IGNOU MMT 9 2026 - Consider a discrete model given byInvestigate the linear stabi

Question

Consider a discrete model given by

$$N_{t+1} = \frac{r N_t}{1 + b N_{t-1}^2} = f(N_t), r > 1.$
Investigate the linear stability about the positive steady state N by setting $N_t = N^</em> + n_t$ . Show that n_t satisfies the equation
$$n_{t+1} - n_t + 2(r - 1)r^{-1}n_{t-1} = 0.$

Hence show that $r = 2$ is a bifurcation value and that as $r \rightarrow 2$ the steady state bifurcates to a periodic solution of period 6.

10 Jan 2026

Answer :

Word Count : 187

The positive steady state (N^*) is obtained by setting (N_{t+1}=N_t=N_{t-1}=N^*) in [ N_{t+1}=\frac{rN_t}{1+bN_{t-1}^2}, ] which gives [ N^*=\frac{rN^*}{1+b(N^*)^2}. ] Since (N^*>0), this reduces to [ 1+b(N^*)^2=r \quad \Rightarrow \quad b(N^*)^2=r-1. ] To investigate linear stability, write [ N_t=N^*+n_t,\qquad |n_t|\ll1. ] Substituting into the model gives [ N_{t+1}=\frac{r(N^*+n_t)}{1+b(N^*+n_{t-1})^2}. ] Expanding the denominator to first order, [ 1+b(N^*+n_{t-1})^2=1+b(N^*)^2+2bN^*n_{t-1}+O(n_{t-1}^2). ] Using (1+b(N^*)^2=r), this becomes [ r+2bN^*n_{t-1}. ]

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