# (Solved): ()/(dt) hi, please do all parts, thank you! dN=(\alpha N(f_(a)-\beta N))/(1+\alpha \gamma N),N(0)=N ...

()/(dt) hi, please do all parts, thank you! dN=(\alpha N(f_(a)-\beta N))/(1+\alpha \gamma N),N(0)=N_(0)

Here,

\alpha ,f_(a),\beta ,\gamma ,N_(0)>0

are constants. Here,

a,b,N_(0)>0

are constants and

N(t)

represents the density of tumour cells. Qualitative analysis (with phase portraits): (a) Plot the velocity field. (b) Find the equilibrium densities. Hint: There are two. (c) Determine the stability of the equilibrium densities. Quantitative analysis (by explicitly solving (5)): (a) (Review of the Chain Rule) Calculate

(d)/(dx)[log(logx)]

. (b) By separating variables from (5) and using the previous part, or otherwise, show that the solution,

N

, to (5) satisfies

-at=log(log(bN))-log(log(bN_(0))).

(c) Using the previous part, write down the formula for the solution,

N

, of (5). (d) Using the previous part, calculate

\lim_(t->\infty )N(t)

.

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