Differential Equation, Phase Plane Solutions and Fish Populations

smiles

Help!

Last year we finished up studying some differential equations and one of the last things we looked at were logistical equations.

ie. dN/dt = aN - bN^2

Then there's stuff about Phase Plane solutions to find the equillibrium points and to find the stability of these equilibrium points.

The question / example given is:

If we assume that a certain species of fish will satisfy the conditions of logistic groth what is the effect of harvesting an amount H of fish every year?

Let the fish satisfy the equation:
dN/dt = rN(1 - (N/K))
K is the limiting population (carrying capacity). Note that the maximum rate of chance for N = K/2 and has value Hc = rK/4.

You have to work out each case where H > Hc, H = Hc and H < Hc and also say what the long term result would be for each of these.

Can anyone explain what the hell you're meant to be doing because it doesn't really make that much sense to me.

<< Fio >>

Capt'n Midnight

I think with the idea behind the fish is that this is supposed to prove fish stocks can be constantly harvested at a certain rate 'cos the equillibrium just depends on rate limiting factor - nutrient or removal rate.
(But it don't in the real world - cos of the effects of competition from other species and overcatch that's not landed. Also many species do not do well if the shoal falls below a certain size.)

look at the nice pictures, with arrows
http://www-rohan.sdsu.edu/~jmahaffy/courses/f01/math636/lectures/competition2/competition2.html



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They used to use a single preditor species and a single prey species as a model for natural systems. Only problem was that it virtually never occured in nature...

Competition Models - Two Species
http://www-rohan.sdsu.edu/~jmahaffy/courses/f01/math636/lectures/competition3/competition3.html