Problem 2 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

a. Show that the equilibrium numbers of the iteration $$ p_{t+1}=F\left(p_{t}\right)=p_{t}+R \times p_{t} \times\left(1-p_{t}\right)-h p_{t} $$ are \(p_{* 1}=0\) and \(p_{* 2}=1-h / R\). b. Show that in order for there to be a positive equilibrium, the fractional harvest rate, \(h,\) must be less than the low density growth rate, \(R\).

Step-by-Step Solution

Answer

Equilibrium points are \( p_{*1} = 0 \) and \( p_{*2} = 1 - \frac{h}{R} \), with \( h < R \) for \( p_{*2} > 0 \).

Step 1: Define Equilibrium

An equilibrium point is where the state of the system doesn't change in subsequent iterations. This means that for equilibrium we have \( p_{t+1} = p_{t} = p_* \). Thus, we can rewrite the function \( p_{t+1}=F(p_{t}) \) so that \( p_* = F(p_*) \).

Step 2: Substitute Equilibrium Condition

We substitute the equilibrium condition into the equation: \[ p_* = p_* + R \times p_* \times (1 - p_*) - h \times p_* \] Simplifying, we can eliminate \( p_* \) from both sides to get: \[ R \times p_* \times (1 - p_*) - h \times p_* = 0 \].

Step 3: Rearrange the Equation

Rearrange the equation to isolate terms on one side: \[ p_* (R(1 - p_*) - h) = 0 \]. This shows there are potentially two solutions: either \( p_* = 0 \) or \( R(1 - p_*) - h = 0 \).

Step 4: Solve for Equilibrium Points

The first solution is straightforward: \[ p_* = 0 \].For the second solution, solve \( R(1 - p_*) = h \): \[ R - Rp_* = h \] \[ Rp_* = R - h \] \[ p_* = \frac{R - h}{R} = 1 - \frac{h}{R} \]. Thus, the equilibriums are \( p_{*1} = 0 \) and \( p_{*2} = 1 - \frac{h}{R} \).

Step 5: Condition for Positive Equilibrium

A positive equilibrium requires \( p_{*2} > 0 \). This implies: \[ 1 - \frac{h}{R} > 0 \] \[ \frac{h}{R} < 1 \] \[ h < R \]. Thus, for a positive equilibrium, \( h < R \) must hold true.

Key Concepts

Fractional Harvest Rate

In the equation, the fractional harvest rate, denoted as \( h \), represents the proportional rate at which resources are removed from the population. Imagine a wildlife reserve where animals such as deer are hunted or harvested. The fractional harvest rate is like setting a quota - it's the fraction of the population that is permitted to be "harvested" or removed during a given period.
To maintain a stable population, it is important that this rate does not exceed the growth capacity of the population. Too high of a harvest rate can lead to declines in population, which is why understanding its equilibrium is essential.

• If the fractional harvest rate is too high, it can push the population to zero, leading to extinction.
• The rate at which harvesting occurs should be less than the growth rate of the population for the number to stabilize around a positive equilibrium.
• This equilibrium ensures that the population can sustain itself over time despite the continuous removal of individuals.

Keeping \( h < R \) ensures a positive equilibrium, meaning the population remains viable and continues to grow or stay constant even after accounting for harvesting.

Low Density Growth Rate

The low density growth rate, denoted by \( R \), is a fundamental concept in population dynamics. It reflects how quickly a population can grow when the population size is small, meaning resources such as food, space, and mates are abundant and not limiting factors.

• Higher low density growth rates imply the population can recover more quickly from reductions due to harvesting or any other impact.
• If the low density growth rate is larger than the fractional harvest rate, it indicates the population can sustain itself and potentially thrive.
• Conversely, when \( R \) is low, populations might struggle to reach a stable equilibrium if there is significant harvesting occurring.

In the context of equilibrium, it's crucial to ensure that \( R > h \) so that the population remains sustainable. This relationship helps in formulating policies for sustainable resource management, ensuring species and ecosystems can endure for future generations.

Iteration Function

The iteration function, \( p_{t+1}=F(p_{t}) \), serves as a mathematical tool to predict the future population size based on the current population size, \( p_t \). This function incorporates key variables such as the low density growth rate \( R \) and the fractional harvest rate \( h \), using these to model the dynamic changes in population over time.

• The iteration function helps determine whether a population will grow, decline, or reach equilibrium based on initial parameters and conditions.
• At an equilibrium point, the population size will stabilize, indicating \( p_{t+1} = p_t = p_* \).
• The mathematical manipulation of this function can identify crucial equilibrium points and conditions necessary for maintaining a stable population.

Understanding iteration functions is vital in ecological modeling, allowing researchers and policymakers to predict and manage changes in wildlife populations responsibly. It emphasizes the importance of aligning growth and harvest rates to achieve sustainability.