anonymous
  • anonymous
how exactly does the standard form of the exponential function y[x] = k E^(r x) relate to this form of the logistic differential equation y'[t] == r y[t] (1 - y[t]/b) and it's counterpart y[t] = (b*E^(r*t + b*k))/(-1 + E^(r*t + b*k)) r is obvious k is not so obvious but probably the same between y[t] and y[x] but b???!!! b is a problem.
Mathematics
  • Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
SOLVED
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
chestercat
  • chestercat
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
UsukiDoll
  • UsukiDoll
omg I feel like I'm in Mathematical Biology again
jagr2713
  • jagr2713
@nincompoop @Australopithecus
anonymous
  • anonymous
lol, its awful, Im in hell

Looking for something else?

Not the answer you are looking for? Search for more explanations.

More answers

jagr2713
  • jagr2713
Please come help
anonymous
  • anonymous
3 days I been trying to work this out
UsukiDoll
  • UsukiDoll
where there are two parameters in the logistic differential equation... r is the growth rate k is carrying capacity. ewww I don't want to do this again!
anonymous
  • anonymous
you any good with mathematica usuki?
UsukiDoll
  • UsukiDoll
too long ago.
UsukiDoll
  • UsukiDoll
but I know this logistic equation
anonymous
  • anonymous
In all the examples they provide b.. but then in the problem.. they dont.. I do have a table of data though.. an exponential sample.
anonymous
  • anonymous
is that required to work out b in the final equation?
jagr2713
  • jagr2713
@Michele_Laino
anonymous
  • anonymous
my best guess is that the logistical curve is going to be an S .. at the middle of that S, in the 2nd derivative will be a 0 marking the point at which the concave changes.. this point is going to be at almost exactly 1/2 of b.. if there is proven to be a relationship between the corresponding exponential curve and the logistical curve then I can find b.
anonymous
  • anonymous
like the slope of the derivatives, might have some relationship that is propotional to r.
anonymous
  • anonymous
at that inflection point.
Astrophysics
  • Astrophysics
What exactly is the problem, this seems like logistic differential equation, I mean like does k represent the carrying capacity, and r,k >0 constant?
anonymous
  • anonymous
My problem is that I was given the top equation.. and asked to turn it into the bottom equation. after they showed me the middle equation.. and they're asking me to work it out.
anonymous
  • anonymous
if we name the equations 1,2,3. I know how to go from equation 2 to 3 .. but not from equation 1 to 2 or 1 to 3.
anonymous
  • anonymous
k as far as I can tell, is the value where x=0 y[0], I do have a value for it.. I have a data table for the exponential points.. and a formula that approximates those data points.
anonymous
  • anonymous
the formula is in the standard exponential form [x] = k E^(r x)
anonymous
  • anonymous
oops y[x] = k E^(r x)
Astrophysics
  • Astrophysics
The counterpart to me seems to be integrated, but I'm not really used to this stuff yet (haven't done much ODE atm), I've done similar but for word problems haha..mhm. I would take an attempt at it, but I don't want to lead you to the wrong place so I'll let the big guns handle it, so to speak haha @ganeshie8 @SithsAndGiggles
anonymous
  • anonymous
thank you astro for taking a look. appreciated.
jagr2713
  • jagr2713
Sorry about the inconvenience @hughfuve i dont want to make false promises but i will try and find someone to help :D @freckles
UsukiDoll
  • UsukiDoll
I've done logistic equation... but like I mentioned I don't want to go back to Mathematical Biology >_<
anonymous
  • anonymous
well I appreciate all help.. no matter how it turns out..
UsukiDoll
  • UsukiDoll
y'[t] == r y[t] (1 - y[t]/b) there are two parameters r and b. r is the growth rate and b is the carrying capacity.
UsukiDoll
  • UsukiDoll
you can't assign values to parameters.. the derivative is a constant.
anonymous
  • anonymous
so if b is the carrying capacity here.. could it just k from the first equation? The problem I have is when I tried that I got a very small S curve.. the data I have represents the US population, and according to a value of b=k, then the US population should not go over 80million.. as that is clearly not the reality.. I am concerned.
UsukiDoll
  • UsukiDoll
hmmm mine had ecology situations
UsukiDoll
  • UsukiDoll
like for bird populations
anonymous
  • anonymous
lol, you studied at the height of environmental hysteria
anonymous
  • anonymous
this course one comes from a time of immigration hysteria
UsukiDoll
  • UsukiDoll
my Mathematical Biology book is still in my backpack since final exam week. I'm not taking it out XD!
anonymous
  • anonymous
touching it probably makes your fingers rash
