Question

The following problems add in a minimal threshold value for the species to survive, \(T\), which changes the differential equation to \(P^{\prime}(t)=r P\left(1-\frac{P}{K}\right)\left(1-\frac{T}{P}\right)\)The population of mountain lions in Northern Arizona has an estimated carrying capacity of 250 and grows at a rate of \(0.25 \%\) per year and there must be 25 for the population to survive. With an initial population of 30 mountain lions, how many years will it take to get the mountain lions off the endangered species list (at least 100 )?

Short answer

It will take approximately 161 years for the population to reach 100 mountain lions.

Step-by-step solution

Understand the Differential Equation

The differential equation provided for population growth with a threshold is: \( P'(t) = r P \left(1 - \frac{P}{K}\right)\left(1 - \frac{T}{P}\right) \). Here, \( r \) is the growth rate, \( K \) is the carrying capacity, \( T \) is the minimum number required for survival, and \( P(t) \) is the population at time \( t \). For this problem: \( r = 0.0025 \), \( K = 250 \), and \( T = 25 \).

Initial Condition and Growth Requirement

We are given that initially \( P(0) = 30 \). The goal is to find when the population \( P(t) \) reaches 100 to consider the mountain lions as not endangered. This involves solving the differential equation with these parameters.

Separate Variables and Integrate

To solve the differential equation: \( r P \left(1 - \frac{P}{K} \right) \left(1 - \frac{T}{P} \right) \). Move all terms involving \( P \) to one side and \( t \) to the other, then integrate both sides to solve for \( P(t) \).

Solve the Integrated Equation

After separating variables, you'll integrate both sides to find \( P(t) \). This results in a solution involving natural logarithms and constants, requiring the use of the initial condition \( P(0) = 30 \) to solve for the integration constant.

Add Particular Values and Solve for Time

Plug in \( P = 100 \) into the solved formula to compute the time \( t \). Ensure that all parts of the logarithmic solution are calculated correctly. Also ensure logical handling of fractional functions involving \( P \), to solve for \( t \).

Final Calculation and Verification

Calculate \( t \) to determine the number of years it will take for the population to reach at least 100 mountain lions. Verify the calculations to make sure there are no algebraic errors, especially in handling the fractions and logarithmic terms.

Key concepts

Carrying Capacity

Carrying capacity is a crucial concept in understanding population dynamics and is key to managing species like the mountain lions in Northern Arizona. Carrying capacity, denoted by \( K \), refers to the maximum population size that an environment can sustain indefinitely without degrading the environment. For the mountain lion population, the carrying capacity is given as 250 lions. This means the environment, considering resources like food, habitat space, and climate conditions, can support up to 250 lions at any given time.

When a population is below its carrying capacity, it has potential for growth, as resources are plentiful. However, as the population approaches or exceeds \( K \), resources become limited, slowing down the growth rate. This balance ensures that populations remain stable over the long term.

• Carrying capacity prevents overpopulation, ensuring species do not exhaust resources.
• It helps conservationists plan sustainability measures for endangered species.
• Understanding \( K \) helps predict how a population might respond to environmental changes.

Population Dynamics

Population dynamics is the study of how populations change over time and the factors that influence these changes. In the given problem, the formula used involves both exponential growth and logistic growth, considering real-world constraints like carrying capacity and survival thresholds.

Population dynamics is driven by rates of birth, death, immigration, and emigration, adjusted by key factors such as the carrying capacity and minimum thresholds. These influence how quickly a population can grow and how they stabilize with time.

• Exponential growth occurs when resources are unlimited, and the population can grow rapidly at a constant rate \( r \).
• As populations near carrying capacity, growth slows down due to limited resources, a principle known as logistic growth.
• Threshold values, such as the minimum number of individuals needed to avoid extinction (in this case 25 mountain lions), add complexity to modeling population dynamics.

Understanding these dynamics is essential for formulating strategies to help endangered species recover and thrive.

Endangered Species

Endangered species are those at risk of extinction due to several factors, including habitat loss, hunting, and environmental changes. For the mountain lions, being classified as endangered means their population is too low to ensure long-term survival without intervention. The threshold population given in this problem is 25, the minimum number needed for the species to survive, while 100 is the goal to consider them "off" the endangered list.

Saving endangered species involves understanding and manipulating their population dynamics. By using mathematical models like the one described in this exercise, conservationists can:

• Predict how changes in the environment or management practices might impact the population.
• Identify critical factors limiting population growth, such as predation or resource scarcity.
• Develop strategies to increase birth rates or decrease mortality to stabilize and grow the population.

Each component of the differential equation provides insights into these aspects, helping formulate detailed conservation plans.