Problem 25 - Calculus for Biology and Medicine

Question

Find the recursion for a population that quadruples in size everv unit of time and that has 30 individuals at time 0 .

Step-by-Step Solution

Answer

The recursive formula is: Initial condition: \( P_0 = 30 \), Recurrence relation: \( P_{n+1} = 4P_n \).

Step 1: Understand the problem

We need to find a recursive formula that describes the growth of a population. The population quadruples every unit of time, and it starts with 30 individuals at time 0.

Step 2: Identify the initial condition

The initial condition is given as 30 individuals at time 0. This means that when the time, denoted by \( n = 0 \), the population is \( P_0 = 30 \).

Step 3: Define the recurrence relation

Since the population quadruples every unit of time, the recurrence relation should capture this pattern. The relation can be defined as \( P_{n+1} = 4P_n \), where \( P_n \) is the population at time \( n \), and \( P_{n+1} \) is the population at time \( n+1 \).

Step 4: Combine initial condition with recurrence relation

The complete recursive formula for the population is given by:- Initial condition: \( P_0 = 30 \).- Recurrence relation: \( P_{n+1} = 4P_n \) for \( n \geq 0 \).

Key Concepts

Population Growth

Population growth in the context of our problem means understanding how the size of a population changes over time. Imagine a scenario where every single individual within a population successfully breeds, leading to a fourfold increase each time unit. This is referred to as quadrupling. In mathematical terms, population growth could be represented by a function showing how the population evolves as time progresses. For this example, the population size grows rapidly because each period (or unit of time) sees a multiplication of four. Understanding this concept is vital, as it sets the stage for establishing the rules (or formula) that describe the population's dynamic nature over time. Consider the implications of such rapid growth - it can lead to exponential increases, significantly affecting resource needs, habitat space, and potentially resulting in environmental impacts.

Initial Condition

The concept of an initial condition is fundamental in understanding recurrence relations like the one in our exercise. In simple terms, the initial condition tells us where we start. It serves as the beginning point or the state from which all future values are derived. In the context of our problem, the initial condition is given as 30 individuals at time zero. Knowing the initial size of the population allows us to predict how the population will change in the future using the provided rules. This is crucial because any projection or model we create depends heavily on this starting point. If our initial condition is incorrect, all subsequent calculations and predictions could be significantly off. Hence, always pay attention to this critical aspect when working with recurrence relations.

Recursive Formula

A recursive formula is a mathematical expression that forms the basis for solving problems related to sequences or series, such as the growth of a population. Essentially, a recursive formula provides a step-by-step recipe for finding specific terms in a sequence based on preceding terms.In our example, the recursive formula used to describe population growth is defined by the relationship: \( P_{n+1} = 4P_n \). This means that to find the population at the next time step, you multiply the current population by 4. This highlights the central mechanism of recursion - each term is directly calculated from the terms that come before it. This step-by-step approach is powerful since it allows us to build complex population forecasts from a simple rule. The recursive formula not only saves time in prediction but also offers elegant solutions to diverse mathematical and real-world problems.