Consider a hypothetical population of rabbits experiencing logistic growth. If the carrying capacity of their environment is 1000 individuals and the current population is 500, what would you expect to happen to the population?

Correct Answer: The population growth will slow down and eventually stabilize near the carrying capacity.

Step-by-Step Explanation:

1. Understanding Logistic Growth:
In ecology, logistic population growth describes how a biological population grows when resources are limited. Unlike exponential growth (where a population continuously increases without limit), logistic growth accounts for environmental resistance such as food availability, space, and competition.

2. The Concept of Carrying Capacity ($K$):
The carrying capacity, represented by $K$, is the maximum population size that a specific environment can sustainably support over time. In this problem, the carrying capacity is given as $K=1000$ individuals.

3. Population Size ($N$) and Growth Rate:
The current population size is $N=500$ rabbits. The mathematical equation for logistic growth is:

$\frac{dN}{dt}$

Where:
- $\frac{dN}{dt}$ is the rate of population change.
- $r$ is the intrinsic growth rate.
- $N$ is the current population size.
- $K$ is the carrying capacity.

4. Analyzing the Growth Dynamics:
- When $N=500$, the factor $\left(1-\frac{500}{1000}\right)=0.5$. At this point, the population is experiencing its maximum absolute growth rate (the inflection point of the S-shaped logistic curve).
- As the population $N$ continues to increase beyond 500 towards the carrying capacity $K=1000$, the term $\left(1-\frac{N}{K}\right)$ approaches $0$.
- This causes the growth rate $\frac{dN}{dt}$ to slow down progressively as resources become more limited.

5. Final Stabilization:
As $N$ reaches 1000, growth approaches zero ($\frac{dN}{dt}\to 0$), and the rabbit population stabilizes near the environment's carrying capacity.