(*p3[N_, A_, B_] := (B*N^2)/(A^2 + N^2)
p2[N_, A_, B_] := (B*N)/(A + N)*)

((\[DifferentialD]N)/(\[DifferentialD]t))=N R (1-(N/K))-(Subscript[p, 2])(N)

Manipulate[Show[
Plot[p2[x,A,B],{x,0,100},PlotRange->{{0,100},{0,10}},
Epilog->{
Blue,Dashed,Line[{{0,B},{100,B}}],
Text[Style["B="<>ToString[B],Blue],{98,B},{2,-1}],
Red, Dashed, Line[{{A,0},{A,9}}],
Text[Style["A="<>ToString[A],Red],{A,10},{0,1}]
}
],
Plot[p3[x,A,B],{x,0,1000},PlotRange->{{0,100},{0,10}},PlotStyle->Green],
ListPlot[{Callout[{A,p2[A,A,B]}]},PlotStyle->Red]
],{{A,5, "Population at Critical Predation (A)"},0,10},{{B,5, "Maximum Predation (B)"},0,20}]

Manipulate[Module[{growth,sol},
growth[n_]:=n[t]*r*(1-n[t]/k);
p2[x_]:=(b*x)/(a+x);
p3[x_]:=(b*x^2)/(a^2+x^2);
sol=NDSolve[{#,n[0]==n0},n[t],{t,0,15}]&/@{n'[t]==growth[n]-predate*p2[n[t]],n'[t]==growth[n]-predate*p3[n[t]]};
(* plots *)
Column[{
(* top *)
Show[
Plot[Evaluate[n[t]/. sol[[2]]],{t,0,15},PlotRange->{{0,15},{0,2000}},ImageSize->Medium,PlotStyle->Green,Frame->True,FrameLabel->{{"Time (t)", None}, {"Number of Insects (N)", "Population"}}],
Plot[Evaluate[n[t]/. sol[[1]]],{t,0,15},PlotRange->{{0,15},{0,2000}},ImageSize->Medium,PlotStyle->Blue]],
(* bottom *)
Show[
Plot[p2[x],{x,0,2000},PlotRange->{{0,2000},{0,500}},ImageSize->Medium,
Epilog->{
Blue,Dashed,Line[{{0,b},{2000,b}}],
Text[Style["B="<>ToString[b],Blue],{1990,b},{2,-1}],
Red, Dashed, Line[{{a,0},{a,500}}],
Text[Style["A="<>ToString[a],Red],{a,500},{-1.5,1}]
},
Frame->True,
FrameLabel->{{"Insects Eaten (-(OverDot[N]))",None}, {"Insects (N)", "Predation"}}
],
Plot[p3[x],{x,0,2000},PlotRange->{{0,2000},{0,500}},PlotStyle->Green,ImageSize->Medium],
ListPlot[{Callout[{a,p2[a]}]},PlotStyle->Red,ImageSize->Medium]]}]],
{{r,1,"Growth Rate (R) (1/Day)"},1,20,0.1},
{{k,400,"Carrying Capacity (K) (Insects)"},1,2000, 1},
{{n0, 5,"Initial Insects ((Subscript[N, 0])) (Insects)"},1,1000,1},
{{predate,1,"Enable Predation"},{0,1}},
{{a,5,"Population at Critical Predation (A) (Insects)"},0,500},
{{b,5,"Maximum Predation (B) (Insects/Day)"},0,500}]

Converting  ((dN)/(dt))=N R (1-(N/K))-((B*N)/(A+N)) into the dimensionless form ((dx)/(dτ))=α x(1-(x/k))-(x/(1+x)).
First take a look at the predation term ((B*N)/(A+N))which will be turned into (x/(1+x)).
Let N=A x: ((B*A*x)/(A+A*x))
Simplifying by eliminating A: ((B*x)/(1+x))
Apply the transformation: ((dN)/(dt))=A((dx)/(dt))
Divide by B on both sides: (A/B)((dx)/(dt))=(R/B)A x (1-((A x)/K))-(x/(1+x))
From this, it can be determined: τ=((B t)/A), α=((R A)/B), k = K/A, x=(N/A)
Plugging these constants in, ((dx)/(dτ))=α x(1-(x/k))-(x/(1+x)) is obtained.

