Problem 26 - Calculus for Biology and Medicine

Question

Find the recursion for a population that has a reproductive rate of \(1 / 3\) and that has 63 individuals at time \(0 .\)

Step-by-Step Solution

Answer

The recursion formula is \( P_{t+1} = P_t \times \frac{1}{3} \) with \( P_0 = 63 \).

Step 1: Understanding the problem

We are tasked with finding the recursion formula for a population where the reproductive rate is \( \frac{1}{3} \) and the initial population size at time \( t = 0 \) is 63 individuals. Our goal is to create a formula that represents the population size at any time \( t \).

Step 1: Define variables

Let \( P_t \) represent the population at time \( t \). Given that the initial population \( P_0 = 63 \). The reproductive rate is given as \( \frac{1}{3} \).

Step 2: Establish the recursive formula

Since the population grows by the factor of the reproductive rate each unit of time, the population at time \( t+1 \) can be expressed as:\[P_{t+1} = P_t \times \frac{1}{3}\]

Step 3: Base case

We know that initially, at time \( t = 0 \), the population size is \( P_0 = 63 \). This forms the base case required for our recursive formula.

Step 4: Recursion formula result

Therefore, the recursive formula for the population size is:\[P_{t+1} = P_t \times \frac{1}{3}, \quad P_0 = 63\]This formula can be applied iteratively to determine the population size at any future time \( t \).

Key Concepts

Recursive Formula: Unraveling the Sequence

The concept of a recursive formula is a cornerstone in understanding how sequences evolve over time. In the context of population dynamics, a recursive formula defines how the population size at one point in time relates to the population size at the previous point. This type of formula is uniquely powerful because it helps model the changes in a population systematically from one time period to the next.

In our original exercise, we established the recursive formula as:

• \[P_{t+1} = P_t \times \frac{1}{3}\]

This formula tells us that to find the population at time \( t+1 \), we multiply the current population \( P_t \) by the reproductive rate \( \frac{1}{3} \). This simple yet effective calculation allows us to predict the population size at any future time by consistently applying the formula. Recursive formulas are particularly helpful in scenarios with consistent proportional changes, as they provide a reliable method for calculating sequences without needing explicit formulas for each period.

Reproductive Rate: Understanding Population Growth

The reproductive rate is an essential factor in determining how a population grows or shrinks over time. It specifies the factor by which a population size changes in each time unit, such as a year or month. Knowing the reproductive rate allows scientists to predict whether a population is growing, shrinking, or remaining stable.

In our exercise, the reproductive rate is \( \frac{1}{3} \). This means that the population size is one-third of its previous size each time unit. This reproductive rate suggests a declining population, as the population decreases to a third of its size each period. The reproductive rate helps us understand the speed and direction of population changes. It's crucial in fields like ecology, conservation, and resource management, where understanding population patterns can be vital for decision-making processes.

Initial Population Size: The Starting Point

Initial population size is the starting level of a population at the beginning of observation, typically noted as \( P_0 \). It's a vital component in any population dynamics model, as it acts as the baseline from which population changes are measured over time. Understanding the starting point allows for accurate modeling and predictions.

In our scenario, the initial population size is 63 individuals at time \( t = 0 \). This number serves as the anchor for our recursive calculations, providing a reference point for all future changes modeled by our recursive formula. Whether you're working with wildlife populations, bacterial growth in a lab, or even studying human demographics, knowing the initial population is crucial for realistic and meaningful forecasting.