MHB Fish Pond Challenge: Show Equilibrium Variation with $R_f$

AI Thread Summary
The discussion centers on a differential equation modeling fish population dynamics in a pond, influenced by fishing rates. For removal rates less than a quarter of the equilibrium population, the fish population stabilizes at an equilibrium between half and the full capacity of the pond. Conversely, if the removal rate is equal to or exceeds this threshold, the population may decline significantly, potentially leading to extinction. Participants seek solutions for both scenarios, emphasizing the implications of fishing on ecological balance. The challenge highlights the importance of understanding population dynamics in resource management.
lfdahl
Gold Member
MHB
Messages
747
Reaction score
0
Suppose a pond contains $x(t)$ fish at time $t$, and $x(t)$ changes according to the DE:
\[\frac{\mathrm{d} x}{\mathrm{d} t} = x\left ( 1-\frac{x}{x_0} \right )-R_f\]
where $x_0$ is the equilibrium amount with no fishing and $R_f > 0$ is the constant rate of removal due to fishing. Assume $x(0) = \frac{x_0}{2}$.

(a). If $R_f < \frac{x_0}{4}$, solve for $x(t)$ and show that it tends to an equilibrium amount between $\frac{x_0}{2}$ and $x_0$.
(b). What happens if $R_f \geq \frac{x_0}{4}$?
 
Mathematics news on Phys.org
There was a https://mathhelpboards.com/calculus-10/help-derivative-22779.html on logistic growth with constant harvesting a few months ago.

In particular, if we start with the system given in post #5 there, and we non-dimensionalize by introducing a new time $\tau := (r K) t$ and a new unknown $x(\tau) := P(t(\tau))$, then in these new terms we have
\[
\frac{dx}{d\tau} = \frac{1}{rK}\frac{dP}{dt} = P\left(1 - \frac{P}{K}\right) - \frac{H}{rK} = x\left(1 - \frac{x}{K}\right) - \frac{H}{r K}.
\]
This is the system given in the challenge if we set $K := x_0$, $R_f := \frac{H}{r K}$ and we abuse notation by denoting non-dimensional time again with $t$ instead of $\tau$.

It is nice to see a problem on population dynamics, by the way.
 
Krylov said:
There was a https://mathhelpboards.com/calculus-10/help-derivative-22779.html on logistic growth with constant harvesting a few months ago.

In particular, if we start with the system given in post #5 there, and we non-dimensionalize by introducing a new time $\tau := (r K) t$ and a new unknown $x(\tau) := P(t(\tau))$, then in these new terms we have
\[
\frac{dx}{d\tau} = \frac{1}{rK}\frac{dP}{dt} = P\left(1 - \frac{P}{K}\right) - \frac{H}{rK} = x\left(1 - \frac{x}{K}\right) - \frac{H}{r K}.
\]
This is the system given in the challenge if we set $K := x_0$, $R_f := \frac{H}{r K}$ and we abuse notation by denoting non-dimensional time again with $t$ instead of $\tau$.

It is nice to see a problem on population dynamics, by the way.

Thankyou very much, Krylov! (Handshake)
May I ask for the answers of (a). and (b). in the challenge, based on the logistic growth model, you refer to?
 
Suggested solution:

\[\frac{\mathrm{d} x}{\mathrm{d} t}=x-\frac{x^2}{x_0} - R_f=-\frac{1}{x_0}\left ( x-\frac{x_0}{2} \right )^2+\left ( \frac{x_0}{4}-R_f \right ).\]

Let $\frac{1}{\sqrt{x_0}}\left ( x-\frac{x_0}{2} \right )=y(t)$, so that $\sqrt{x_0}dy = dx$ and $y(0) = 0$.
Also, let $a^2 = \left | \frac{x_0}{4}-R_f \right |$. In these terms the D.E. is

\[-dt = \frac{\sqrt{x_0}dy}{y^2\mp a^2}\]
where $a^2 = 0$ if $R_f = \frac{x_0}{4}$, negative if $R_f < \frac{x_0}{4}$, and positive if $R_f > \frac{x_0}{4}$.
(a). When $R_f < \frac{x_0}{4}$, $-t = \frac{\sqrt{x_0}}{2a}\ln \left ( \frac{a-y}{a+y} \right )+c$. Since $y(0) = 0$, we have $c = 0$, and so
\[e^{\frac{-2at}{\sqrt{x_0}}}= \frac{a-y}{a+y} = \frac{2a}{a+y}-1\]
i.e. \[y = \frac{2a}{1+e^{\frac{-2at}{\sqrt{x_0}}}}-a.\]

As $t \rightarrow \infty$, clearly $y \rightarrow a$, i.e.
\[\frac{1}{\sqrt{x_0}}\left ( x-\frac{x_0}{2} \right )\rightarrow \sqrt{\frac{x_0}{4}-R_f}\]

and so \[x \rightarrow \frac{x_0}{2}+\sqrt{\frac{x_0^2}{4}-R_fx_0}.\]

(b). When $R_f = \frac{x_0}{4}$, the original equation is

\[\frac{\mathrm{d} x}{\mathrm{d} t}=x-\frac{x^2}{x_0} - R_f=-\frac{1}{x_0}\left ( x-\frac{x_0}{2} \right )^2,\]
which has the obvious constant solution $x(t) = \frac{x_0}{2} = x(0)$.
When $R_f > \frac{x_0}{4}$, \[-t = \frac{\sqrt{x_0}}{a}\arctan \left ( \frac{y}{a} \right ) + c.\]

Again $c = 0$. Now, $-\tan \left ( \frac{at}{\sqrt{x_0}}\right )=\frac{y}{a}$, or $-\sqrt{R_f-\frac{x_0}{4}}\tan \left ( \frac{\sqrt{R_f-\frac{x_0}{4}}}{\sqrt{x_0}}t\right ) = \frac{1}{\sqrt{x_0}}\left ( x-\frac{x_0}{2} \right )$, and so
\[x = \frac{x_0}{2}-\sqrt{R_fx_0-\frac{x_0^2}{4}}\: \tan \left ( \sqrt{\frac{R_f}{x_0}-\frac{1}{4}}\: \cdot t\right ).\]
This is a decreasing function of $t$, which becomes $0$, when $\tan \left ( \sqrt{\frac{R_f}{x_0}-\frac{1}{4}}\: \cdot t\right ) = \frac{x_0}{2\sqrt{R_fx_0-\frac{x_0^2}{4}}}$.
So, the fish population becomes $0$ in a finite time.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Back
Top