The size of an undisturbed fish population has been modeled...

The size of an undisturbed fish population has been modeled by the formula $$ p_{n+1}=\frac{b p_{n}}{a+p_{n}} $$ where $p_{n}$ is the fish population after $n$ years and $a$ and $b$ are positive constants that depend on the species and its environment. Suppose that the population in year 0 is $p_{0} > 0$ . (a) Show that if $\left\{p_{n}\right\}$ is convergent, then the only possible values for its limit are 0 and $b-a$ . (b) Show that $p_{n+1}<(b / a) p_{n}$ (c) Use part (b) to show that if $a > b,$ then $\lim _{n \rightarrow \infty} p_{n}=0$ in other words, the population dies out. (d) Now assume that $a < b .$ Show that if $p_{0} < b-a$ , then $\left\{p_{n}\right\}$ is increasing and $0 < p_{n}< b-a$ . Show also that if $p_{0} >b-a$ , then $\left\{p_{n}\right\}$ is decreasing and $p_{n} > b-a$ Deduce that if $a < b$ , then $\lim _{n \rightarrow \infty} p_{n}=b-a$

The size of an undisturbed fish population has been modeled
by the formula
$$
p_{n+1}=\frac{b p_{n}}{a+p_{n}}
$$
where $p_{n}$ is the fish population after $n$ years and $a$ and $b$ are
positive constants that depend on the species and its environment. Suppose that the population in year 0 is $p_{0} > 0$ .
(a) Show that if $\left\{p_{n}\right\}$ is convergent, then the only possible
values for its limit are 0 and $b-a$ .
(b) Show that $p_{n+1}<(b / a) p_{n}$
(c) Use part (b) to show that if $a > b,$ then $\lim _{n \rightarrow \infty} p_{n}=0$
in other words, the population dies out.
(d) Now assume that $a < b .$ Show that if $p_{0} < b-a$ , then
$\left\{p_{n}\right\}$ is increasing and $0 < p_{n}< b-a$ . Show also that
if $p_{0} >b-a$ , then $\left\{p_{n}\right\}$ is decreasing and $p_{n} > b-a$
Deduce that if $a < b$ , then $\lim _{n \rightarrow \infty} p_{n}=b-a$

Key Concepts

Fixed Points

This concept involves finding values that remain unchanged under the recurrence—that is, values L for which L = f(L). In the context of iterative models, determining fixed points helps in understanding the long?term behavior by solving the equation L = (bL)/(a+L), which yields possible limits of the sequence.

Monotonic Sequences

Monotonic sequences are those that are either non-decreasing or non-increasing. Establishing that a sequence is monotonic, together with boundedness, guarantees convergence. In the context of population models, showing that the sequence is increasing or decreasing under certain conditions is crucial for determining the limit behavior.

Comparison Techniques in Recurrences

Comparison techniques involve deriving inequalities or bounds that relate successive terms of a recurrence. For instance, showing that p??? is bounded above by (b/a) p? allows one to compare the sequence with a geometric progression. This method is important for assessing whether the sequence grows, decays, or remains bounded in the long run.

Stability of Equilibria

Stability analysis studies whether small deviations from an equilibrium point will diminish (leading back to the equilibrium) or amplify (leading away from it). In iterative models, understanding the stability of fixed points (or equilibria) informs us which of the fixed points will attract the sequence over time and which will not.

Recurrence Relations Analysis

This concept covers the study of sequences defined recursively. Analyzing recurrence relations involves identifying fixed points, checking for monotonicity, deriving bounds, and applying convergence tests. This systematic approach is foundational for understanding the long-term dynamics of discrete-time models in various fields.

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