Differential Equation : Word Problems

[refid]

Hi, I Understand how to find the equilibrium point(s) and by graphical or using eigenvalues method to show the stablity and unstability, but when present by a model i dont understand what is it

Question:
Suppose we are studying a given species in a metapopulation using levin's model. That is, the fraction of patches occuppied by the species at time t, p(t) satisifies

\(\displaystyle \L\\dp/dt = 2p(1-p)-p\)

t greater than or equal to 0
Part A.

At p=1/2 locally stable
At p=0 Unstable

Heres where im stuck:

Part B.
Find lim t-> Infinity p(t) if p(0) using result in part a without actually solving the differential equation. Will the species survive or be extinct in the long run? provide clear answer and explanation.

Is there some trick to grasping the meaning of these word problem and equations? I know this isn't an English class

 

[Guest]

Hi, I Understand how to find the equilibrium point(s) and by graphical or using eigenvalues method to show the stablity and unstability, but when present by a model i dont understand what is it

Question:
Suppose we are studying a given species in a metapopulation using levin's model. That is, the fraction of patches occuppied by the species at time t, p(t) satisifies

\(\displaystyle \L\\dp/dt = 2p(1-p)-p\)

t greater than or equal to 0
Part A.

At p=1/2 locally stable
At p=0 Unstable

Heres where im stuck:

Part B.
Find lim t-> Infinity p(t) if p(0) using result in part a without actually solving the differential equation. Will the species survive or be extinct in the long run? provide clear answer and explanation.

Is there some trick to grasping the meaning of these word problem and equations? I know this isn't an English class

\(\displaystyle \L\\dp/dt >0\) if \(\displaystyle \L 1/2>p>0\), and \(\displaystyle \L\\dp/dt <0\) if \(\displaystyle \L p>1/2\), therefore all time trajectories of \(\displaystyle \L p\) converge to \(\displaystyle \L p=1/2\).

That is if \(\displaystyle \L 1/2>p>0\) \(\displaystyle \L p\) increases and it moves towards \(\displaystyle \L p=1/2\), if \(\displaystyle \L p>1/2\) then it decreases
towards \(\displaystyle \L p=1/2\).

RonL

 

[refid]

Multivariable calculus

Thanks !

I have another question: how do i finding the domain and range of function:

i can find the domain and range in a single variable function like this \(\displaystyle \L\\ y = x^2 +3\)

range: 2
domain all reals


but how do i find like this :

\(\displaystyle \L\\f(x,y) = sqroot(y^2 - x)\) ??

 

[Guest]

Re: Multivariable calculus

Thanks !

I have another question: how do i finding the domain and range of function:

i can find the domain and range in a single variable function like this \(\displaystyle \L\\ y = x^2 +3\)

range: 2
domain all reals


but how do i find like this :

\(\displaystyle \L\\f(x,y) = \sqrt {y^2 - x}\) ??

The \(\displaystyle \L \sqrt {u}\) is defined when its argument is positive so the domain is the region of \(\displaystyle \L \mathbb{R}^2\) where \(\displaystyle \L y^2-x \ge 0\), also the range is set of non-negative reals as \(\displaystyle \L y^2 - x\) can take any non-negative value for some point in the domain of \(\displaystyle \L F\)

RonL