(Subscript[A, u]):="Insects"
(Subscript[B, u]):="Insects"/"Days"
(Subscript[K, u]):="Insects"
(Subscript[R, u]):=1/"Days"
(Subscript[t, u]):="Days"
(Subscript[N, u]):="Insects"
((Subscript[α, u])=(((Subscript[R, u]) (Subscript[A, u]))/(Subscript[B, u])))==1
((Subscript[k, u])=((Subscript[K, u])/(Subscript[A, u])))==1
((Subscript[τ, u])=(((Subscript[B, u])(Subscript[t, u]))/(Subscript[A, u])))==1
((Subscript[x, u])=((Subscript[N, u])/(Subscript[A, u])))==1
((Subscript[x, u])/(Subscript[τ, u]))==(Subscript[α, u])(Subscript[x, u])(1-((Subscript[x, u])/(Subscript[k, u])))-((Subscript[x, u])/(1+(Subscript[x, u])))

Manipulate[Module[{sol},
sol=NDSolve[{x'[τ]==α*x[τ](1-x[τ]/k)-x[τ]/(1+x[τ]),x[0]==x0},x[τ],{τ,0,30}];
Plot[Evaluate[x[τ]/.sol],{τ,0,30},PlotRange->{0,10},Frame->True,
FrameLabel->{{"Scaled Population (x)",None}, {"Scaled Time (τ)", "Dimensionless"}}]],
{{x0,10,"Initial Scaled Population ((Subscript[x, 0]))"},0,10},
{{α,1,"Growth vs Predation Strength (α)"},0,5},
{{k,0.5,"Scaled Carrying Capacity (k)"},0.5,3,0.25}]

Manipulate[Module[{sol},
sol=NDSolve[{x'[τ]==If[τ>15,α2,α1]*x[τ](1-x[τ]/k)-x[τ]/(1+x[τ]),x[0]==x0},x[τ],{τ,0,30}];
Plot[Evaluate[x[τ]/.sol],{τ,0,30},PlotRange->{0,3},Frame->True,
FrameLabel->{{"Scaled Population (x)",None}, {"Scaled Time (τ)", "Dimensionless"}}]],
{{x0,10,"Initial Scaled Population ((Subscript[x, 0]))"},0,10},
{{α1,1,"Growth vs Predation Strength (α1)"},0,10},
{{α2,1,"Growth vs Predation Strength (α2)"},0,10},
{{k,2,"Scaled Carrying Capacity (k)"},0.5,3,0.25}]

Bifurcation Points (k=1)

Clear[a,α,xs]
xs=Solve[0==α*x*(1-x)-(x/(1+x)),x]
Show[
Plot[Evaluate[x/.xs[[2]]],{α,1,5},PlotStyle->{Thick,Red},PlotRange->{{0,2},{-1.5,1.5}},Frame->True,FrameLabel->{{"x",None},{"α","Bifurcations (k=1)"}}],
Plot[Evaluate[x/.xs[[3]]],{α,1,5},PlotStyle->Thick,PlotRange->{{0,2},{-1.5,1.5}}],
Plot[Evaluate[x/.xs[[1]]],{α,0,1},PlotStyle->Thick,PlotRange->{{0,2},{-2,2}}],
Plot[Evaluate[x/.xs[[1]]],{α,1,5},PlotStyle->Dashed,PlotRange->{{0,2},{-2,2}}],
Graphics[{PointSize[Large],Green,Point[{1,0}]}]
]

Bifurcation Points (k=2)

Clear[a,α,xs]
xs=Simplify[Solve[0==α*x*(1-(x/2))-(x/(1+x)),x]]
f[x_,alpha_]:=alpha*x*(1-x/2)-x/(1+x);
bifPoint=Solve[{f[x,alpha]==0,D[f[x,alpha],x]==0,x>0,alpha>0},{x,alpha}]
ContourPlot[f[x,alpha]==0,{alpha,0,2},{x,0,4},FrameLabel->{"Growth Rate (α)","Population Density (x)"},PlotLabel->"Bifurcations (k=2)",GridLines->Automatic,ContourStyle->{Thick,Blue},Epilog->{Red,PointSize[0.02],Point[{alpha,x}/. bifPoint]},PlotRange->Full]

(Limit2Arg[((B N)/(A+N)), N, A])=(B/2)