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If the original fixed-point multiplier has modulus greater than one, its square also exceeds one, so the same fixed point is unstable for the twice-iterated map. New stable fixed points of the twice-iterated map, if present, must be different points; they can represent a genuine two-cycle of the original map.

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At low density the per-generation growth factor is approximately

α / β

. The power

γ

controls how sharply crowding reduces recruitment or survival when resources become limiting; at high density

F (u) \sim α u^{1 - γ}

. Larger

γ

therefore means a stronger overcompensation in a population model.

The flip threshold says that sufficiently strong density feedback can destabilize a constant positive biological population and create alternating generations, under the usual nondegeneracy conditions. A local loss of fixed-point stability alone does not establish chaos, nor does it prove every ecological system with stronger crowding must follow the same route.

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The fixed-point multiplier decreases continuously with

γ

and first reaches the unit circle at

- 1

. Solving

1 - γ_{*} (1 - β / α) = - 1

gives

γ_{*} = \frac{2 α}{α - β} > 2.

(1)

It is stable for

1 < γ < γ_{*}

and unstable for

γ > γ_{*}

. At the critical value linear stability is inconclusive. The crossing at

- 1

is the local signature of a period-doubling bifurcation, rather than a crossing through

+ 1

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A positive fixed point requires

β + u_{*}^{γ} = α

, so it exists exactly when

α > β

and is

u_{*} = (α - β)^{1/ γ}

. Differentiating the map at it gives

F^{'} (u_{*}) = 1 - γ (1 - \frac{β}{α}) .

(1)

For

1 < γ \leq 2

and

0 < β / α < 1

, this fixed-point multiplier is strictly between

- 1

and

1

. Hence the positive fixed point is linearly asymptotically stable, including

γ = 2

; positivity of

β

makes the lower bound strict.

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The iteration of a map is

F (u) = u / (2 + u^{2})

. For every

u > 0

0 < F (u) < u /2

, so

0 < u_{n} \leq 2^{- n} u_{0}

and

u_{n} \to 0

. A cobweb alternates vertical steps to the graph and horizontal steps to the diagonal, approaching the origin monotonically.

Figure 1.
Resource-limited population cobweb: every positive orbit approaches extinction
. **Details**.

More precisely

u_{n + 1} / u_{n} \to 1/2

. The product

2^{n} u_{n} = u_{0} \prod_{j < n} (1 + u_{j}^{2} /2)^{- 1}

has a finite strictly positive limit because

\sum_{j} u_{j}^{2} < \infty

. Thus

u_{n} \sim C (u_{0}) 2^{- n}

for a positive constant depending on the initial biological population.

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 Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.

Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

. Source.

We have two killer features:

topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
Articles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1.
Screenshot of the "Derivative" topic page
. View it live at: ourbigbook.com/go/topic/derivative
Video 2.
OurBigBook Web topics demo
. Source.
local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
Figure 2.
You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website
.
Figure 3.
Visual Studio Code extension installation
.
Figure 4.
Visual Studio Code extension tree navigation
.
Figure 5.
Web editor
. You can also edit articles on the Web editor without installing anything locally.
Video 3.
Edit locally and publish demo
. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
. Source.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade

All our software is open source and hosted at: github.com/ourbigbook/ourbigbook

Further documentation can be found at: docs.ourbigbook.com

Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact

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 Pinned article: Introduction to the OurBigBook Project