(n=k+1\). We have the given inequality: $$ \operatorname{Pr}\left[X_{k+2}=0 \mid X_{1}, \ldots, X_{k+1}\right] \geq \delta, \quad \text { if } \quad X_{k+1} \leq N $$ Now, given the assumption for \(n=k\), it follows that with probability 1, either \(\varepsilon\) occurs at an earlier time, or the population \(X_{k+1} \rightarrow \infty\) when \(k\rightarrow \infty\). In either case, the statement holds for \(n=k+1\). This completes the induction. #Step 4: Connect the induction result to the conclusion#

From the induction on \(n\), we have proven that either the event of eventual extinction \(\varepsilon\) occurs, or the population size \(X_n\) tends to infinity as the time goes to infinity. Since the environment is bounded, it is not possible for the population size to go to infinity. Therefore, the only possibility is that the event of eventual extinction (\(\varepsilon\)) occurs with probability one. This proves that eventual extinction is certain in a bounded environment.

Key concepts

Stochastic Processes

A stochastic process is a mathematical concept that represents a collection of random variables ordered in time. These variables can describe a system or phenomenon that is subject to random changes. In the case of our population of organisms, the stochastic process is represented by the sequence \(X_n\), where each \(X_n\) is the random variable indicating the number of organisms alive at time \(n\).

The study of stochastic processes provides valuable insights into the behavior of dynamic systems that evolve over time under the influence of random factors. Examples include stock market fluctuations, signal processing, and the growth or decline of populations, as seen in the given exercise. By analyzing stochastic processes, we're able to make probabilistic statements about future events, such as the likelihood of a population’s extinction.

Absorbing States

In the domain of stochastic processes, an absorbing state is a condition that, once entered, cannot be left. In our population example, the state where \(X_n = 0\) signifies that the population has become extinct, and naturally, it’s impossible for this state to transition to a state with a living population.

It’s crucial to understand the concept of absorbing states to correctly interpret models of real-world phenomena where certain outcomes are final. For instance, in Markov processes, which are a specific type of stochastic process without memory of past states, the absorbing state holds particular importance as it dictates the ultimate fate of the system. In many models, the long-term behavior is characterized by the probability of reaching such absorbing states.

Mathematical Induction

Mathematical induction is a proof technique used to establish that a statement holds true for all natural numbers. It operates in two steps: first, proving the 'base case' to show the statement is true for the initial number, often \(n=1\); then, proving the 'inductive step' by assuming the statement holds for \(n=k\) and showing it must also hold for \(n=k+1\).

In the context of our exercise, mathematical induction is employed to show that the stochastic process concerning the population's extinction will conclude in one of two ways: either the population extinguishes (\