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 $p_{x+1}<(b / a) p_{x}$ (c) Use part (b) to show that if $a>b$, then $\lim _{x \rightarrow \infty} p_{x}=0$ : 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 $\left\langle p_{0}\right\}$ is decreasing and $p_{n}>b-a$ Deduce that if $a<b,$ then $\lim _{x \rightarrow \infty} p_{x}=b-a$

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 $p_{x+1}<(b / a) p_{x}$
(c) Use part (b) to show that if $a>b$, then $\lim _{x \rightarrow \infty} p_{x}=0$ :
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 $\left\langle p_{0}\right\}$ is decreasing and $p_{n}>b-a$ Deduce that if $a<b,$ then $\lim _{x \rightarrow \infty} p_{x}=b-a$

Key Concepts

Fixed Points in Recurrence Relations

This concept refers to the equilibrium or steady state values that a sequence (or dynamical system) eventually approaches. In the context of a recurrence relation, a fixed point is a value where, when substituted into the recurrence, yields the same value. Analyzing fixed points provides a way to understand the long-term behavior of the system by solving for the condition where p(n+1) = p(n).

Stability Analysis in Discrete Dynamical Systems

Once fixed points are identified, it is essential to determine whether these points are stable or unstable—that is, whether nearby initial values converge to the fixed point or diverge away from it. Stability analysis involves evaluating whether the system's recurrence will bring the sequence closer to or farther from these equilibria over time, thus predicting the eventual fate of the system (e.g., persistence or extinction).

Monotonicity and Boundedness

Monotonicity refers to a sequence that is either non-decreasing or non-increasing, while boundedness means the sequence is confined within upper and lower limits. These concepts are critical because monotone and bounded sequences are guaranteed to converge by the Monotone Convergence Theorem. Demonstrating that a recurrence relation yields a monotone sequence bounded by a fixed point helps in proving convergence to that fixed point.

Inequalities in Recurrence Relations

Using inequalities in recurrence relations is a powerful technique to compare the given recurrence with simpler or well-known sequences, such as geometric sequences. These comparisons can provide upper or lower bounds for the terms in the sequence, aiding in the demonstration of convergence or divergence of the sequence, and in establishing conditions under which the population described by the recurrence dies out or stabilizes.

Transcript

00:01 Here, pn represents a fish population after n years, and that population is modeled by this formula over here.

00:12 Okay, so for part a, we'll show the following.

00:21 If the sequence converges, then the limit, its n goes to infinity, is either zero or b minus a.

00:41 So let's go ahead and verify this fact.

00:51 Solution.

00:52 So let's just go ahead and suppose, so we're supposing that it converges.

00:56 So let's just call that limit l.

01:10 Now, using the given formula for pn up here, let's take a limit on both sides of this equation as n goes to infinity.

01:31 So we're applying the limit on both sides.

01:34 So this becomes, and this can be simplified now to a quadratic.

01:56 Okay, so now at this step, l equals 0 is a solution to this equation over here.

02:06 However, if l is not equal to 0, then we can divide it to obtain, and there are two solutions to this quadratic.

02:18 Zero or b minus a, and that was what we wanted to establish for part a.

02:27 So if it converges, the limit, it either dies off at 0 or it stabilizes at b minus a, which is a constant.

02:37 Okay, with that said, let's go on to the next page for part b.

02:45 So this is where we'd like to show the following inequality.

02:58 So let's go ahead and give a solution for this.

03:02 We have pn plus 1.

03:07 By definition or by the recursion formula, given for pn, we can write this.

03:15 Now let's divide top and bottom by a.

03:25 A over a is just 1, and then we get pn over a.

03:30 Now i can just go ahead and ignore that denominator.

03:38 The reason i can do this, we're dealing with positive numbers here.

03:52 In the beginning, a was a positive fish population.

03:55 Pn has to be non -negative.

03:57 So this is bigger than or equal to one.

04:02 Or in this case, just equal to one.

04:11 So that justifies this inequality here.

04:15 And that resolves part b because this is what we wanted.

04:19 Pn plus 1 and then b over a pn and here this was justify this okay let me go on i'll need some more room here so let me go on to part c but on the next page so here we like to use part b to show that if a is bigger than b then limit a pn goes to zero as n goes to infinity okay, so one way to show this is the five so if the series pn converges, then by the test for divergence, we have the limit of pn is zero.

05:51 So let's just go ahead and try to establish this fact here, and then we'll finish the problem.

05:57 So let's try the ratio test for this using part c in mind.

06:03 Because in part c, remember, excuse me from part b, we had p to the n plus 1, it was less than b over a.

06:21 Pn.

06:23 So now we try the ratio test here.

06:36 Since we're just dealing with positive numbers, we could drop the absolute value.

06:41 This is b over a and since now in part c we're assuming a is larger than b.

06:49 That will imply b over a is less than one so that the limit of pn plus 1 over pn is less than 1...

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