Problem 17 - Calculus for Biology and Medicine

Question

Suppose that the size of a fish population at generation \(t\) is given by $$ N_{t+1}=10 N_{t} e^{-0.01 N_{t}} $$ for \(t=0,1,2, \ldots\) (a) Assume that \(N_{0}=100\). Find the size of the fish population at generation \(t\) for \(t=1,2, \ldots, 20\). (b) Show that if \(N_{0}=100 \ln 10\), then \(N_{t}=100 \ln 10\) for \(t=\) \(1,2,3, \ldots ;\) that is, show that \(N^{*}=100 \ln 10\) is a nontrivial fixed point, or equilibrium. How would you find \(N^{*}\) ? Are there any other equilibria? (c) On the basis of your computations in (a), make a prediction about the long-term behavior of the fish population when \(N_{0}=\) \(100 .\) How does your answer compare with that in (b)? (d) Use the cobwebbing method to illustrate your answer in (c). In Problems \(18-20\), consider the following discrete-time dynamical system, which is called the discrete logistic model and which models the size of a population over time: $$ N_{t+1}=N_{t}\left[1+R\left(1-\frac{N_{t}}{100}\right)\right] $$ for \(t=0,1,2, \ldots\)

Step-by-Step Solution

Answer

The fish population eventually stabilizes around the equilibrium \(N^* = 100 \ln 10\). This agrees with both the calculations and the cobweb diagram analysis.

Step 1: Understanding the Model for Part (a)

The given model is a recursive equation for the fish population: \[ N_{t+1} = 10 N_{t} e^{-0.01 N_{t}}. \]We start with \(N_0 = 100\) and need to calculate \(N_t\) for \(t = 1, 2, \ldots, 20\). This is done by substituting the previous population size \(N_t\) into the equation to find \(N_{t+1}\), repeatedly for each time step up to \(t=20\).

Step 2: Calculate Population for Each Generation (Part a continued)

Begin with \(N_0 = 100\). Calculate \(N_1\) using the model:\[ N_1 = 10 imes 100 imes e^{-0.01 imes 100} = 10 imes 100 imes e^{-1} \approx 367.8794. \]Repeat this process iteratively using \(N_1\) to find \(N_2\), and so forth until \(N_{20}\) using the same formula. It involves recalculating \(N_{t+1}\) based on the current \(N_t\).

Step 3: Verify Fixed Point for Part (b)

A fixed point satisfies \(N_{t+1} = N_t = N^*\). Given \(N_0 = 100 \ln 10\), we need to show\[ N_1 = 10 N_0 e^{-0.01 N_0} = N_0. \]Substituting \(N_0 = 100 \ln 10\) gives:\[ 10 \times 100 \ln 10 \times e^{-0.01 \times 100 \ln 10} = 100 \ln 10. \]This confirms \(N^* = 100 \ln 10\) is an equilibrium since:\[ 10 e^{-\ln 10} = 1, \]hence \(N^*\) remains constant over time.

Step 4: Finding Additional Equilibria for Part (b)

Equilibria occur when \(N_{t+1} = N_t = N^*\). For other potential values of \(N^*\), solve:\[ 10 N^* e^{-0.01 N^*} = N^*. \]To solve, divide both sides by \(N^*\), if \(N^* eq 0\), to get:\[ 10 e^{-0.01 N^*} = 1. \]Taking the natural log, we obtain:\[-0.01 N^* = -\ln 10. \]Thus, \(N^* = 100 \ln 10\) is the primary non-trivial equilibrium. The trivial solution is \(N^* = 0\).

Step 5: Predict Long-term Behavior for Part (c)

Using the computations from part (a), observe the trend in \(N_t\). Initially, \(N_t\) grows but later stabilizes with repetitive values close to the equilibrium calculated in part (b), suggesting the long-term behavior settles around \(N^* = 100 \ln 10\). This matches with the fixed point found, verifying that the population stabilizes.

