Problem 14 - Thomas Calculus

Question

Controlling a population The fish and game department in a certain state is planning to issue hunting permits to control the deer population (one deer per permit). It is known that if the deer population falls below a certain level \(m,\) the deer will become extinct. It is also known that if the deer population rises above the carrying capacity \(M,\) the population will decrease back to \(M\) through disease and malnutrition. \begin{equation}\begin{array}{l}{\text { a. Discuss the reasonableness of the following model for the }} \\ {\text { growth rate of the deer population as a function of time: }}\end{array}\end{equation} $$\frac{d P}{d t}=r P(M-P)(P-m)$$ \begin{equation}\begin{array}{l}{\text { where } P \text { is the population of the deer and } r \text { is a positive con- }} \\ {\text { stant of proportionality. Include a phase line. }}\end{array}\end{equation} \begin{equation}\begin{array}{l}{\text { c. Show that if } P > M \text { for all } t \text { , then } \lim _{t \rightarrow \infty} P(t)=M .} \\ {\text { d. What happens if } P < m \text { for all } t ?} \\ {\text { e. Discuss the solutions to the differential equation. What are }} \\ {\text { the equilibrium points of the model? Explain the dependence }} \\ {\text { of the steady-state value of } P \text { on the initial values of } P . \text { About }} \\ {\text { how many permits should be issued? }}\end{array}\end{equation}

Step-by-Step Solution

Answer

The model supports maintaining deer population at \( M \) to avoid extinct and overburden. Too many permits may risk extinction if population falls below \( m \).

Step 1: Analyze the Model

The given model is \( \frac{dP}{dt} = rP(M-P)(P-m) \), where \( P \) is the population, \( r \) is a positive constant, \( M \) is the carrying capacity, and \( m \) is the minimum population level to avoid extinction. This model suggests that the growth rate depends on how far the population is below the carrying capacity \( M \) and above the minimum viable population \( m \).

Step 2: Finding Equilibrium Points

Set the growth rate \( \frac{dP}{dt} \) to zero: \( rP(M-P)(P-m) = 0 \). Solving this gives the equilibrium points: \( P = 0, P = m, \) and \( P = M \). These are the points where the population stops changing.

Step 3: Determine Stability with a Phase Line

Sketch a phase line with the equilibrium points \( P = 0, m, \) and \( M \). Analyze each region: 1. For \( P < m \), \( (P-m) < 0 \) causing population decline. 2. For \( m < P < M \), both \( (M-P) \) and \( (P-m) \) are positive so the population grows. 3. For \( P > M \), \( (M-P) < 0 \) causing decline back to \( M \). Hence, \( M \) is stable, \( m \) is unstable, and \( P = 0 \) is stable.

Step 4: Behavior Above Carrying Capacity

If \( P > M \) for all \( t \), then \( (M-P) < 0 \), indicating the population decreases until \( P = M \). Hence, the solution \( \lim_{t \to \infty} P(t) = M \), showing \( M \) as the stable carrying capacity.

Step 5: Behavior Below Minimum Level

If \( P < m \) for all \( t \), \( (P-m) < 0 \) causing the population to continue decreasing to extinction \( P = 0 \). As \( t \to \infty \), \( P(t) \to 0 \), showing extinction risk.

Step 6: Discuss Solutions and Permit Strategy

Multiple equilibria demonstrate \( P = M \) as the target stable state for sustainability. The initial population \( P \) determines the solution path. Issuing permits to maintain \( P \approx M \) without dropping below \( m \) is essential. The number of permits should ensure the target of \( P = M \) by managing population pressure and disease control.

Key Concepts

Equilibrium Points

In the context of differential equations, equilibrium points are values where the system is in a steady state. Here, the deer population's growth rate given by \( \frac{dP}{dt} = rP(M-P)(P-m) \) can be zero, indicating that the population is at an equilibrium point. By setting the differential equation to zero, we find:

• \(P = 0\): Represents extinction, a stable equilibrium as the population cannot fall below zero.
• \(P = m\): This is an unstable equilibrium. If the population falls below this minimum viable level \(m\), it risks extinction.
• \(P = M\): Denotes the carrying capacity, and is a stable equilibrium as the population tends to return to this level through natural self-regulation.

Equilibrium points help us understand where the population might end up long-term based on its current state.

Phase Line

A phase line is a valuable tool for visualizing the stability of equilibrium points within a differential equation. It helps determine the behavior of different solution paths. Consider the phase line for our deer population model with equilibrium points at \(P = 0, m, \text{ and } M\):

• For \(P < m\), the factor \((P-m) < 0\) makes the population decline, moving towards extinction.
• In the range \(m < P < M\), both \((M-P)\) and \((P-m)\) are positive, leading to population growth towards the carrying capacity \(M\).
• For \(P > M\), \((M-P) < 0\), resulting in a decline back towards \(M\), the stable carrying capacity.

The phase line clearly shows that \(M\) is a stable equilibrium point, while \(m\) is unstable. This visualization helps in understanding the system dynamics and the likely future state of the population.

Carrying Capacity

Carrying capacity \(M\) is a crucial concept in population dynamics. It represents the maximum population size that the environment can sustain indefinitely. In our model, if the deer population exceeds \(M\), natural factors like disease and malnutrition will decrease the population back to \(M\). If the population is below \(M\), it will naturally grow towards this level as resources allow.

• Above \(M\), the equation \((M-P) < 0\) ensures the population decreases.
• Below \(M\), the population increases since the growth terms \((M-P)(P-m) > 0\).

Carrying capacity ensures balance as the system self-regulates the population size. Understanding \(M\) helps in planning how many permits to issue for hunting, ensuring wildlife management strategies maintain a balance, neither allowing overpopulation nor underpopulation.

Population Dynamics

Population dynamics is the study of how populations change over time through births, deaths, and migration. In this context, the differential equation \( \frac{dP}{dt} = rP(M-P)(P-m) \) describes how the deer population behaves in relation to equilibrium points, growth rates, and environmental thresholds such as carrying capacity.

• Stable points like \(P = M\) highlight long-term population sustainability.
• Unstable points like \(P = m\) show critical thresholds that risk extinction.
• The interplay of growth factors \((M-P)(P-m)\) drives populations toward equilibrium points based on initial conditions.

Understanding these dynamics is key for wildlife management, ensuring the deer population remains viable and sustainable. Strategic control methods, like hunting permits, must align with the dynamics described by the model to maintain ecological balance.