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\(\begin{array}{|c|c|}\hline \text { Date } & {N} \\ \hline 5 / 1 /12 & {56} \\\ \hline 6 / 1/12 & {98} \\ \hline 7 / 2 / 12 & {203} \\ \hline 8 / 10 / 12 & {421} \\ \hline\end{array}\) These data were collected on a population of beetles in Florida. Based on the data, how would you describe population growth in this case and what do you predict about growth of this population in the future? Explain your reasoning. a. Population shows logistic growth, as number of individuals doubles every month and will likely continue to grow logistically until its resources become depleted. At that point, the population growth rate will slow down and level off to zero. b. The population shows exponential growth, as the number of individuals doubles every month and will likely grow logistically in the future when the resources become limited. At that point, the population growth rate will slow down and level off to zero. c. The population shows exponential growth, as number of individuals doubles every month and will likely continue to grow exponentially until its resources become limited. At that point, the growth will become logistic; the population growth rate will slow down and level off to zero. d. The population shows logistic growth and is likely to grow exponentially as the resources are probably increasing. The population growth rate will increase in the future as well.

Step by step solution

01

Understanding the Data

Observe the given data to understand the population numbers of beetles over specific dates. The population increases from 56 to 98, then to 203, and finally to 421.

02

Checking for Doubling Pattern

Determine if the population approximately doubles each month. Check: 56 doubled is 112 (close to 98). 98 doubled is 196 (close to 203). 203 doubled is 406 (close to 421).

03

Identifying the Growth Type

Recognize the pattern where the population approximately doubles each month. This doubling suggests an exponential growth rather than a logistic growth, as exponential growth is characterized by consistent proportional increases.

04

Predict Future Growth

Infer that the population is showing exponential growth based on the doubling pattern. However, in the future, as resources become limited, the growth will transition to logistic growth, where the rate slows down and eventually levels off.

05

Choose the Correct Option

Among the given options, the best description aligns with identifying the current exponential growth and predicting a future transition to logistic growth when resources are limited. Therefore, select option c.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

exponential growth

Exponential growth is a phase where a population doubles in size over consistent periods. Observing the provided beetle data, we notice the numbers jump from 56 to 98, 98 to 203, and 203 to 421 in quick intervals. This pattern strongly indicates exponential growth.
Such growth is mathematically represented as \[ N(t) = N_0 e^{rt} \] where\( N(t) \)is the population size at time\( t \),\( N_0 \)is the initial population size,\( r \) is the growth rate, and\( e \) is the base of natural logarithms (approximately equal to 2.718).
The key characteristic of exponential growth is the consistent proportional increase, causing numbers to rise quickly. However, this phase is typically short-lived as resources like food and space tend to become limited.

logistic growth

Logistic growth occurs when a population's growth slows down as it reaches the environment's carrying capacity. This type of growth pattern is realistic in natural settings as resources are finite.
Initially, the population grows rapidly, following an exponential phase. Over time, the growth rate decreases, and the population size levels off, creating an S-shaped curve referred to as the logistic curve. This pattern is represented by the equation: \[ N(t) = \frac{K}{1 + \left(\frac{K - N_0}{N_0}\right)e^{-rt}} \] Here:\( K \) is the carrying capacity, or the maximum population size that the environment can sustain.
In the context of beetles, while they show exponential growth currently by doubling almost monthly, it is predicted that this growth will transition to logistic mode. As resources get used up, the rapid growth will stabilize, eventually leveling off to a point where the population size remains steady.

population dynamics

Population dynamics studies the changes in population sizes and compositions over time. Various factors influence these dynamics, including birth rates, death rates, immigration, and emigration. For the beetles, the doubling in numbers monthly suggests a high birth rate, contributing to the evident exponential growth.
In natural populations, growth phases transit from exponential to logistic as resources become limiting. Initially, with fewer individuals, resources are ample, leading to rapid, exponential growth. Gradually, as the population increases, the per capita resource availability diminishes, and the growth rate reduces, transitioning into logistic growth.
Other aspects of population dynamics also consider interspecies interactions (like predation) and environmental changes, which can significantly impact population trends. Understanding these dynamics helps predict future changes and manage species and ecosystems more effectively.