How I wish logistic growth was taught to me in Calc 2

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[bloom]

At this point you should just say whenever the video is NOT sponsored by Brilliant 😂

[supernovaxs]

It’s so weird seeing Zach go back into serious STEM mode from his second channel

[mgm]

Zach is still able and willing to teach use, thanks bro

[ChalkyWhiteChalkyWhite]

mate just make your own series' of videos courses your teaching is quality.

[no-bkzx]

The timing is impeccable. I was just taught population modelling this semester but never really understood exactly what the differential equation meant. I knew how to solve it and how to model the curves but not what it meant. Thanks a lot this was genuinely insightful.

[ShinDMitsuki]

I was waiting for the punchline and then remembered I originally subbed to you because you are an engineer who made math videos 😂

[ekandrot]

I remember a few years ago on the news, everyone was explaining virus spread using exponential growth rather than logistic growth. At that time and still, I believe logistic is what should have been used and your video is a great explanation as to why and what the correct curves should have been.

[anindyaguria]

I'm able AND willing to solve this differential eqn.

[_mishi]

Gotta mention existence and uniqueness theorem there. For this exact differential equation its fine, but generally this rule can be used only if the existence and uniqueness is satisfied, otherwise further analysis is required

[stokedfool]

Logistics growth explained with visuals. Great example involving the effects of hunting on deer populations.

[natanprzybylko]

Saw this video a couple weeks ago before I learned about logistic growth, now I'm coming back while I'm learning it lol

[mattias]

One of the most fun courses i had in University was non linear dynamics which was essentially doing analysis of differential equations which are non linear (hard to solve usually) in a qualitative way. There is so much information you can infer about solutions without ever solving anything

This video reminded me a lot about how we approached things there

[northernlight]

Great video Zach! and yes often times in math, the higher level you learn certain fundamental topics a lot of the time people really stop looking for actual explicit solutions to things for the most part and focus much more on trying to find all the possible qualitative information they can about that certain problem

[iteerrex]

Love differential equations. They are so powerful and it applies to so much. 👍😁

[Stacee-jxyz]

1) Calculus Foundations:

Contradictory:
Newtonian Fluxional Calculus
dx/dt = lim(Δx/Δt) as Δt->0

This expresses the derivative using the limiting ratio of finite differences Δx/Δt as Δt shrinks towards 0. However, the limit concept contains logical contradictions when extended to the infinitesimal scale.

Non-Contradictory:
Leibnizian Infinitesimal Calculus
dx = ɛ, where ɛ is an infinitesimal
dx/dt = ɛ/dt

Leibniz treated the differentials dx, dt as infinite "inassignable" infinitesimal increments ɛ, rather than limits of finite ratios - thus avoiding the paradoxes of vanishing quantities.

2) Foundations of Mathematics

Contradictory Paradoxes:
- Russell's Paradox, Burali-Forti Paradox
- Banach-Tarski "Pea Paradox"
- Other Set-Theoretic Pathologies

Non-Contradictory Possibilities:
Algebraic Homotopy ∞-Toposes
a ≃ b ⇐⇒ ∃n, Path[a, b] in ∞Grpd(n)
U: ∞Töpoi → ∞Grpds (univalent universes)

Reconceiving mathematical foundations as homotopy toposes structured by identifications in ∞-groupoids could resolve contradictions in an intrinsically coherent theory of "motive-like" objects/relations.

3) The Unification of Physics

Contradictory Barriers:
- Clash between quantum/relativistic geometric premises
- Infinities and non-renormalizability issues
- Lack of quantum theory of gravity and spacetime microphysics

Non-Contradictory Possibilities:
Algebraic Quantum Gravity
Rμν = k [ Tμν - (1/2)gμνT ] (monadic-valued sources)
Tμν = Σab Γab, μν (relational algebras)
Γab, μν = f(ma, ra, qa, ...) (catalytic charged mnds)

Treating gravity/spacetime as collective phenomena emerging from catalytic combinatorial charge relation algebras Γab, μν between pluralistic relativistic monadic elements could unite QM/QFT/GR description.

4) Formal Limitations and Undecidability

Contradictory Results:
- Halting Problem for Turing Machines
- Gödel's Incompleteness Theorems
- Chaitin's Computational Irreducibility

Non-Contradictory Possibilities:
Infinitary Realizability Logics
|A> = Pi0 |ti> (truth of A by realizability over infinitesimal paths)
∀A, |A>∨|¬A> ∈ Lölc (constructively locally omniscient completeness)

Representing computability/provability over infinitary realizability monads rather than recursive arithmetic metatheories could circumvent diagonalization paradoxes.

5) Computational Complexity

Contradictory:
Halting Problem for Turing Machines
There is no general algorithm to decide if an arbitrary program will halt or run forever on a given input.

This leads to the unsolvable Turing degree at the heart of computational complexity theory.

Non-Contradictory:
Infinitary Lambda Calculus
λx.t ≝ {x→a | a ∈ monadic realizability domain of t}

Representing computations via the interaction of infinitesimal monads and non-standard realizers allows non-Church/Turing computational models avoiding the halting problem paradox.

[Impatient_Ape]

Very nice! Somewhere between taking the course decades ago and now, "DiffEQ" courses started to use visualization tools from non-linear dynamics analysis (like the flow fields you show here), which makes it WAYYYY easier to understand. They're even useful for *partial* differential equations. Without the visualization, you're just memorizing a bunch of procedures for each of dozens of cases, which doesn't motivate deeper understanding.

[JxH]

BEWARE !! In the real world, it should (sometimes) be possible to cross the equilibrium lines. For example, if the logistics equation includes an equilibrium line at 4.5, but (for example) human babies arrive in integer units. So the curve might presently be at 4, and then this example could 'tunnel' straight over to 5, leaping over the equilibrium line at 4.5. Imagine if the 'litter' was a larger number, such as six puppies or piglets. It could overshoot the line. What happens then depends on the rest of it. ...Sorry, I always see the exceptions. It's in my nature.

[tobybartels]

This kind of thing is often covered in an introductory Differential Equations course, and you might even get the higher-order version of this (where a non-linear autonomous equation or system of equations usually can't be solved exactly, but you can still analyse its qualitative features). But we could certainly put this in Calculus 2, with maybe one extra day on the subject.

[jamesjohn]

Thanks, for clarity. Visual interpretation of big idea logistics ODE telling us! Cheers and happy Easter

[sonnyward]

Learned Logistic Growth today in pre-calc. After I took the test, and got a 98%, I wanted to relax and watch some YouTube. The first video I see is this. What are the odds?