For a population that grows exponentially in the time interval (t, t+1), we have N_t $+1=RN$_t, where NNN denotes population size and RRR denotes the growth rate. Under intraspecific competition where births and deaths are density-dependent, we expect the population to stabilize at carrying capacity, KKK. In the figure below, N_t / N_t+1 is plotted as a linear function of N_t

We may write down the linear equation for the line joining A with B and derive a model for density-dependent population growth under intraspecific competition. Denoting $(R-\; 1)K\; as\; a$, which of the following is the correct relationship that describes population growth?​(R−1)K\frac{(R-1)}{K}

Correct option is A

The given equation describes a population growing exponentially but modified by intraspecific competition, where the growth rate decreases as the population size increases. The linear equation joining A and B in the graph represents the relationship between_​ N_t and $$\frac{N_{t}}{N_{t+1}}$$​ reflecting density-dependent growth.

The general equation for such a model, considering the effect of competition, is given by:

Where:

• NtN_tNt​ is the current population size,
  ​
• $R$ is the intrinsic growth rate,
• $a$ is a constant that represents the density-dependent competition effect.

This equation implies that as the population size increases, the growth rate slows down due to intraspecific competition, stabilizing when the population reaches the carrying capacity $K$.

Information Booster:

• Density-Dependent Growth: This occurs when the rate of population growth depends on the population density. As the population increases, resources become limited, and the growth rate decreases.
• Carrying Capacity $K$: The carrying capacity is the maximum population size that the environment can sustain. At this point, the growth rate becomes zero because the resources are fully utilized.
• Parameter $a$: In the equation $$\frac{(R-1)}{K}$$​, $a$ represents the strength of density dependence. It influences how quickly the growth rate slows as the population size increases.
• Exponential Growth: The equationNt+1=RNtN_{t+1} = R N_tNt+1​=RNt​ represents exponential growth, where the population increases without any limitations. The presence of density dependence modifies this equation to reflect more realistic population dynamics.
• Population Stabilization: The equation $$N{t+1}$$ = $$\frac{N_tR}{1+aN_t}$$ models a population that initially grows exponentially but then slows as it approaches carrying capacity, eventually stabilizing.