## Mathematical Systems Biology

Beiträge: 11
Registriert: 28.05.2013, 19:01

### Mathematical Systems Biology

Hi I have following assignment and I have not really a clue on how to solve it, because it is completely different then what we did at class.

The simple fishery model reads $\dot{N} = rN(1-N/K)-H$, where N is the fish population, K is a carrying capacity, r is the growth rate, and H is the term considering the effect of fishing.
In a refined version, the model is improved to $\dot{N}=rN(1-N/K)-HN/(A+N)$, with parameters H and A being larger than zero.

(a) Why is the refined model more realistic than the original, simple one?

(b) Give a biological interpretation of parameter A; what does it measure?

(c) Show that the refined system can be rewritten in dimensionless form as
$\frac{dx}{d\tau} = x(1-x) - \frac{hx}{a+x}$

(d) Show that the refined system can have one, two, or three fixed points, depending on the numerical values of parameters a and h. Classify the stability of the fixed points.

1(e) Analyze the dynamics of the system near $x=0$, and show that a bifurcation occurs if $h=a$. Which kind of bifurcation is it?

(f) Plot the stability diagram of the system in the pa, hq parameter space. Can hysteresis occur in any of the stability regions?

d and e I have some idea but with the others i am clueless Greetings Max