Null Clines

ol98

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1596385603733.png

I've got two equations here that I need to work out the null clines for.
I've set both equations to 0.

For the du/dt equation, after setting the equation to 0 I attempted to solve for u but wasn't able to factorise any further than getting the line:
a+u-au-u^2 - v=0

I've tried to attempt the dv/dt equation.
Factorising v out to get
v(bu-c)=0
I got v=0 and bu-c=0

Does anyone know anything further about null clines and able to tell me where I'm supposed to go with this?

Thanks
 
You can't draw them without knowing values for a, b, and c. It's like asking you to plot the parabola [MATH]y = ax^2+bx+c[/MATH]. In that example you could say the parabola opens upward if [MATH]a>0[/MATH], downward if [MATH]a<0[/MATH], and is a straight line (degenerate parabola) if [MATH]a=0[/MATH]. Then you might say more about what you know for cases for [MATH]b[/MATH] and [MATH]c[/MATH]. But you need the actual numbers to actually plot the parabola. Perhaps that is what is what is wanted in your problem -- a general discussion about what the nullclines look like as the variables change, maybe with some particular examples. It's hard for me to say.
 
You can't draw them without knowing values for a, b, and c. It's like asking you to plot the parabola [MATH]y = ax^2+bx+c[/MATH]. In that example you could say the parabola opens upward if [MATH]a>0[/MATH], downward if [MATH]a<0[/MATH], and is a straight line (degenerate parabola) if [MATH]a=0[/MATH]. Then you might say more about what you know for cases for [MATH]b[/MATH] and [MATH]c[/MATH]. But you need the actual numbers to actually plot the parabola. Perhaps that is what is what is wanted in your problem -- a general discussion about what the nullclines look like as the variables change, maybe with some particular examples. It's hard for me to say.

In my question it's asked about sketching the phase planes in the two cases: i. b>c and ii b<c

1596455094216.png

I got this for my u null clines, if they look correct?
 
Your arithmetic looks correct. If your phase plane has [MATH]u[/MATH] and [MATH]v[/MATH] as the horizontal and vertical axes, respectively, then those [MATH]u[/MATH] equations would give the locus where [MATH]u'=0[/MATH] so the curves in the phase plane have horizontal tangents. Remember you also have [MATH]v'= v(bu-c)[/MATH] so [MATH]v=0[/MATH] and [MATH]u=\frac c b[/MATH] give the locus where the curves have vertical tangents.
 
Your arithmetic looks correct. If your phase plane has [MATH]u[/MATH] and [MATH]v[/MATH] as the horizontal and vertical axes, respectively, then those [MATH]u[/MATH] equations would give the locus where [MATH]u'=0[/MATH] so the curves in the phase plane have horizontal tangents. Remember you also have [MATH]v'= v(bu-c)[/MATH] so [MATH]v=0[/MATH] and [MATH]u=\frac c b[/MATH] give the locus where the curves have vertical tangents.

Yes, u and v are the horizontal and vertical axes, respectively. I'm just going over the null clines again.
1596636651181.png

for the du/dt equation, I'm looking for the u null clines.

I'm unsure whether the correct null cline solutions are:

u=0, v=-u^2 - au+u+a

or

1596636722101.png
 
Was just wondering if anyone was able to clarify which null cline solutions are correct, as I'm a bit unsure.

I've got the differential equation:

du/dt = u(1-u)(a+u) - uv

I know to find the u null cline I set du/dt to 0.

I've got u=0 as one solution.

However, I'm not sure whether I try to find a solution in terms of v, so I got v=-u^2 -au +u + a

Or if I find another u solution, and use the quadratic formula to get:
1596637113910.png

Any help would be great, cheers
 
I think you are misunderstanding what "cline" means here. It is NOT a specific value of u or v. A "cline" is a line or curve. "u= 0" is not the number 0 but is the straight line in the uv plane of all points (0, v), the v-axis. The other "cline" is the parabola v= -u^2+ (1- a)u+ a.
 
I think you are misunderstanding what "cline" means here. It is NOT a specific value of u or v. A "cline" is a line or curve. "u= 0" is not the number 0 but is the straight line in the uv plane of all points (0, v), the v-axis. The other "cline" is the parabola v= -u^2+ (1- a)u+ a.

Nice, yeh so I've got my clines as:

u=0, v=0, u=c/b and the parabola v= -u^2 -u(a-1) +a

Do you know much about how you would draw these on the phase plane?
 
They are just straight lines and parabolas. LIke I said before, you need to put values of the variables in to plot them. Here's a picture of a few parabolas for various values of [MATH]a[/MATH]. I used Maple to plot them. It all depends on what you want:

clines.JPG
But, remember, for a particular value of your constants, you just get one parabola and a couple of straight lines for your clines. I would think you would also want a phase portrait of your solutions to see how they agree with the clines. You haven't made it very clear in this thread what you are really trying to do.
 
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