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 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. a. Discuss the reasonableness of the following model for the growth rate of the deer population as a function of time: $$ \begin{aligned} \frac{d P}{d t}=r P(M-P)(P-m) \\ \text { where } P \text { is the population of the deer and } r \text { is a positive } \\ \text { constant of proportionality. Include a phase line. } \end{aligned} $$ b. Explain how this model differs from the logistic model \(d P / d t=r P(M-P) .\) Is it better or worse than the logistic model? c. Show that if \(P > M\) for all \(t\) , then \(\lim _{t \rightarrow \infty} P(t)=M\) d. What happens if \(P < m\) for all \(t ?\) e. Discuss the solutions to the differential equation. What are the equilibrium points of \(P\) on the initial values of \(P .\) About how many permits should be issued?

Step-by-Step Solution

Answer

The enhanced model is more precise for small populations near extinction thresholds. When \( P > M \), \( P(t) \to M \). When \( P < m \), the population declines toward extinction. Issue permits to maintain \( m < P < M \).

Step 1: Analyze the Proposed Model

The given model is \( \frac{d P}{d t}=r P(M-P)(P-m) \). Here, the growth rate of the population \( P \) is depended on three factors: \( r \) is a growth constant, \( M \) is the carrying capacity, and \( m \) is the extinction threshold. This model suggests that if \( P > M \), the growth rate is negative, causing \( P \) to decrease, aligning with disease and malnutrition effects. If \( P < m \), the growth rate is also negative, risking extinction. This model is suitable for representing populations influenced by both a carrying capacity and extinction threshold.

Step 2: Phase Line Analysis

A phase line helps visualize the stability of equilibrium points. Plotting \( \frac{dP}{dt} = rP(M-P)(P-m) \), we find equilibrium points when \( P = m \), \( P = M \), and \( P = 0 \). On the phase line, arrows between these points indicate the direction of change in \( P \). For \( P \in (m, M) \), arrows point towards \( M \); for \( P < m \) or \( P > M \), arrows lead away from stability, suggesting possible extinction if \( P < m \).

Step 3: Compare with Logistic Model

The logistic model \( \frac{dP}{dt} = rP(M-P) \) accounts only for the carrying capacity influence, ignoring any extinction threshold. The proposed model \( \frac{dP}{dt} = rP(M-P)(P-m) \) incorporates both limits, making it potentially more realistic for populations subject to an extinction boundary as well as upper constraints, hence more practical for managing small but biologically critical populations.

Step 4: Evaluate Long-term Behavior for \( P > M \)

To show \( \lim_{t \rightarrow \infty} P(t) = M \) when \( P > M \), observe that \( (M-P)(P-m) < 0 \), implying \( \frac{dP}{dt} < 0 \). This forces \( P \) downward towards \( M \). Over time, with persistent negative growth as \( P \) exceeds \( M \), the population stabilizes around \( M \).

Step 5: Analyze Scenario \( P < m \)

When \( P < m \), observe \( (M-P)(P-m) > 0 \) turns negative since \( r > 0 \), slanting \( \frac{dP}{dt} < 0 \) further downward, suggesting that \( P \) continues to decrease, risking extinction due to continual negative growth without upward pressure until falling to extinction or remedial intervention.

Step 6: Discuss Equilibrium Points and Permits

Equilibrium points occur at \( P = 0 \), \( P = m \), and \( P = M \). Stability analysis shows \( P = M \) is stable under typical growth, \( P = m \) an unstable equilibrium potentially visualizing a threshold before an extinction spiral downward, and \( P = 0 \) illustrating extinction. Permits should aim to keep \( P \) safely within \( (m, M) \), precisely choosing within \( P \) near \( M \) without excessively depleting below \( m \); calculate based on maintaining the balance between threats of disease and extinction without suppressing beneficial biodiversity.

Key Concepts

Population Dynamics

Population dynamics is a scientific discipline focused on studying the changes in population sizes and the factors driving these changes. In this exercise, we see how the deer population evolves in response to several ecological factors. The growth or reduction of the deer population hinges on the pressure exerted by biological and environmental constraints. It is influenced by various factors such as food availability, predation, disease, and climatic conditions. When modeling such scenarios, differential equations serve as effective tools to simulate dynamics.
A notable model that captures population changes over time is proposed in the given exercise. This model reflects realistic circumstances, including thresholds below which the deer may face extinction and above which their numbers may decline due to stressors like disease. This approach provides a more comprehensive view than simpler models and facilitates a better understanding of how populations can be managed through interventions like hunting permits.

Logistic Growth Model

The logistic growth model is a classical representation of population growth that accounts for the carrying capacity of the environment. It is defined by the equation \( \frac{dP}{dt} = rP(M-P) \), where:

• \(P\) is the population size.
• \(r\) is the intrinsic rate of increase.
• \(M\) represents the carrying capacity, the maximum population size the environment can sustain.

This model arises from the assumption that growth rate decreases as the population approaches the carrying capacity. It initially reflects exponential growth but slows as resources become scarcer, ideally reaching a stable point at \(M\).
In comparing this classical model to the exercise's proposed approach, the latter adds an extinction threshold \(m\), suggesting a more nuanced view. It accounts for both upper and lower limits, offering richer insight into scenarios where populations might dip into critical low levels or exceed sustainable numbers. Hence, while the logistic model is foundational, incorporating additional constraints like \(m\) can yield more pragmatic management insights.

Equilibrium Points

In differential equations concerning population dynamics, equilibrium points represent conditions where the population stabilizes and does not change over time. For the deer population model given, the equilibrium points are derived by setting \( \frac{dP}{dt} = rP(M-P)(P-m) \) to zero.
This yields equilibrium points at:

• \(P = 0\) - indicating extinction
• \(P = m\) - the extinction threshold
• \(P = M\) - the carrying capacity

Analyzing the stability of these points provides insights into long-term population behavior:

• \(P = M\) is a stable equilibrium, suggesting a balance when the population doesn't face major perturbations.
• \(P = m\) is unstable, representing a critical threshold where unmanaged decline could spiral to extinction.
• \(P = 0\) implies that total extinction is a stable endpoint below \(m\) if mitigation actions are absent.

Thus, understanding equilibrium points help in strategy formulation for sustainable population control measures.

Phase Line Analysis

Phase line analysis is a qualitative tool used to understand the behavior of solutions to differential equations. By examining a phase line for the equation \( \frac{dP}{dt} = rP(M-P)(P-m) \), you can visualize the stability and direction of population growth or decline.
In this analysis, arrows on the phase line indicate how the population moves relative to equilibrium points:

• For \(P < m\), arrows point towards \(P = 0\), signaling a risk of extinction if the population remains below the threshold.
• Within \((m, M)\), the arrows point towards \(M\), showing an attraction to stability at the carrying capacity.
• When \(P > M\), arrows again lead towards \(M\), driven by environmental pressures like resource depletion and disease.

This method illustrates potential outcomes based on initial conditions and helps visualize trajectories over time, highlighting how adjustments like issuing permits might maintain values within safe population bounds.