Population Modelling / Non-Dimensionalisation

pomp · #1 $Old$ November 8th, 2009, 11:21 AM

Hey guys, I'm stuck on a mathematical modelling problem and could do with a push in the right direction.

Here is the question:

Consider a lake with some fish attractive to fisherman. We wish to model the fish-fisherman interaction under the following assumptions:
1. the fish population grows logistically in the absence of fishing;
2. the presence of fishermen depresses the fish growth rate at a rate jointly proportional to the size of the fish and fisherman populations;
3. fishermen are attracted to the lake at a rate directly proportional to the number of fish in the lake;
4. fishermen are discouraged from the lake at a rate directly proportional to the number of fishermen already there.

(a) Write down the model for this situation, clearly defining your terms.
(b) Show that a non-dimensionalised version of the model is given by:

$\frac{du}{d \tau} = ru(1-u) -uv$

$\frac{dv}{d \tau} = \beta u - v$

The rest of the question I can deal with (stuff about singularities/nullclines etc.)

Here's my attempt so far.

Let N be the fish population and F the fishermen.

Assumption 1.

$\Rightarrow$ $\frac{dN}{dt} = r \left( 1-\frac{N}{K} \right)$

This is just logistic growth in the absence of fishing.

Assumption 2.

$\Rightarrow$ $\frac{dN}{dt} = r \left( 1-\frac{N}{K} \right) - aFN$

Where a is some constant and the term -aFN is the fishermen's effect on the fish's growth and is directly proportional to the size of the fish and fishermen populations.

Assumption 3.

$\Rightarrow$ $\frac{dF}{dt} = bN$

Obviously.

Assumption 4.

$\Rightarrow$ $\frac{dF}{dt} = bN - cF$

Giving me the model:

$\frac{dN}{dt} = r \left( 1-\frac{N}{K} \right) - aFN$

$\frac{dF}{dt} = bN - cF$

My trouble then lies in casting it into non-dimensionalised form. I've tried different ways and failed. Either my model is wrong or I'm missing a trick when non-dimensionalising.

Applied math is definitely not my forte and any help would be much appreciated.

Pomp.

pomp · #2 $Old$ November 8th, 2009, 12:07 PM

I've realised my model was wrong.

I think corrected it should be.

$\frac{dN}{dt} = rN \left( 1 - \frac{N}{K} \right) - aNF$

$\frac{dF}{dt} = bNF - cF$

libabao · #3 $Old$ November 25th, 2009, 03:20 PM

I accept information:
Here is the question:

Consider a lake with some fish attractive to fisherman. We wish to model the fish-fisherman interaction under the following assumptions:
1. the fish population grows logistically in the absence of fishing;
2. the presence of fishermen depresses the fish growth rate at a rate jointly proportional to the size of the fish and fisherman populations;
3. fishermen are attracted to the lake at a rate directly proportional to the number of fish in the lake;
4. fishermen are discouraged from the lake at a rate directly proportional to the number of fishermen already there.

(a) Write down the model for this situation, clearly defining your terms.
(b) Show that a non-dimensionalised version of the model is given by:

The rest of the question I can deal with (stuff about singularities/nullclines etc.)

Here's my attempt so far.

Let N be the fish population and F the fishermen.

Assumption 1.

This is just logistic growth in the absence of fishing.

Assumption 2.

Where a is some constant and the term -aFN is the fishermen's effect on the fish's growth and is directly proportional to the size of the fish and fishermen populations.

Assumption 3.

Obviously.

Assumption 4.

Giving me the model:

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