Question

What is the carrying capacity of a species in a habitat?
Explain the growth curve that takes this capacity into account.

Hint:

Remember that if the term $\left(\frac{K - N}{K}\right)$ is included in a growth equation, it represents environmental resistance. As the population size ($N$) approaches the carrying capacity ($K$), this value drops toward zero, mathematically flattening the growth curve into an S-shape.

Answer 1

Step 1: Understanding the Question:
The question asks for a definition of the ecological term carrying capacity ($K$). It also requires an explanation of the population growth curve (logistic growth) that factors in this environmental limit.

Step 2: Key Formula or Approach:
When resources like food and space are limited, population growth follows a Verhulst-Pearl Logistic Growth pattern. This pattern is mathematically defined by the differential equation: $$\frac{dN}{dt} = rN \left(\frac{K - N}{K}\right)$$ Where $N$ is population density, $r$ is the intrinsic rate of natural increase, and $K$ is the carrying capacity.

Step 3: Detailed Explanation:
(a) Carrying Capacity ($K$): Carrying capacity is defined as the maximum population size of a given biological species that a specific ecosystem can sustainably support over time, given the available resources like food, water, nesting territory, and space. Beyond this point, resource depletion increases mortality rates, causing population growth to level off.
(b) Logistic Growth Curve: When resources are limited, a population exhibits a Sigmoid (S-shaped) growth curve. This pattern consists of four distinct phases: 1. Lag Phase: Initial growth is slow while the population acclimates to the new environment.
2. Acceleration Exponential Phase: As individuals reproduce, the population grows rapidly because resources are still relatively abundant.
3.
Deceleration Phase: Growth begins to slow down as resources become scarce and competition intensifies.
4.
Asymptote Phase: The population size stabilizes and reaches a plateau when it matches the carrying capacity ($N = K$), bringing net growth ($\frac{dN}{dt}$) to zero.
This model provides a more realistic representation of natural populations than exponential models, because resources in the real world are always limited.

Step 4: Final Answer:
(a) Carrying capacity is the maximum number of individuals of a species that a habitat can sustainably support with its limited resources .
(b) This model produces a Sigmoid (S-shaped) growth curve containing a lag phase, a rapid acceleration phase, a deceleration phase, and a flat asymptote phase where the population stabilizes at the carrying capacity ($K$).