As n approaches k for a certain population, which of the following is predicted by the logistic equation?

General
1

as n approaches k for a certain population, which of the following is predicted by the logistic equation?

2

As ( n ) approaches ( k ) for a certain population, which of the following is predicted by the logistic equation?

Answer:

In population dynamics, the logistic equation is used to model population growth while accounting for environmental constraints such as limited resources. The logistic growth equation is written as:

\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)

Here:

  • ( N ) represents the population size.
  • ( r ) represents the intrinsic growth rate (rate of reproduction per individual).
  • ( K ) represents the carrying capacity of the environment, which is the maximum population size that the environment can sustain indefinitely.
  • ( \frac{dN}{dt} ) represents the rate of population growth over time.

Key Dynamics of Logistic Growth:

When ( N ) approaches ( K ), the term ( \left(1 - \frac{N}{K}\right) ) becomes close to zero. This leads to the following predictions:

  1. Population Growth Rate Slows Down:
    As ( N ) gets closer to ( K ), the growth rate ( \frac{dN}{dt} ) decreases significantly because the factor ( \left(1 - \frac{N}{K}\right) ) approaches zero. Growth effectively halts when ( N = K ).

    Example Calculation:
    Let’s say ( r = 1.5 ), ( K = 1000 ), and ( N = 990 ). Substituting these values into the equation:

    \frac{dN}{dt} = 1.5 \times 990 \times \left(1 - \frac{990}{1000}\right)
    \frac{dN}{dt} = 1.5 \times 990 \times (1 - 0.99)
    \frac{dN}{dt} = 1.5 \times 990 \times 0.01 = 14.85

    Thus, the population growth rate becomes very small as ( N ) approaches ( K ).

  2. Population Levels Off to Carrying Capacity:
    When ( N = K ), the term ( \left(1 - \frac{N}{K}\right) ) becomes zero, resulting in ( \frac{dN}{dt} = 0 ). This means the population stops growing because the resources needed for further growth are fully utilized.

  3. Stable Equilibrium:
    At ( N = K ), the population reaches a stable equilibrium. The number of births equals the number of deaths, and the population size remains constant barring external influences.

Interpretations of Logistic Growth Model:

  • Density-Dependent Limitation:
    The logistic model predicts that as ( N ) approaches ( K ), density-dependent factors such as competition for resources, predation, and disease increasingly limit growth.

  • S-Shaped Curve:
    The logistic equation produces an S-shaped curve when population size is plotted over time. Growth is initially exponential (when ( N << K )), slows as ( N ) approaches ( K ), and levels off when ( N = K ).

Correct Prediction:

The population growth rate slows as ( N ) approaches ( K ), eventually stopping when ( N ) equals ( K ).

Let me know if you’d like me to break this down further or include additional examples, @anonymous13!