Problem 50 - Calculus for Biology and Medicine

Question

Graph the Ricker's curve $$ N_{t+1}=N_{t} \exp \left[R\left(1-\frac{N_{t}}{K}\right)\right] $$ in the \(N_{t}-N_{t+1}\) plane for the given values of \(R\) and \(\bar{K} .\) Find the points of intersection of this graph with the line \(N_{t+1}=N_{t}\). R=4, K=20

Step-by-Step Solution

Answer

The graph intersects the line \(N_{t+1} = N_{t}\) at \(N_{t} = 20\).

Step 1: Understand the Equation

The Ricker's curve is: \[ N_{t+1} = N_{t} \exp\left[R\left(1 - \frac{N_{t}}{K}\right)\right] \]This curve represents the population at time \(t+1\) based on the population at time \(t\), with a growth factor \(R\) and carrying capacity \(K\). Here, \(R=4\) and \(K=20\).

Step 2: Set Up the Graph

To graph the function \(N_{t+1} = N_{t} \exp\left[4 \left(1 - \frac{N_{t}}{20}\right)\right]\), use the plane where the x-axis is \(N_{t}\) and the y-axis is \(N_{t+1}\). Plot points for distinct values of \(N_{t}\) to obtain the curve.

Step 3: Plot Points on the Graph

Calculate \(N_{t+1}\) for specific values of \(N_{t}\). Let's calculate a few examples:- For \(N_{t} = 0\), \(N_{t+1} = 0 \times \exp(4) = 0\).- For \(N_{t} = 10\), \(N_{t+1} = 10 \exp\left(4\left(1 - \frac{10}{20}\right)\right)\). Compute the result.- For \(N_{t} = 20\), \(N_{t+1} = 20 \exp(4(1 - 1)) = 20\).- Continue this for various values to sketch the curve.

Step 4: Draw the Line of Intersection

The line \(N_{t+1} = N_{t}\) represents the points where the next population is equal to the current population. This is a straight line with a slope of 1 passing through the origin.

Step 5: Find Points of Intersection

To find the intersection, set \(N_{t+1} = N_{t}\) in the Ricker's curve equation:\[ N_{t} = N_{t} \exp\left[4\left(1 - \frac{N_{t}}{20}\right)\right] \]Simplify this to:\[ 1 = \exp(4 - \frac{4N_{t}}{20}) \]Taking the natural logarithm:\[ 0 = 4 - \frac{4N_{t}}{20} \]\[ 4N_{t} / 20 = 4 \]\[ N_{t} = 20 \]The curve intersects the line at \(N_{t} = 20\).

Key Concepts

Population Dynamics

Population dynamics explore how populations of organisms change over time and are an essential concept in biology and ecology. Ricker's curve is a popular model used in understanding these changes. It's a type of mathematical function that helps predict future population sizes based on current conditions and biological constraints.

• The equation for Ricker's curve is: \[N_{t+1} = N_{t} imes \exp\left[R\left(1 - \frac{N_{t}}{K}\right)\right]\]
• Here, \(N_{t}\) is the population at the current time \(t\) and \(N_{t+1}\) is the population at the next time step.
• \(R\) represents the intrinsic growth rate, showing how fast the population can grow under ideal conditions.
• \(K\) is the carrying capacity, which is the maximum population size that an environment can sustain given the resources available.

Ricker's curve accounts for the fact that as a population reaches its carrying capacity, the growth will slow down and eventually stabilize if the population reaches equilibrium. This model is particularly helpful in projects involving conservation, resource management, and understanding ecosystem changes.

Graphing Techniques

Graphing Ricker's curve effectively requires understanding how to represent an equation graphically in a plane. Using specific graphing techniques makes it easier to visually interpret how the population will evolve over time for different scenarios.

• The axes: In the graph, the x-axis represents \(N_{t}\) and the y-axis represents \(N_{t+1}\). This shows the relationship between the population at two consecutive times.
• Selecting values: Calculate \(N_{t+1}\) for a range of \(N_{t}\) values to draw the curve. For instance:
  • At \(N_{t} = 0\), no growth occurs, so \(N_{t+1} = 0\).
  • Near \(N_{t} = 20\), the curve flattens, suggesting the population stabilizes at this point.
  • Choosing intermediate values such as \(N_{t} = 5, 10, 15\) helps in obtaining the overall shape of the curve.
• Drawing the line of intersection \(N_{t+1} = N_{t}\): This is a simple linear equation with a slope of 1, indicating where current and future populations are the same.

By connecting the values of \(N_{t}\) and \(N_{t+1}\) through plotting, one can sketch the curve and observe dynamic population changes visually.

Intersection Points

Finding intersection points in this context relates to finding where two functions meet on the graph. For Ricker's curve, we specifically look for where the curve intersects the line \(N_{t+1} = N_{t}\). This is crucial because it signifies points where the population remains stable between two time periods.

• To find intersection points, set the right sides of both equations equal:\[N_{t} = N_{t}\exp\left[4\left(1 - \frac{N_{t}}{20}\right)\right]\]
• Simplify this to find the value of \(N_{t}\) that satisfies both the curve and the line equation. This leads to:\[1 = \exp\left(4 - \frac{4N_{t}}{20}\right),\]
• Taking the natural logarithm on both sides:\[0 = 4 - \frac{4N_{t}}{20},\]
• Solving it results in \(N_{t} = 20\).

The solution indicates that at \(N_{t} = 20\), the populations at two consecutive time points will be equal, meaning the population size isn't changing, thus achieving stability.