Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Endangered Species A conservation organization releases 100 animals of an endangered species into a game preserve. The organization believes that the preserve has a carrying capacity of 1000 animals and that the growth of the herd will be modeled by the logistic curve \(p=\frac{1000}{1+9 e^{-k t}}, \quad t \geq 0\) where \(p\) is the number of animals and \(t\) is the time (in years). The herd size is 134 after 2 years. Find \(k\). Then find the population after 5 years.

Step-by-Step Solution

Answer

After calculating, \(k\) is found to be approximately 0.15. Substituting the \(k\) value into the original equation gives the population after 5 years as approximately 382 animals.

Step 1: Find k

Given that the herd size is 134 after 2 years, substitute \(p = 134\) and \(t = 2\) into the equation and solve for \(k\). The equation becomes \(134=\frac{1000}{1+9 e^{-2k}}\). In order to isolate \(e^{-2k}\), first multiply both sides by \(1+9 e^{-2k}\), then subtract 134 from both sides to get \(9 e^{-2k}=1000-134\), divide by 9 to get \(e^{-2k}=\frac{1000-134}{9}\). Taking natural logarithm on both sides we have \(-2k = \ln\left(\frac{1000-134}{9}\right)\). Finally, solve for \(k\) by dividing both sides by -2.

Step 2: Substitute k in the original equation

Plug the calculated \(k\) value into the original logistic curve to find the population after 5 years, i.e., \(p=\frac{1000}{1+9 e^{-5k}}\) when \(t = 5\). This should give the population after 5 years.

Key Concepts

Carrying Capacity

The concept of carrying capacity is pivotal in understanding population dynamics within an ecosystem. It refers to the maximum number of individuals of a particular species that an environment can sustain indefinitely without degrading the habitat.

Imagine a game preserve where resources such as food, water, and shelter are finite. When the population of an animal species exceeds the preserve's carrying capacity, the resources become insufficient, leading to increased competition, starvation, or disease. This limits the population growth, ensuring it does not exceed the available resources. The carrying capacity for the endangered species in our example is 1000 animals. Recognizing this limit is essential for effective conservation strategies and ecological management.

Logistic Curve

The logistic curve is a common S-shaped curve used to model the growth of populations. It starts with an initial exponential growth phase, where the population increases rapidly. However, as resources become limited, the growth rate slows, and the population size begins to plateau towards the carrying capacity of the environment.

In mathematical terms, the logistic curve is typically represented by the equation: \(p = \frac{K}{1 + Ce^{-rt}}\), where:

• \(p\) is the population size at time \(t\),
• \(K\) is the carrying capacity,
• \(C\) is a constant determined by initial conditions,
• \(e\) is the base of natural logarithms,
• \(r\) is the intrinsic growth rate,
• and \(t\) is time.

The logistic curve accurately describes how our endangered species' population is expected to grow within the game preserve.

Mathematical Modeling in Ecology

Mathematical modeling in ecology offers a powerful tool for predicting and understanding population dynamics. It involves the creation of equations that represent biological processes, like population growth, predation, competition, and disease spread.

Models like the logistic growth model help ecologists assess potential population sizes over time, the impact of environmental pressures, and the efficacy of conservation efforts. These models are based on careful observations and data collection, allowing ecologists to simulate scenarios and decide on action plans for biodiversity preservation. By finding the value of \(k\) in our exercise, conservationists can predict how quickly the endangered species' population might reach its carrying capacity.

Solving Exponential Equations

Exponential equations are common in logistic growth models and require specific methods to solve. To solve for unknown variables, one often isolates the exponential expression and then employs the natural logarithm, as it is the inverse operation of the exponential function.

In our exercise, to find the growth rate \(k\), we rearranged the given logistic equation into an exponential form and then applied the natural logarithm to both sides to solve for \(k\). This technique transformed an unwieldy exponential equation into a linear one that's much easier to manage. Comprehending how to manipulate and solve exponential equations is crucial in many fields of study, including ecology, economics, and physics.