For a population that grows exponentially in the time interval (t, t + 1), we have Nt+1 = R Nt, where N denotes population size and R denotes the growth rate. Under intraspecific competition where births and deaths are density dependent, we expect the population to stabilize at carrying capacity, K. In the figure below, Nt/Nt+1 is plotted as a linear function of Nt.
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?
1
\(N_{t+1}=\frac{N_t R}{\left(1+a N_t\right)}\)
2
\(N_{t+1}=\frac{a N_t}{\left(1+R N_t\right)}\)
3
\(N_{t+1}=\frac{N_t R}{\left(a+N_t\right)}\)
4
\(N_{t+1}=\frac{a N_t R}{\left(1+a N_t\right)}\)