Given population doubles in 20 minutes, what is intrinsic growth rate r?

Attempt: Given population doubles, using exponential growth rate we have

hapantad2j
2022-05-02
Answered

Question in population dynamics using exponential growth rate equation

Given population doubles in 20 minutes, what is intrinsic growth rate r?

Attempt: Given population doubles, using exponential growth rate we have$\frac{dN}{dt}=2N$ so $N\left(t\right)={N}_{0}{e}^{2t}$ therefore $r=2$ , but I have a feeling this is wrong since 20 minutes should be used somewhere around here.

Given population doubles in 20 minutes, what is intrinsic growth rate r?

Attempt: Given population doubles, using exponential growth rate we have

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Alice Harmon

Answered 2022-05-03
Author has **12** answers

First part.

Since the population doubles in 20 minutes, assuming your t variable is in minutes, you should have that:

$N\left(20\right)=2N\left(0\right)$

Second part.

If you have an exponential growth rate by assumption,$\frac{dN}{dt}=\lambda N$ , which results in $N={N}_{0}{e}^{\lambda t}$ .

If you evaluate this at times$t=0\text{}\text{and}\text{}t=20$ , you'll find that $N\left(0\right)={N}_{0}\text{}\text{and}\text{}N\left(20\right)={N}_{0}{e}^{20\lambda}$ . You can then plug in these expressions into the initial equation comparing the population at these two times to solve for $\lambda$ , which should not be equal to 2.

Since the population doubles in 20 minutes, assuming your t variable is in minutes, you should have that:

Second part.

If you have an exponential growth rate by assumption,

If you evaluate this at times

asked 2021-01-02

Find

asked 2022-04-30

Removal of absolute signs

An object is dropped from a cliff. The object leaves with zero speed, and t seconds later its speed v metres per second satisfies the differential equation

$\frac{dv}{dt}=10-0.1{v}^{2}$

So I found t in terms of v

$t=\frac{1}{2}\mathrm{ln}\left|\frac{10+v}{10-v}\right|$

The questions goes on like this: Find the speed of the object after 1 second. Part of the answer key shows this

$t=\frac{1}{2}\mathrm{ln}\left|\frac{10+v}{10-v}\right|$

$2t=\mathrm{ln}\left|\frac{10+v}{10-v}\right|$

$e}^{2t}=\frac{10+v}{10-v$

So here's my question: why is it not like this?

$\pm {e}^{2t}=\frac{10+v}{10-v}$

Why can you ignore the absolute sign?

An object is dropped from a cliff. The object leaves with zero speed, and t seconds later its speed v metres per second satisfies the differential equation

So I found t in terms of v

The questions goes on like this: Find the speed of the object after 1 second. Part of the answer key shows this

So here's my question: why is it not like this?

Why can you ignore the absolute sign?

asked 2022-04-22

How do I get an estimate for this nonlocal ODE?

Consider the following nonlocal ODE on $[1,\mathrm{\infty})$:

$r}^{2}f{}^{\u2033}\left(r\right)+2r{f}^{\prime}\left(r\right)-l(l+1)f\left(r\right)=-\frac{({f}^{\prime}\left(1\right)+f\left(1\right))}{{r}^{2}$

$f\left(1\right)=\alpha$

$\underset{r\to \mathrm{\infty}}{lim}f\left(r\right)=0$

where l is a positive integer and $\alpha$ is a real number.

Define the following norm $\Vert \xb7\Vert $

$\Vert f{\Vert}^{2}:={\int}_{1}^{\infty}{r}^{2}f\text{'}(r{)}^{2}dr+l(l+1\left){\int}_{1}^{\infty}f\right(r{)}^{2}dr$

I want to prove the estimate:

$\Vert f\Vert \le C\sqrt{l(l+1)}\left|\alpha \right|$

for some constant C independent of $\alpha$, l and f. But I am stuck.

Here is what I tried. Multiply both sides by f and integrate by parts to get:

$$\begin{array}{rl}\Vert f{\Vert}^{2}={\int}_{1}^{\mathrm{\infty}}{r}^{2}{f}^{2}+{\int}_{1}^{\mathrm{\infty}}l(l+1){f}^{2}& ={f}^{\prime}(1)f(1)+({f}^{\prime}(1)+f(1)){\int}_{1}^{\mathrm{\infty}}\frac{f}{{r}^{2}}\\ & \le {f}^{\prime}(1)\alpha +\frac{({f}^{\prime}(1)+\alpha )}{3}\sqrt{{\int}_{1}^{\mathrm{\infty}}{f}^{2}}\\ & \le {f}^{\prime}(1)\alpha +\frac{({f}^{\prime}(1)+\alpha )}{3}\Vert f\Vert \end{array}$$

where I used Cauchy-Schwartz in the before last line. I am not sure how to continue and how to get rid of the f'(1) term.

Any help is appreciated.

asked 2021-10-23

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution.

$\mathrm{cos}x\frac{dy}{dx}+\left(\mathrm{sin}x\right)y=1$

asked 2022-04-03

Solving homogenous system with complex eigenvalues

$\frac{dx}{dt}=2x+8y$

$\frac{dy}{dt}=-x-2y$

When I solve the determinant of the matrix, I get$\lambda =\pm 2i$ . Then , I plug it in the matrix, and get the for the first eigenvalue, $\lambda =2i$ :

$0=(2-2i)x+8y$

$0=-x-(2+2i)y$

This gives

$x=\frac{-8}{2-2i}y$

$x=-(2+2i)y$

When I solve the determinant of the matrix, I get

This gives

asked 2022-03-27

How am I supposed to find the solution to the non-homogeneous ode ?

$\overrightarrow{Y}\text{'}=\left(\begin{array}{cc}-4& 2\\ 2& -1\end{array}\right)\overrightarrow{Y}+\left(\begin{array}{c}{x}^{-1}\\ 2{x}^{-1}+4\end{array}\right)$

asked 2021-12-13

Find the differential.

a)$z=\frac{1}{2}({e}^{{x}^{2}+{y}^{2}}-{e}^{-{x}^{2}-{y}^{2}})$

b)$w=2{z}^{3}y\mathrm{sin}x$

a)

b)