anonymous
  • anonymous
my book is going to give me shingles one day.. I just know it.
UsukiDoll
  • UsukiDoll
well it's not a discrete model (that's for difference equations) it's a continuous model.. what I 'm saying may be a different point of view for the logistic equation
anonymous
  • anonymous
yes this is one of the first problems of my chapter introducing differential equations. what an intro.. geeez.
UsukiDoll
  • UsukiDoll
pfffffffft you should be doing integrating factor as the first topic
UsukiDoll
  • UsukiDoll
forgot to mention that the logistic equation is separable -_-!
Astrophysics
  • Astrophysics
Does that look right Hugh?
Astrophysics
  • Astrophysics
|dw:1435638707408:dw|
anonymous
  • anonymous
looks good to me.
anonymous
  • anonymous
I am on chapter 6.. integration isn't until chapter 10. I have a hunch that perhaps this course is not for teaching calculus, but instead for teaching mathematica to those who already know calculus.
UsukiDoll
  • UsukiDoll
your curriculum is backwards -_-
UsukiDoll
  • UsukiDoll
Populations tend to get larger until there is no longer enough food or space to support so many individuals. This type of growth is called logistic population growth What Is Logistic Population Growth? A group of individuals of the same species living in the same area is called a population. The measurement of how the size of a population changes over time is called the population growth rate, and it depends upon the population size, birth rate and death rate. As long as there are enough resources available, there will be an increase in the number of individuals in a population over time, or a positive growth rate. However, most populations cannot continue to grow forever because they will eventually run out of water, food, sunlight, space or other resources. As these resources begin to run out, population growth will start to slow down. When the growth rate of a population decreases as the number of individuals increases, this is called logistic population growth.
anonymous
  • anonymous
I have the textbook calculus concepts and contexts by james stewart.. I just found a section in there on this formula.. Its almost word for word
anonymous
  • anonymous
doing some reading
UsukiDoll
  • UsukiDoll
James Stewart? I had a Single Variable Calculus Book from that author, but it was only Calculus I and II.
UsukiDoll
  • UsukiDoll
logistic equation is a 100% ODE though
Astrophysics
  • Astrophysics
Is this in james ste 7e?!
anonymous
  • anonymous
I got the larson hostetler, edwards video seriies too.. maybe theres something in there.
anonymous
  • anonymous
7e?
Astrophysics
  • Astrophysics
james stewart 7th edition
anonymous
  • anonymous
oh no.. its 2nd edition
UsukiDoll
  • UsukiDoll
OH THAT IS OLD !
anonymous
  • anonymous
2001, it was on our bookshelf.. my g/f used it way back when
UsukiDoll
  • UsukiDoll
:/
anonymous
  • anonymous
lol, no wonder Im struggling.. I need some new stuff..
anonymous
  • anonymous
I have pearsons college mathematics 13th edition if that helps
anonymous
  • anonymous
but they dont say much on this topic
anonymous
  • anonymous
actually this looks promising http://reference.wolfram.com/language/tutorial/DSolveOverview.html
anonymous
  • anonymous
might have found a possible solution, just have to work it backwards. if y[x] = logarithmic S curve f[x] = corresponding exponential curve with same r then y''[n]==0 solve for n finds the mid point of the S curve. Then take that n of midpoint of S curve and f[n]+f[0] = value for b
anonymous
  • anonymous
well well y[n]/2 =f[n]+f[0]
anonymous
  • anonymous
might have found a possible solution, just have to work it backwards. if y[x] = logarithmic S curve f[x] = corresponding exponential curve with same r then y''[n]==0 solve for n finds the mid point of the S curve. Then take that n of midpoint of S curve and f[n]+f[0] = value for b well well y[n]/2 =f[n]+f[0] so (y[n]/2)-f[0] =f[n]
UsukiDoll
  • UsukiDoll
You are right about the logistic equation producing scenarios that can't happen. Like the one for your 80 million people in the US. Obviously to sustain 80 million, you have to force people to not have plenty of kids. It's like the logistic equation is saying ok your world must sustain exactly 80 million people or less and if it goes 1 over, you'll have a shortage of supplies. Same with the birds. Either 100 birds or less...otherwise if there are 1000 birds there will be less resources and eventually the birds will die.
anonymous
  • anonymous
so does that mean then that b is arbitrary? and you can set it to whatever you want? say if you determine that you are going to put some cap on on growth and force it?
UsukiDoll
  • UsukiDoll
you can try to force it .. but in reality that's not possible... humans are going to mate anyways... same with birds.
