Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.


a. Suppose P=10, A=20, and r=0. If the initial population is N(0)=10, does the population ever become extinct? Explain.

Verified step by step guidance

1. Identify the given growth rate function: \(N'(t) = r + A \sin \left( \frac{2\pi t}{P} \right)\), with \(P=10\), \(A=20\), and \(r=0\).
2. Write the differential equation explicitly with the given values: \(N'(t) = 0 + 20 \sin \left( \frac{2\pi t}{10} \right) = 20 \sin \left( \frac{\pi t}{5} \right)\).
3. To find the population function \(N(t)\), integrate the growth rate \(N'(t)\) with respect to \(t\): \(N(t) = N(0) + \int_0^t N'(s) \, ds = 10 + \int_0^t 20 \sin \left( \frac{\pi s}{5} \right) ds\).
4. Compute the integral: \(\int 20 \sin \left( \frac{\pi s}{5} \right) ds = - \frac{100}{\pi} \cos \left( \frac{\pi s}{5} \right) + C\). Apply the limits from 0 to \(t\) to get: \(N(t) = 10 - \frac{100}{\pi} \left[ \cos \left( \frac{\pi t}{5} \right) - \cos(0) \right]\).
5. Analyze the expression for \(N(t)\) to determine if it ever reaches zero or below for \(t > 0\). Since \(\cos(0) = 1\), rewrite \(N(t)\) as: \(N(t) = 10 - \frac{100}{\pi} \left( \cos \left( \frac{\pi t}{5} \right) - 1 \right) = 10 + \frac{100}{\pi} \left( 1 - \cos \left( \frac{\pi t}{5} \right) \right)\). Because \$1 - \cos(\theta) \geq 0\( for all \)\theta\(, \)N(t)$ is always greater than or equal to 10, so the population never reaches zero and thus does not become extinct.

Key Concepts

Differential Equations and Population Growth

This concept involves understanding how the rate of change of a population, given by N'(t), relates to the population size N(t). The differential equation describes how the population evolves over time based on growth rates, which can be constant or variable. Solving or analyzing such equations helps predict population behavior, including extinction events.

Oscillatory Functions and Periodicity

Oscillatory functions like sine introduce periodic fluctuations in growth rates, characterized by a fixed period P. Understanding how the sine function affects the growth rate over time is crucial to analyze when and how the population growth rate changes sign, influencing population increases or decreases cyclically.

Extinction Criteria and Population Thresholds

Extinction occurs if the population reaches zero at any time after t=0. This concept requires analyzing the population trajectory to determine if and when N(t) hits zero. It involves integrating the growth rate and considering initial conditions to assess whether the population can sustain itself or decline to extinction.