The problem is solved with the help of Maple (You can find the worksheet here ).

> restart;

THE SYSTEM OF EQUATIONS IS:
dN/dt = -a*C*N/(b+C) + k*N

dC/dt = -alpha*C*N/(b+C)+C0*F/V-uC/V 
Nt:= -a*C*N/(b+C)+k*N;
Ct:=-alpha*(a*C*N/(b+C))+C0*F/V-u*C/V;

We next try to diminish the number of parameters (which may be thought of as “non dimensionalizing”).
Let us write C instead of C*, N instead of N*, t instead of t*, and Un, Uc, Ut instead of the “hats” (the “u” stands for “units of”, if we think in terms of units).

> C:=c*Uc; N:=n*Un;T:=t*Ut;

Let us see now what this gives in our equations, by plugging-in everywhere, right and left.

Nt:=expand(Ut/Un*Nt);
Ct:=expand(Ut/Uc*Ct);

Let us try (looking at the first equation), Uc=b and Ut=1/k:

Simp1:={Ut=1/k,Uc=b};
Nt:=expand((subs(Simp1,Nt)));
Ct:=expand((subs(Simp1,Ct)));

Looking at the first equation, apart of the b that should simplify, there is a first constant that I cannot get rid of, a/k. Let us name it beta= a/k (alpha is already taken).
Looking at the second equation, there seems to be two constants, C0*F/(k*b*V) and u/(k*V) that I cannot simplify.

They will be gamma and delta.

Finally, let us pick Un = k*b/(alpha*a).

Simp2:={a=beta*k};
Simp3:={Un=b*k/(alpha*a),C0=Gamma*k*b*V/F,u=Delta*k*V};
Nt:=subs(Simp2,Nt);
Ct:=subs(Simp3,Ct);

The "b" should disappear..(that what we use the command collect for, so that Maple realizes that it forgot something):

Nt:=collect(Nt,[b]);
Ct:=collect(Ct,[b]);

Let us know solve this equation to find the steady states
Steady:=solve({Nt,Ct},{n,c});

Set the values of these solutions to Sol1 and Sol2
Sol1:=Steady[1];
Sol2:=Steady[2];

The first solution is nice: the cancer cells (n) is killed. It would be great if is was a stable solution
We must as well see when the second solution is of interest, i.e. when it is in the first quadrant

Let us do the inverse susbtitution
Usimp:={beta=a/k,Gamma=C0*F/(k*b*V),Delta=u/(k*V)};
S2:=subs(Usimp,Sol2);

beta is a measure of the drug effectiveness (kill/reprod)
gamma is a measure of the inflow concentration of drug (CoF/V)
with respect to the cancer cell reproduction (k b)
delta is a measure of the outflow with respect to cancer reproduction (k)

So for the concentration c to be positive (a negative concentration would make no sense), we need a>k,
i.e. the kill rate to be greater than the reproductive rate
Let us look now at what it implies for the second solution

> solve(subs(S2,n),u);

That is, we need u to be smaller than this quantity for this equilibrium to be in the first quadrant.

> with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

Let us now study the stability of these solutions: we compute the jacobian
Jac:=jacobian([Nt,Ct],[n,c]);

The jacobian at the first steady point (Sol1) is J1
J1:=matrix(2,2):
for i from 1 to 2 do
for j from 1 to 2 do
J1[i,j]:=subs(Sol1,Jac[i,j])
od od;
print(J1);

It is a diagonal matrix: we only need for the trace and determinant to be positve (or the two eigenvalues) that
the top left term be negative.
that is

Cond1:=k1-1>0;
Cond2:=simplify(J1[1,1]<0);

> simplify(subs(Usimp,Cond1));

i.e. we need the drug to be very effective (beta >>1) or the inflow drug concentration (gamma)
to be a lot bigger than the outflow quantity (delta), delta/gamma<<1.
By the way, what would it mean if the N2<0

> collect(subs(Sol2,n),gamma);
I.e. delta>gamma(beta-1). But then
beta*gamma/(delta+gamma) < 1 ! ) is not stable, blow up.

The jacobian at the 2nd steady point (Sol2) is J2
J2:=matrix(2,2):
for i from 1 to 2 do
for j from 1 to 2 do
J2[i,j]:=simplify(subs(Sol2,Jac[i,j]))
od od;
print(J2);

> D2:=factor(det(J2));T2:=simplify(trace(J2)); Sol2;

We cannot have a the same time n>0 , c>0 and det>0 So yes it is unstable (good news: otherwise we would be in trouble)

> beta:=2;Gamma:=3;Delta:=1;

>
Steady[1];Steady[2];Cond1:=k1-1>0;
Cond2:=simplify(J1[1,1]<0);

> with(DEtools):
Title:=cat("Condition 1 TRUE and Condition 2 TRUE");
dfieldplot([diff(n(t),t)=-beta*c(t)*n(t)/(1+c(t))+n(t), diff(c(t),t)=-c(t)*n(t)/(1+c(t))+Gamma-Delta*c(t)],
[n(t),c(t)],t=0..10, n=0..5, c=0..4, title = Title);