# Population modelling

#### missbooty87

1. The problem statement, all variables and given/known data

"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."

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.

2. Relevant equations
3. The attempt at a solution

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 - I can see that its a first order non linear differential equation. But where do I go with this?

d - a regrowth rate represented by a logarithmic function?

The total marks for the 4 questions is 5 marks - if that helps

Thanks to all help received :)

Last edited:

#### HallsofIvy

1. The problem statement, all variables and given/known data

"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."

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.

2. Relevant equations
3. The attempt at a solution

a - Is the immigration denoted as "I", because as immigration is constant, the I has 'constant effects' on the equation?
Yes, I is the only thing that does not depend upon the current mass- it "comes from the outside".

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

c - I can see that its a first order non linear differential equation. But where do I go with this?
If there is a "limiting population", the rate of change as M nears it must decrease to 0. Is there a value of M such that dM/dt= 0 (so that M is constant)? Set M(S-M) + I = 0 and solve for M.

d - a regrowth rate represented by a logarithmic function?
In other words, solve the problem M'= M(S-M)+ I, M(0)= 0, a separable differential equation. No, M is not logarithmic.

The total marks for the 4 questions is 5 marks - if that helps

Thanks to all help received :)

#### missbooty87

If there is a "limiting population", the rate of change as M nears it must decrease to 0. Is there a value of M such that dM/dt= 0 (so that M is constant)? Set M(S-M) + I = 0 and solve for M.
So what I've done is, I've found the value of M using the quadratic formula, this is in terms of S. Is this all I have to do? It feels inadequate...not that i'm undermining your way of thinking... I probably stopped short...

In other words, solve the problem M'= M(S-M)+ I, M(0)= 0, a separable differential equation.
so if i directly integrate M'= -M^2 + SM + I, which will presumably give the function for M, which is: $$M = \frac{-M^3 }{3} + \frac{SM^2}{2} + IM + C = 0$$ >>> is this what you're talking about? if so, I then substitute M=0? I'm getting lost...

Thank you for your help HallsofIvy :) Much appreciated :)

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