UsukiDoll
  • UsukiDoll
@Michele_Laino is here!
UsukiDoll
  • UsukiDoll
I was told that Mathematical Biology is harder ... it actually takes Calculus III, IV, ODE, and Linear Algebra... puts it in a blender and messes everything up.
UsukiDoll
  • UsukiDoll
plus I had to use Matlab for assignments and a project... I still think my project about Killer Zombies saved my grade.
anonymous
  • anonymous
maybe I got thrown off because the examples show exponential plots vs logarithmic plots that are proportional to each other in respect to the value for b. They created a log plot, then used that to make the exponential plot. But now I am thinking that maybe the two plots are always proportional in this way no matter what the value for b.. and b is arbitrary, and it depends on what controls you want to place on the model. So Im looking for a shadow
anonymous
  • anonymous
crap.. I think it is the case
Michele_Laino
  • Michele_Laino
if we try this substitution: \[y\left( t \right) = \frac{1}{{z\left( t \right)}}\] where z is the new variable, then our starting equation, namely equation #2 can be rewritten as follows: \[\frac{{dz}}{{dt}} + rz = \frac{r}{b}\]
anonymous
  • anonymous
Hi michele.. welcome :)
UsukiDoll
  • UsukiDoll
now it looks like the integrating factor in the form of \[\frac{dy}{dx} =p(x)y=q(x)\]
Michele_Laino
  • Michele_Laino
Hi :) @hughfuve
UsukiDoll
  • UsukiDoll
NUGH NOT MY NIGHT! \[\frac{dy}{dx} +p(x)y=q(x)\]
UsukiDoll
  • UsukiDoll
I can't do math in the heat D:!
Michele_Laino
  • Michele_Laino
the solutions to that ODE are: \[\Large z\left( t \right) = {e^{ - rt}}\left( {c + \int {\frac{r}{b}{e^{rt}}dt} } \right)\]
UsukiDoll
  • UsukiDoll
so you do need integrating factor right?
Michele_Laino
  • Michele_Laino
right! so we have: \[\Large z\left( t \right) = c{e^{ - rt}} + \frac{1}{b}\]
Michele_Laino
  • Michele_Laino
now we have to return to our old variable y(t): \[\Large y\left( t \right) = \frac{b}{{bc{e^{ - rt}} + 1}}\] where c is an arbitrary real constant
UsukiDoll
  • UsukiDoll
@hughfuve didn't you mention earlier that integrating factor was not until real later in your book?
UsukiDoll
  • UsukiDoll
"hughfuve Best Response Medals 1 I am on chapter 6.. integration isn't until chapter 10. I have a hunch that perhaps this course is not for teaching calculus, but instead for teaching mathematica to those who already know calculus."
anonymous
  • anonymous
yes, but that's okay.. I can still use the Iintegrate[] mathematica command in this problem at least..
UsukiDoll
  • UsukiDoll
I would complain to your instructor if I were you. How can you do this problem without integrating factor?
anonymous
  • anonymous
Its okay.. Im learning for now.. this will at least put me in a place where I can put this one to rest..
UsukiDoll
  • UsukiDoll
.___. last time I checked all of differential equations come first...then the Mathematical Biology comes second
Michele_Laino
  • Michele_Laino
now we can multiply both numerator and denominator by \[\Large {{e^{rt}}}\] so we get: \[\Large y\left( t \right) = \frac{{b{e^{rt}}}}{{bc + {e^{rt}}}}\]
Michele_Laino
  • Michele_Laino
now we have to know the initial conditions, since the formula above gives us infinite solutions
anonymous
  • anonymous
so the exponential function I started with.. r = 0.01300265909581666; k = E^(4.382968909053888); yExp[x] = k E^(r x) I can basically pull the variables here as the starting conditions and then b is whatever we want to set the limit of the function too?
anonymous
  • anonymous
I assume c =k r = r b = whatever
anonymous
  • anonymous
and I will have an exponential version and a logarithmic version that start out the same, but end up going in different directions.
anonymous
  • anonymous
hence the idea that.. logistical growth is controlled growth ?
Michele_Laino
  • Michele_Laino
the coefficient b is called "carrying capacity"
UsukiDoll
  • UsukiDoll
I got that one too
anonymous
  • anonymous
ah gotcha, thanks.. the language helps
Michele_Laino
  • Michele_Laino
it is suffice to use this substitution: \[\Large \frac{1}{c} = {e^{bk}}\]
Michele_Laino
  • Michele_Laino
I have integrated that ODE using the separation of variable method, and I got this solutions: \[\Large y\left( t \right) = \frac{{b{c_1}{e^{rt}}}}{{1 + {c_1}{e^{rt}}}}\] where c_1 is the integration constant. Now it is suffice apply this substitution: \[\Large {c_1} = {e^{bk}}\]
UsukiDoll
  • UsukiDoll
yes that's correct... though these days I prefer integrating factor... I know it's more steps but taking the antiderivative is so much easier than integrating on both sides with separation of variables and end up getting stuck on the left side.

Looking for something else?

Not the answer you are looking for? Search for more explanations.