Step 6: Cobweb Diagram Method for Part (d)

The cobwebbing method involves plotting the function \(N_{t+1} = 10 N_t e^{-0.01 N_t}\) and the line \(N_{t+1} = N_t\) on a graph. Start at \(N_0 = 100\), and draw vertical lines to the curve, then horizontal lines to the line \(N_{t+1} = N_t\). Repeat this process iteratively. This visually shows \(N_t\) converging to the equilibrium at \(100 \ln 10\) as described in part (c).

Step 7: Discuss the Logistic Model (Additional Problems)

The discrete model given is:\[ N_{t+1} = N_t\left[1 + R\left(1 - \frac{N_t}{100}\right)\right]. \]This model's behavior is similar—initial exponential growth followed by stabilization or oscillation depending on the rate \(R\). The critical difference from Part (a) is inclusion of an additional term representing logistic growth dynamics.

Key Concepts

Recursive Equation

A recursive equation is used to generate a sequence of values by applying a mathematical formula repeatedly. In the context of the fish population example given, the recursive equation is: \[ N_{t+1} = 10 N_{t} e^{-0.01 N_{t}}. \]

• "Recursive" means using the result of one step as the starting point for the next.
• The equation defines how each population size depends on the previous one.
• Starting with an initial value, you can calculate future values step by step.
• This process is similar to programming loops where each iteration depends on the previous result.

For this specific model, the given starting population is substituted into the equation, and the same process is done iteratively to get the population size at each step. This approach helps in modeling scenarios where future states depend on past events, such as population growth over generations.

Fixed Point

A fixed point in a dynamical system is a value of the variable that remains unchanged when the function is applied. For the fish population problem, a fixed point is observed when:\[ N_{t+1} = N_t = N^*. \]

• This means when the recursive function is applied, the result is the same as the input.
• In mathematical terms, the fixed point satisfies the equation without a change across iterations.
• For the equation \(10 N^* e^{-0.01 N^*} = N^* \), solving for \(N^*\) gives us the equilibrium point.

The non-trivial fixed point in this problem is \(N^* = 100 \ln 10\), meaning that if the population starts at this point, it will remain constant over time.

Equilibrium

In dynamical systems, an equilibrium point is a condition where the system remains unchanged over time. In population models, equilibrium can indicate stability where population sizes do not fluctuate from one generation to the next. For the model given:

• An equilibrium occurs when \(N_{t+1} = N_t = N^*\).
• The primary equilibrium found was \(N^* = 100 \ln 10\).
• Another trivial equilibrium is \(N^* = 0\), where the population dies out.

Equilibrium points help in understanding the long-term behavior of a system, where achieving equilibrium means stability, and diversions might indicate growth or decline depending on other factors included in a more complex model.

Population Model

Population models simulate the dynamics of a biological population over time. These models are built to understand how populations grow, stabilize, or decline based on certain assumptions. In the exercise at hand, the model:\[ N_{t+1} = 10 N_t e^{-0.01 N_t} \]

• Demonstrates growth potential followed by regulation due to limiting factors, represented by exponential decay.
• Represents a theoretical but simplified view of how populations might behave.
• Allows for predictive understanding by extrapolating trends and can be adjusted based on empirical data.

Such models are crucial in ecology and resource management to make predictions about fish populations, plan conservation efforts, and study environmental impacts.

Cobwebbing Method

The cobwebbing method is a graphical technique used to visualize iterative processes in dynamical systems. It provides a way to understand how iterations affect convergence to fixed points. For our fish population example:

• Plot the function \( N_{t+1} = 10 N_{t} e^{-0.01 N_{t}} \) along with the line \( N_{t+1} = N_{t} \).
• Start with an initial value and draw a vertical line to the curve, followed by a horizontal line to the diagonal line.
• This helps in illustrating how the population sizes move toward or away from equilibrium.

Cobwebbing gives an intuitive understanding of how different starting points approach a fixed point, showing visually the convergence or divergence of population sizes over time.