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:
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.
This is just logistic growth in the absence of fishing.
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.
Giving me the model:
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.