b = 0.3;
d = 0.2;

pop[0] = 10;
tot[0] = pop[0];
cells = Table[1,{pop[0]}];

tmax = 50;
Do[
pop[t] = pop[t-1];
tot[t] = tot[t-1];
Do[
If[cells[[i]] == 1,
If[RandomReal[]<b,
AppendTo[cells,1];
pop[t] = pop[t]+1;
tot[t] = tot[t]+1
];
If[RandomReal[]<d,
cells[[i]] = 0;
pop[t] = pop[t]-1
]
],
{i,1,tot[t-1]}
],
{t,1,tmax}
];

plotdata = Table[{t,pop[t]},{t,0,tmax}];
plot1 =
ListPlot[
plotdata,
PlotRange->All,
AxesOrigin-> {0,0},
AxesLabel->{"t","pop"},
PlotStyle->Red,
Filling->Axis,
FillingStyle->Red
]

Epop[t_] := (1-d+b)^t * pop[0];
plot2= 
Plot[Epop[t],{t,0,tmax},PlotStyle->Blue];

Show[
plot1,plot2
]