Problem 1 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Two kilos of a fish poison, rotenone, are mixed into a lake which has a volume of \(100 \times 20 \times 2=4000\) cubic meters. No water flows into or out of the lake. Fifteen percent of the rotenone decomposes each day. a. Write a mathematical model that describes the daily change in the amount of rotenone in the lake. b. Let \(R_{0}, P_{1}, R_{2}, \cdots\) denote the amounts of rotenone in the lake, \(P_{t}\) being the amount of poison in the lake at the beginning of the \(t \underline{h}\) day after the rotenone is administered. Write a dynamic equation representative of the mathematical model. c. What is \(R_{0} ?\) Compute \(R_{1}\) from your dynamic equation. Compute \(R_{2}\) from your dynamic equation. d. Find a solution equation for your dynamic equation.

Step-by-Step Solution

Answer

a) Daily change: \( R_{t+1} = 0.85R_t \). c) \( R_0 = 2 \) kg, \( R_1 = 1.7 \) kg, \( R_2 = 1.445 \) kg. d) Solution equation: \( R_t = 2 \times 0.85^t \).

Step 1: Understanding Daily Decomposition

The problem states that 15% of the rotenone decomposes each day. Therefore, 85% of the rotenone remains each day. If we let \( R_t \) represent the amount of rotenone at the beginning of day \( t \), then the relationship from one day to the next is given by \( R_{t+1} = 0.85 R_t \). This equation models the daily change in the amount of rotenone in the lake.

Step 2: Determine Initial Amount \( R_0 \)

The initial amount of rotenone, \( R_0 \), is the total amount introduced to the lake initially, which is 2 kilos. Therefore, \( R_0 = 2 \) kg.

Step 3: Compute \( R_1 \)

Using the dynamic equation \( R_{t+1} = 0.85 R_t \), calculate \( R_1 \):\[ R_1 = 0.85 \times R_0 = 0.85 \times 2 = 1.7 \text{ kg} \]

Step 4: Compute \( R_2 \)

Now compute \( R_2 \) using the same equation: \[ R_2 = 0.85 \times R_1 = 0.85 \times 1.7 = 1.445 \text{ kg} \]

Step 5: Find General Solution

The general solution for the dynamic equation \( R_{t+1} = 0.85 R_t \) can be expressed using the initial amount \( R_0 \):\[ R_t = R_0 \times (0.85)^t \]This equation describes how the amount of rotenone changes on any given day \( t \).

Key Concepts

Exponential Decay

Exponential decay is an important concept in mathematical modeling, describing how quantities reduce over time at a rate proportional to their current value. Imagine something decreasing by a certain percentage each day, like how 15% of the poison rotenone was decomposing daily in our scenario.
This means that 85% remains each day. The pattern of decay in exponential decay can be represented by the formula:

• For day 0: initial amount, say 2 kg.
• For day 1: 85% of the initial amount remains, calculated as \(0.85 \times 2 \text{ kg}\).
• Continue similarly for day 2, day 3, and so on.

The formula becomes \( R_t = R_0 \times (0.85)^t \), which captures the essence of exponential decay.
Understanding exponential decay is crucial in various fields including chemistry, physics, and economics, where it helps predict how substances lose their effectiveness over time.

Differential Equations

Differential equations are a mathematical way to relate some quantity to its rate of change. They are the heart of modeling continuous processes like radioactive decay, population dynamics, or in our case, the decomposition of rotenone. Instead of looking at how things change in steps, differential equations allow us to see this change continuously.
While our exercise looked at a simpler, discrete model, the continuous equivalent describes a similar concept. For exponential decay in the continuous realm, the differential equation is written as:

• \( \frac{dR}{dt} = -kR \)

Here, \( k \) is a constant rate of decay. Solving this equation gives us a continuous form of the exponential decay equation. Knowing this helps you appreciate the power of differential equations in modeling dynamic systems even if they are hidden behind the scenes of simple algebraic models.

• They help model not only physical processes but also anything that changes continuously over time, making them versatile tools in science and engineering.

Dynamic Systems

Dynamic systems, as the name suggests, deal with systems that change over time. They can be modeled using differential equations, as mentioned before, or using simpler difference equations for discrete changes. Every dynamic system, whether the economy, climate, or the rotenone in our lake, can be analyzed to understand how they evolve over time.
Our exercise used a discrete dynamic model to describe how the poison concentration changes day by day, which is especially useful when dealing with step-by-step processes.
Dynamic systems are often characterized by feedback loops, where the outcome depends not only on the current state but also previous ones. This feedback mechanism is what the equation \( R_{t+1} = 0.85R_t \) represents.

• Understanding these systems can help manage resources, predict future states, and optimize processes.
• By building models, we can simulate different scenarios and see how they affect outcomes.

Thus, dynamic systems form the backbone of decision-making processes in complex environments.