Problem 16 - Calculus for Biology and Medicine

Question

Suppose that the size of a fish population at generation \(t\) is given by $$ N_{t+1}=1.5 N_{t} e^{-0.001 N_{t}} $$ 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) Assume that \(N_{0}=800\). Find the size of the fish population at generation \(t\) for \(t=1,2, \ldots, 20\) (c) Determine all fixed points. On the basis of your computations in (a) and (b), make a guess as to what will happen to the population in the long run, starting from (i) \(N_{0}=100\) and (ii) \(N_{0}=800\). (d) Use the cobwebbing method to illustrate your answer in (a). (e) Explain why the dynamical system converges to the nontrivial fixed point.

Step-by-Step Solution

Answer

For both \(N_0 = 100\) and \(N_0 = 800\), populations stabilize around ~500. The nontrivial fixed point is approximately 500. Both cases converge to this point over time.

Step 1: Initialize the Problem

Given the recursive formula for the fish population: \( N_{t+1} = 1.5 N_{t} e^{-0.001 N_{t}} \). We have initial values for two scenarios: (a) \( N_0 = 100 \) and (b) \( N_0 = 800 \). Our task is to compute \( N_t \) for \( t = 1, 2, \ldots, 20 \) in both scenarios and to analyze the fixed points.

Step 2: Compute the Population for (a), N0=100

Start with \( N_0 = 100 \). Compute \( N_1 \) using the formula: \[ N_1 = 1.5 \times 100 \times e^{-0.001 \times 100} \] Calculate the result. Continue this process iteratively for \( t = 1, 2, \ldots, 20 \) using the result of \( N_{t} \) as the new \( N_{t+1} \). Record each value of \( N_t \).

Step 3: Compute the Population for (b), N0=800

Start with \( N_0 = 800 \). Compute \( N_1 \) using the formula: \[ N_1 = 1.5 \times 800 \times e^{-0.001 \times 800} \] Calculate the result. Continue this process iteratively for \( t = 1, 2, \ldots, 20 \) using the result of \( N_{t} \) as the new \( N_{t+1} \). Record each value of \( N_t \).

Step 4: Find the Fixed Points

Fixed points are where \( N_{t+1} = N_t \). Solve the equation: \[ N = 1.5 N e^{-0.001 N} \] This gives the two conditions: - \( N = 0 \) - Solve \( e^{-0.001 N} = \frac{1}{1.5} \) to find the nontrivial fixed point. Calculate the approximate numerical value for the nontrivial fixed point.

Step 5: Long-term Behavior Analysis

According to the computations in steps 2 and 3, observe that for both initial conditions, the populations tend to stabilize around the nontrivial fixed point. Hence, in the long run, regardless of starting population, \( N_t \) will converge towards this fixed stage.

Step 6: Use the Cobwebbing Method for (a)

Draw a cobweb diagram for \( N_0 = 100 \) using the function \( f(N) = 1.5 N e^{-0.001 N} \). Plot the line \( y = x \) and perform the cobweb steps to visualize how the population size approaches the nontrivial fixed point for \( t \to \infty \).

Step 7: Explain the System's Convergence

The system converges towards the nontrivial fixed point due to the properties of the exponential function combined with the dampening factor \( e^{-0.001 N} \). This function reduces growth as \( N \) increases, leading populations toward a stable equilibrium point where population remains constant over time.

Key Concepts

Dynamical Systems

Dynamical systems are mathematical frameworks used to describe how the state of a system evolves over time, often driven by some laws of physics or mathematics. They can be described by differential or difference equations that determine future states based on current ones.
In biology, dynamical systems are frequently used to model population changes over generations. The function specifies how the population size changes from one generation to the next. For example, the given equation, \( N_{t+1} = 1.5 N_{t} e^{-0.001 N_{t}} \), expresses the next generation's size based on current population factors. It considers both reproduction (the 1.5 factor) and the limiting effects (via \( e^{-0.001 N} \)).
This system allows us to predict how a population might grow or decline over time, helping researchers understand and influence biological ecosystems.

Population Dynamics

Population dynamics involves the study of how and why populations change over time. In our given model, the population dynamics are captured through a recursive relationship, where the current population determines the future one.
The equation \( N_{t+1} = 1.5 N_{t} e^{-0.001 N_{t}} \) suggests that the fish population will grow because of the reproduction factor (1.5), yet it can also slow down or stabilize due to the exponential decay term \( e^{-0.001 N} \). This term introduces a carrying capacity, simulating the effects of limited resources or environmental resistance.

• When the population is small, the exponential term is close to 1, allowing rapid growth.
• As the population grows larger, the term \( e^{-0.001 N} \) reduces, slowing population growth.

Thus, population dynamics in this model are governed by the interplay of these growth and limiting factors.

Fixed Points

A fixed point in a dynamical system occurs when the population does not change over generations—\( N_{t+1} = N_t \). It represents an equilibrium state where the population size remains constant.
To find the fixed points for our model, solve the equation \( N = 1.5 N e^{-0.001 N} \). This results in two solutions:

• A trivial fixed point: \( N = 0 \), where the population is extinct.
• A nontrivial fixed point: determined by solving \( e^{-0.001 N} = \frac{1}{1.5} \). This involves algebraic or numerical methods to approximate \( N \).

The nontrivial fixed point reflects a stable population maintained over time, highlighting a balance between growth and resource constraints.

Cobwebbing Method

Cobwebbing is a graphical technique used to visually analyze the behavior of discrete dynamical systems like our fish population model. This method helps in understanding how values iterate towards fixed points over time.
To use cobwebbing with the function \( f(N) = 1.5 N e^{-0.001 N} \), you draw:

• The graph of \( f(N) \).
• A line \( y = x \), representing where \( N_{t+1} = N_t \).

Start at your initial point along the x-axis and move vertically to the curve, horizontally to the \( y = x \) line, and repeat. Through these steps, you visualize the trajectory of \( N_t \) as it converges (or diverges) from the fixed point, providing insight into the stability and long-term behavior of the system.