Problem 45 - Applied Calculus

Question

An island has a carrying capacity of 1 million rabbits. (That is, no more than 1 million rabbits can be supported by the island.) The rabbit population is two at time \(t=1\) day and grows at a rate of \(r(t)\) thousand rabbits/day until the carrying capacity is reached. For each of the following formulas for \(r(t)\), is the carrying capacity ever reached? Explain your answer. (a) \(r(t)=1 / t^{2}\) (b) \(r(t)=t\) (c) \(r(t)=1 / \sqrt{t}\)

Step-by-Step Solution

Answer

(a) No, (b) Yes, (c) Yes.

Step 1: Evaluate formula (a)

For formula (a) where the growth rate is given by \(r(t) = \frac{1}{t^2}\), we want to calculate the cumulative growth over time. The growth of the population is modeled by the integral of \(r(t)\) from \(t=1\) to \(t=\infty\), because the initial population at \(t=1\) is 2 rabbits (or 0.002 thousand rabbits). The integral is: \[\int_{1}^{\infty} \frac{1}{t^2} \, dt = \left[ -\frac{1}{t} \right]_1^{\infty} = -\frac{1}{\infty} + \frac{1}{1} = 1\]This shows that the population can only gain up to \(0.001\) thousand more rabbits, far short of the 1 million (or 1000 thousand) limit. Therefore, the carrying capacity is never reached.

Step 2: Evaluate formula (b)

For formula (b) where the growth rate is \(r(t) = t\), the cumulative growth of the population is the integral of \(r(t)\) from \(t=1\). That is:\[\int_{1}^{T} t \, dt = \left[ \frac{t^2}{2} \right]_1^{T} = \frac{T^2}{2} - \frac{1^2}{2} = \frac{T^2}{2} - \frac{1}{2}\] As \(T\) approaches a sufficiently large value, the term \(\frac{T^2}{2}\) grows without bound and will surpass 1000 thousand rabbits. Thus, in this case, the carrying capacity is reached.

Step 3: Evaluate formula (c)

For formula (c) where the growth rate is \(r(t) = \frac{1}{\sqrt{t}}\), we need to find the cumulative growth:\[\int_{1}^{\infty} \frac{1}{\sqrt{t}} \, dt = \left[ 2\sqrt{t} \right]_1^{\infty} = 2\cdot\infty - 2 = \infty\] This shows the population will grow indefinitely, ensuring that beyond a certain point, the total number of rabbits will surpass the 1 million limit. Therefore, the carrying capacity will be reached.

Key Concepts

Population Growth

Population growth is a critical concept in ecology and biology, reflecting how a species increases in number over time. On the island with a carrying capacity of 1 million rabbits, population growth can vary based on natural conditions and available resources. The carrying capacity is the maximum number of individuals an environment can sustainably support. When a population size surpasses this limit, it's often due to temporary conditions that allow for overpopulation, resulting in stress on the ecosystem.
Factors influencing population growth include:

• Availability of resources such as water and food
• Environmental conditions like climate and habitat
• Predation and disease
• Reproductive rates

Understanding these elements can help predict and manage the population size, ensuring a balance within the ecosystem.

Integral Calculus

Integral calculus is a branch of mathematics that deals with accumulation or summation of quantities, often used to find areas, volumes, and in this case, growth over time. In ecological modeling, integrals are used to calculate cumulative population growth. Given a rate of growth function, the integral provides a way to find the total growth over a specific period.
In the context of the rabbit population:

• The integral of the growth rate function, such as \(r(t)=1/t^2\), models how the population increases over a continuous range of time.
• It measures the entire quantity of new rabbits added to the population from a start time \(t=1\) until a limit (like infinity or a specific point).
• The result of this integration reveals whether the population could reach the carrying capacity or not.

Therefore, integral calculus serves as a valuable tool in predicting long-term population trends and potential environmental impacts.

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives, helping to express how a changing system evolves over time. They play a crucial role in population dynamics by modeling how populations develop given certain conditions.
Here's how differential equations tie into the rabbit problem:

• The growth rate \(r(t)\) represents the derivative of the population size with respect to time.
• By integrating these differential equations, we connect the rate of change (growth rate) to the actual population size at any time \(t\).
• This relationship is fundamental in determining if and when a population might hit the carrying capacity.

Differential equations are essential for understanding and forecasting population behaviors, allowing ecologists to make informed decisions based on mathematical predictions.