The problem asks to explain population growth changes in response to resources using a logistical graph curve. It also asks to identify the part of the graph that exhibits exponential growth, list at least three limiting factors, and explain what happens when a population overshoots its carrying capacity.

1. Problem Description

The problem asks to explain population growth changes in response to resources using a logistical graph curve. It also asks to identify the part of the graph that exhibits exponential growth, list at least three limiting factors, and explain what happens when a population overshoots its carrying capacity.

2. Solution Steps

*Explanation of the logistical graph curve and population growth changes in response to resources:*

A logistical growth curve represents population growth that is initially exponential but slows down as the population approaches the carrying capacity ($K$) of the environment. Carrying capacity ($K$) is the maximum population size that the environment can sustain given available resources. When resources are abundant, the population grows rapidly, but as resources become limited, the growth rate decreases.

The general equation for logistic growth is:

$\frac{dN}{dt} = r_{\text{max}} N \frac{(K-N)}{K}$

Where:

$N$ is the population size.

$t$ is time.

$r_{\text{max}}$ is the intrinsic rate of increase.

$K$ is the carrying capacity.

As $N$ approaches $K$, the term $\frac{(K-N)}{K}$ approaches 0, thus reducing the rate of population growth, $\frac{dN}{dt}$.

*Identifying the part of the graph exhibiting exponential growth:*

The early portion of the logistical growth curve, where the population size is small relative to the carrying capacity, exhibits exponential growth. During this phase, resources are abundant, and the population grows at a rate close to its intrinsic rate of increase. This is the initial, nearly vertical, segment of the "S" shaped curve.

*Listing at least 3 limiting factors:*

Limiting factors are environmental conditions that restrict population growth. Examples include:

1. Availability of food and water.

2. Space or habitat availability.

3. Predation pressure.

4. Disease.

5. Competition for resources.

6. Accumulation of waste products.

*Answering the question: What happens when a population "overshoots" its carrying capacity?:*

When a population overshoots its carrying capacity, it exceeds the resources available in the environment to sustain that population size. This leads to a decline in population size, often drastically. The decline occurs due to increased mortality (death rate) and/or decreased natality (birth rate) resulting from resource scarcity, increased competition, increased disease transmission, or other density-dependent factors. The population may then fluctuate around the carrying capacity or experience a die-off, where a large portion of the population dies.

3. Final Answer

*Population growth changes in response to resources according to a logistical graph curve because initially, resources are abundant, leading to exponential growth. As the population increases, resources become limited, slowing the growth rate until it reaches a stable point at the carrying capacity.*

*The part of the graph exhibiting exponential growth is the initial segment, where the population size is small relative to the carrying capacity.*

*Three limiting factors are: availability of food and water, space or habitat availability, and predation pressure.*

*When a population "overshoots" its carrying capacity, it exceeds the available resources, leading to increased mortality and/or decreased natality, resulting in a population decline, potentially a die-off, and fluctuations around the carrying capacity.*