1. ## [SOLVED] Population model

Hello hello,

I have a few assignment question, which I cannot comprehend. I'd appreciate it if someone could rephrase what the assignment may be asking ... I've supplied my answers/ideas too. Although, I think they may be incorrect, due to the lack of my understanding of what the question wants ....
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Preliminary information:

"The very simple population model for a resource limited population with constant immigration, and no breeding, M'(t) = M(S-M) + I attempts to describe the growth of corals on a reef. Function M(t) represents the biomass of corals."

Question:

"a - Explain which term gives the immigration of juveniles onto the reef.

b - Describe the presumptions being made about the growth rate of corals at their different ages and sizes.

c - Determine if the biomass of corals tends to a limiting amount as $t \rightarrow \infty$ .

d - Suppose a coral reef has completely died, due to excessive cyanide fishing. Find and describe what this model suggests will be the pattern of its recovery."
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"a - Is the immigration denoted as "I", because as immigration is constant, the I has 'constant effects' on the equation?

b - Is it correct to presume that the growth rate of the corals are constant, irregardless of their age and size?

c - Do we have to integrate the M'(t)?

d - a regrowth rate represented by a logarithmic function?"

Thank you for all help provided.

2. Originally Posted by tsal15
Hello hello,

I have a few assignment question, which I cannot comprehend. I'd appreciate it if someone could rephrase what the assignment may be asking ... I've supplied my answers/ideas too. Although, I think they may be incorrect, due to the lack of my understanding of what the question wants ....
----------
Preliminary information:

"The very simple population model for a resource limited population with constant immigration, and no breeding, M'(t) = M(S-M) + I attempts to describe the growth of corals on a reef. Function M(t) represents the biomass of corals."

Question:

"a - Explain which term gives the immigration of juveniles onto the reef.

b - Describe the presumptions being made about the growth rate of corals at their different ages and sizes.

c - Determine if the biomass of corals tends to a limiting amount as $t \rightarrow \infty$ .

d - Suppose a coral reef has completely died, due to excessive cyanide fishing. Find and describe what this model suggests will be the pattern of its recovery."
-----------

"a - Is the immigration denoted as "I", because as immigration is constant, the I has 'constant effects' on the equation?

b - Is it correct to presume that the growth rate of the corals are constant, irregardless of their age and size?

c - Do we have to integrate the M'(t)?

d - a regrowth rate represented by a logarithmic function?"

Thank you for all help provided.
Have you given the complete problem statement? Surely you have been given definitions of S and I?

3. Originally Posted by awkward
Have you given the complete problem statement? Surely you have been given definitions of S and I?
Well unfortunately, all the information I was given, I provided on the thread

Even the first question asks me to find which term is representing 'immigration'...Any ideas though?

4. Originally Posted by tsal15
Well unfortunately, all the information I was given, I provided on the thread

Even the first question asks me to find which term is representing 'immigration'...Any ideas though?
It looks like I may be the rate of biomass increase due to immigration, since we are told immigration is constant. The spelling of "immigration" (starts with I) is suggestive.

And I suppose S may be something like the maximum capacity of the reef in some sense or other, like available food, because the term M(S - M) drops to zero when M = S, and we are told the population is resource-constrained. M(S - M) doesn't make sense dimensionally, though, because the result has to be a rate of increase of biomass. Maybe we are missing a multiplicative coefficient which converts M(S - M) to a rate, as in cM(S - M).

Frankly, I don't think it makes much sense to throw down an equation like that without defining the terms.

5. Thank you very much for your help, awkward!!!

Originally Posted by awkward
Maybe we are missing a multiplicative coefficient which converts M(S - M) to a rate, as in cM(S - M).
the equation initially given was M'(t) (which I gave in the 1st post) - do you not think that this equation is a rate? maybe its a non linear Differential Equation?

Originally Posted by awkward
Frankly, I don't think it makes much sense to throw down an equation like that without defining the terms.
I TOTALLY AGREE WITH YOU ON THIS COMMENT!!!

6. Originally Posted by tsal15
Thank you very much for your help, awkward!!!

the equation initially given was M'(t) (which I gave in the 1st post) - do you not think that this equation is a rate? maybe its a non linear Differential Equation?
[snip]
M'(t) is the rate of change of biomass, so the right hand side of the equation must also be in those units-- including M(S - M). But M(S - M) appears to be in the units of mass^2, so we must be missing something.

The equation is non-linear, because if we expand M(S - M) we have SM - M^2.

7. Originally Posted by awkward
M'(t) is the rate of change of biomass, so the right hand side of the equation must also be in those units-- including M(S - M). But M(S - M) appears to be in the units of mass^2, so we must be missing something.

The equation is non-linear, because if we expand M(S - M) we have SM - M^2.
I've sent an email to my teacher in regards to the inconsistency of the question... hopefully there is more to the question than what was originally given.

So in regards to question d, what are your suggestions, awkward?

8. Originally Posted by tsal15
I've sent an email to my teacher in regards to the inconsistency of the question... hopefully there is more to the question than what was originally given.

So in regards to question d, what are your suggestions, awkward?
It looks to me like you should be able to make some qualitative remarks based on what happens to M' when M=0 and when M >= S. Or you can just solve the equation for M as a function of t, for that matter.

9. Originally Posted by awkward
It looks to me ... you can just solve the equation for M as a function of t, for that matter.
Thats only possible if I generate an equation for M, yes?

10. Originally Posted by awkward
M(S - M) doesn't make sense dimensionally, though, because the result has to be a rate of increase of biomass. Maybe we are missing a multiplicative coefficient which converts M(S - M) to a rate, as in cM(S - M).

My teacher said that the c was present, just it was invisible to see because c = 1. does this help? where do we go from here?

Thank you for your continued assistance awkward

11. If you want to actually solve the equation (which I'm not sure is necessary, given the problem statement), you can use separation of variables: write

$\frac{dM}{M(S - M) + I} = dt$

and integrate. You would have to consider a couple of cases depending on whether the roots of $M(S - M) + I = 0$, when solved for M, are real or complex.

But as I said earlier, I think you can tell a lot by just considering where $\frac{dM}{dt}$ is positive or negative.