91Ó°ÊÓ

An exponential model of growth follows from the assumption that the yearly rate of change in a population is \((b-d) N\), where \(b\) is births per year, \(d\) is deaths per year, and \(N\) is current population. The increase is in fact to some degree probabilistic in nature. If we assume that population increase is normally distributed around \(r N\), where \(r=b-d\), then we can discuss the probability of extinction of a population. a. If the population begins with a single individual, then the probability of extinction by time \(t\) is given by $$ P(t)=\frac{d\left(e^{r t}-1\right)}{b e^{r t}-d} $$ If \(d=0.24\) and \(b=0.72\), what is the probability that this population will eventually become extinct? (Hint: The probability that the population will eventually become extinct is the limiting value for \(P\).) b. If the population starts with \(k\) individuals, then the probability of extinction by time \(t\) is $$ Q=P^{k}, $$ where \(P\) is the function in part a. Use function composition to obtain a formula for \(Q\) in terms of \(t, b, d, r\), and \(k\). c. If \(b>d\) (births greater than deaths), so that \(r>0\), then the formula obtained in part b can be rewritten as $$ Q=\left(\frac{d\left(1-a^{t}\right)}{b-d a^{t}}\right)^{k} $$ where \(a<1\). What is the probability that a population starting with \(k\) individuals will eventually become extinct? d. If \(b\) is twice as large as \(d\), what is the probability of eventual extinction if the population starts with \(k\) individuals? e. What is the limiting value of the expression you found in part \(\mathrm{d}\) as a function of \(k\) ? Explain what this means in practical terms.

Step by step solution

01

Calculate the Limiting Value of P(t)

For part (a), we need to find the limit of \( P(t) \) as \( t \to \infty \). Given \( P(t)=\frac{d(e^{rt}-1)}{be^{rt}-d} \), substitute \( d=0.24 \) and \( b=0.72 \) into the function. As \( t \to \infty \), the terms involving \( e^{rt} \) dominate, and \( P(t) \to \frac{d}{b} = \frac{0.24}{0.72} = \frac{1}{3} \). Therefore, the population's eventual probability of extinction starting from one individual is \( \frac{1}{3} \).

02

Compose Function for Multiple Individuals

For part (b), use the formula \( Q = P^k \) where \( P \) is the extinction probability from part (a). Substitute \( P = \frac{1}{3} \) to obtain \( Q = \left(\frac{1}{3}\right)^k \) for any integer \( k \) individuals. This expresses \( Q \) as a composition of functions based on \( P \).

03

Rewriting Q Using Function Derived from Part c

Part (c) states to rewrite the formula into \( Q=\left(\frac{d(1-a^{t})}{b-da^{t}}\right)^{k} \). Since this is for cases where \( b > d \) leading \( r > 0 \), which is already given, rewrite using \( P \) from part (a). As \( a = e^{-r} = 0.5 \) (given by problem logic), thus confirming the given structure of \( Q \) aligns with previous portions of the solution.

04

Probability Calculation with Modified Parameters

In part (d), \( b = 2d \) or \( b = 2 \times 0.24 = 0.48 \). Consequently, the limiting extinction probability becomes \( P = \frac{d}{b} = \frac{0.24}{0.48} = \frac{1}{2} \). Hence \( Q = \left(\frac{1}{2}\right)^k \) for a starting population of \( k \) individuals.

05

Understanding the Limiting Expression as k Increases

For part (e), as \( k \to \infty \), \( Q = \left(\frac{1}{2}\right)^k \) approaches zero, indicating that with a sufficiently large number of starting individuals, the probability of eventual extinction decreases towards zero. This outcome implies that larger initial populations have extremely low chances of eventual extinction even with \( b = 2d \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability of Extinction

Understanding the probability of extinction is crucial in studying population dynamics. In simpler terms, the probability of extinction is the chance that a population will eventually disappear. Let's say we start with one individual in a population. We use a mathematical formula to find out this probability over time.

In this context, the formula given for the population's extinction probability is:

• \( P(t) = \frac{d(e^{rt}-1)}{be^{rt}-d} \)

Here:

• \(d\) is the death rate per year.
• \(b\) is the birth rate per year.
• \(t\) is time.
• \(r = b-d\).

The idea is to find this probability when time \(t\) becomes very large (approaches infinity). As time progresses, the influence of exponential growth or decay factors affects these probabilities. If the number \(d/b\) is high, the probability of extinction also rises.

For calculating this probability when starting with multiple individuals, we take a power of \(P\). This part of the solution is where function composition plays a role and leads to finding \(Q = P^k\). The probability decreases as the initial population size increases.

Population Dynamics

The study of population dynamics involves understanding how populations grow, shrink, and evolve over time. Exponential growth is a central concept here, especially when birth rates exceed death rates, allowing populations to grow.

In this scenario, the differential equation behind this model is \(\frac{dN}{dt} = rN\), with:

• \(N\) being the current population.
• \(r = b-d\), the difference between birth and death rates.

This describes how the size of the population changes over time. When \( r > 0 \), the population is expected to grow.

However, real-world populations might face challenges such as limited resources or high predation, which could affect their growth or lead them towards extinction. The dynamics are, therefore, more than just mathematical equations and include biological and ecological considerations.
Additionally, when projecting future population scenarios, environmental factors and random events impact predictions. This unpredictability is modeled using probabilistic approaches to capture the uncertainty in outcomes such as extinction.

Function Composition

Function composition is a key mathematical concept used extensively in different fields, including population dynamics. In this exercise, function composition helps determine the extinction probability when starting with multiple individuals.

The basic idea is to apply a function to the results of another function. For instance, if we have:

• The function \(P(t)\) gives the extinction probability for one individual.
• Then, \(Q = P^k\) gives the probability for \(k\) individuals.

This is a clear example of function composition, where the entire population's probability depends on the initial individual's probability raised to the power of the number of individuals.

In mathematical terms, if you have a function \(f\) and another function \(g\), the composition \(f \circ g\) means to apply \(g\) first and then \(f\) to \(g\)'s output. Through this composition, we capture how a small, individual probability influences a larger collective scenario. Understanding this helps to accurately predict population outcomes over time.