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:

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

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

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.

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

This is just logistic growth in the absence of fishing.

Assumption 2.

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

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.

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

Obviously.

Assumption 4.

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

Giving me the model:

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

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

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.