1.8: Solution to the dynamic equation Pₜ₊₁-Pₜ=rPₜ + b

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The general form of the dynamic equation of the previous section is,

\[P_{t+1}-P_{t}=r P_{t}+b, \quad t=0,1, \cdots, \quad P_{0} \quad \text { a known value } \label{1.24}\]

The constant b represents influence from outside the system, such as flow or immigration into (b > 0) or flow or emigration out of (b < 0) the system.

Suppose \(r \neq 0\) and \(b \neq 0\). We convert

\[P_{t+1}-P_{t}=r P_{t}+b\]

to iteration form

\[P_{t+1}=R P_{t}+b, \quad \text { where } \quad R=r+1 \label{1.25}\]

Next we look for an equilibrium value E satisfying

\[E=R E+b \label{1.26}\]

(Think, if \(P_{t} = E\) then \(P_{t+1} = E\).) We can solve for E

\[E=\frac{b}{1-R}=-\frac{b}{r} \label{1.27}\]

Now subtract Equation \ref{1.26} from Equation \ref{1.25}.

\[\begin{aligned}
P_{t+1} &=R P_{t}+b \\
E &=R E+b
\end{aligned}\]

\[P_{t+1}-E=R\left(P_{t}-E\right) \label{1.28}\]

Let \(D_{t} = P_{t} − E\) and write

\[D_{t+1}=R D_{t}\]

This is iteration Equation 1.2.4 with DT instead of \(P_t\) and the solution is

\[D_{t}=R^{t} D_{0}\]

Remembering that \(D_{t} = P_{t} − E\) and \(E = \frac{b}{(1 − R)} = −\frac{b}{r}\) and \(R = 1 + r\) we write

\[P_{t}-E=R^{t}\left(P_{0}-E\right), \quad P_{t}=-\frac{b}{r}+\left(P_{0}+\frac{b}{r}\right)(1+r)^{t} \label{1.29}\]

You are asked to show in Exercise 1.8.2 that the solution to

\[P_{t+1}-P_{t}=b \quad \text { is } \quad P_{t}=P_{0}+t b \label{1.30}\]

Exercises for Section 1.8, Solution to the dynamic equation, \(P_{t+1}-P_{t}=r P_{t}+b\)

Exercise 1.8.1 Write a solution equation for the following initial conditions and difference equations or iteration equations. In each case, compute \(B_{100}\).

\(B_{0}=100 \quad B_{t+1}-B_{t}=0.2 B_{t}+5\)
\(B_{0}=138 \quad B_{t+1}-B_{t}=0.05 B_{t}+10\)
\(B_{0}=138 \quad B_{t+1}-B_{t}=0.5 B_{t}-10\)
\(B_{0}=100 \quad B_{t+1}-B_{t}=10\)
\(B_{0}=100 \quad B_{t+1}=1.2 B_{t}-5\)
\(B_{0}=100 \quad B_{t+1}-B_{t}=-0.1 B_{t}+10\)
\(B_{0}=100 \quad B_{t+1}=0.9 B_{t}-10\)
\(B_{0}=100 \quad B_{t+1}=-0.8 B_{t}+20\)

Exercise 1.8.2 Equation \ref{1.30}

\[P_{t+1}-P_{t}=b\]

represents a large number of equations

\[\begin{aligned}
P_{1}-P_{0} &=b \\
P_{2}-P_{1} &=b \\
P_{3}-P_{2} &=b \\
\vdots \\
P_{n-1}-P_{n-2} &=b \\
P_{n}-P_{n-1} &=b
\end{aligned}\]

Add these equations to obtain

\[P_{n}=P_{0}+n b\]

Substitute t for n to obtain

\[P_{t}=P_{0}+t b\]

Exercise 1.8.3 Suppose a quail population would grow at 20% per year without hunting pressure, and 1000 birds per year are harvested. Describe the progress of the population over 5 years if initially there are

5000 birds
6000 birds
4000 birds