Parasite growth rate, with permissive and cargo-bearing hosts

Transition Matrix

Parameters : 
α = P(offspring remain on parental host type)
(Subscript[b, 1]) = fecundity on host with cargo
(Subscript[b, 0]) = fecundity in permissive host
x0 = P(migrant settles on permissive host) = 1-P(migrant settles on host with cargo)

A = {{(Subscript[b, 1])(α  + (1-α)(1-x0)),(Subscript[b, 0])(1-α)(1-x0)},{(Subscript[b, 1]) (1-α)x0,(Subscript[b, 0])(α+(1-α)x0)}}

Eigenvalues[A]

Leading Eigenvalue

λmax[α_,b1_,b0_,x0_]:=(1/2)(α (b1 x0 + b0(1-x0))+b1(1-x0)+b0 x0) +(1/2)Sqrt[(((b1((1-α)(1-x0)+α)+b0((1-α)x0+α)))^2)-4α b1 b0]

Cases

α=0.2, x0=0.05

p1=ContourPlot[λmax[0.2,b1,b0,0.05],{b1,0,2},{b0,1,6},FrameLabel ->{{Style["Fecundity in cargo-\nfree hosts ((Subscript[b, 00]))",{FontSize->20}],None},{Style["Fecundity in cargo-\nbearing hosts ((Subscript[b, 01]))",{FontSize->20}],Style["(A) α = 0.2, (Subscript[x, 0]) = 0.05",{FontSize->20,Black}]}},Epilog->{Text[Style["(Subscript[λ, max])",26],{1.1,5.5}]},
Contours->{0.5,1.0,1.5,2.0},
ContourStyle->{Thickness[0.001],Thickness[0.01],Thickness[0.001],Thickness[0.001]},ContourLabels->(Text[#3,Max[x/.Solve[λmax[0.3,x/3,x,0.05]==#3]]{1/2.6,1.1}]&),
ContourShading-> {RGBColor[.9,1,.8],RGBColor[.8,.95,.8],RGBColor[.8,.9,.8],RGBColor[.8,.85,.8],RGBColor[.8,.8,.8]},ColorFunctionScaling->False,
BaseStyle->{FontSize -> 16},LabelStyle->Directive[Blue,16]]

α=0.4, x0=0.05

p2 = ContourPlot[λmax[0.4,b1,b0,0.05],{b1,0,2},{b0,1,6},FrameLabel ->{
{Style["(Subscript[b, 00])",{FontSize->20}],None},{"(Subscript[b, 01])",
Style["(B) α = 0.4, (Subscript[x, 0]) = 0.05",{FontSize->20,Black}]}
},
Epilog->{Text[Style["(Subscript[λ, max])",26],{1.,5.5}]},
Contours->{0.5,1.0,1.5,2.0},
ContourStyle->{Thickness[0.001],Thickness[0.01],Thickness[0.001],Thickness[0.001]},ContourLabels->(Text[#3,Max[x/.Solve[λmax[0.4,x/3,x,0.05]==#3]]{1/3,1.08}]&),
ContourShading-> {RGBColor[.9,1,.8],RGBColor[.8,.95,.8],RGBColor[.8,.9,.8],RGBColor[.8,.85,.8],RGBColor[.8,.8,.8]},ColorFunctionScaling->False,
BaseStyle->{FontSize -> 16},LabelStyle->Directive[Blue,16]]

α=0.2, x0=0.01

p3=ContourPlot[λmax[0.2,b1,b0,0.01],{b1,0,2},{b0,1,6},FrameLabel ->{{"(Subscript[b, 00])",None},{"(Subscript[b, 01])",Style["(C) α = 0.2, (Subscript[x, 0]) = 0.01",{FontSize->20,Black}]}},Contours->{0.5,1.0,1.5,2.0},
Epilog->{Text[Style["(Subscript[λ, max])",26],{1.,5.5}]},
ContourStyle->{Thickness[0.001],Thickness[0.01],Thickness[0.001],Thickness[0.001]},ContourLabels->(Text[#3,Max[x/.Solve[λmax[0.2,x/3,x,0.01]==#3]]{.32,1.0}]&),
ContourShading-> {RGBColor[.9,1,.8],RGBColor[.8,.95,.8],RGBColor[.8,.9,.8],RGBColor[.8,.85,.8],RGBColor[.8,.8,.8]},ColorFunctionScaling->False,
BaseStyle->{FontSize -> 16},LabelStyle->Directive[Blue,16]]

α=0.4, x0=0.01

p4=ContourPlot[λmax[0.4,b1,b0,0.01],{b1,0,2},{b0,1,6},FrameLabel ->{{"(Subscript[b, 00])",None},{"(Subscript[b, 01])",Style["(D) α = 0.4, (Subscript[x, 0]) = 0.01",{FontSize->20,Black}]}},Contours->{0.5,1.0,1.5,2.0},
Epilog->{Text[Style["(Subscript[λ, max])",26],{1.,5.5}]},
ContourStyle->{Thickness[0.001],Thickness[0.01],Thickness[0.001],Thickness[0.001]},ContourLabels->(Text[#3,Max[x/.Solve[λmax[0.4,x/3,x,0.01]==#3]]{.323,1.06}]&),
ContourShading-> {RGBColor[.9,1,.8],RGBColor[.8,.95,.8],RGBColor[.8,.9,.8],RGBColor[.8,.85,.8],RGBColor[.8,.8,.8]},ColorFunctionScaling->False,
BaseStyle->{FontSize -> 16},LabelStyle->Directive[Blue,16]]

p5=GraphicsGrid[{{p1,p2},{p3,p4}}]

Export["/Users/bull/Box\ Sync/mss/cargo/test.pdf",p5,"AllowRasterization"->True,ImageSize->660,ImageResolution->600]