Coral reef population model with immigration and logistic growth

[missbooty87]

Homework Statement

"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.

Homework Equations

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, regardless 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 :)

 

[HallsofIvy]

Homework Statement

"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.

Homework Equations

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, regardless 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 :)