Contour plot of greatest real root of an equation y^4=f(a,b)

Clear["Global`*"]

Λ1 = 100;
β1 = 0.0000006;
β2 = 0.00011;
α2 = 359;
μ1 = 0.0004;
δ1 = .038;
μ2 = 0.33;
δ2 = 0.0227;
k1 = 100000;
r = 100;
η1 = .6;
μ3 = 0.142;

A function that uses a numeric technique should have its arguments restricted to numeric values.

R0[σ_?NumericQ, α1_?NumericQ] := 
 NSolve[y^4 - Λ1 k1 β1 β2 α1 α2 (1 - 
σ)/(3 μ1 μ2 μ3 δ2 (η1 + μ1 + δ1)) ==
    0, y, Reals]

A Blank (pattern object) should not be used when calling a function.

ContourPlot[Max[y /. R0[σ, α1]],
 {σ, 0.01, .99}, {α1, 300, 350},
 PlotLegends -> Automatic,
 LabelStyle -> {Black, 14},
 FrameLabel -> {"σ", "α1"},
 ColorFunction -> "RedBlueTones"]

I reduced the number of contours so that the contours can be seen. Rather than using a high number of contours use DensityPlot.

DensityPlot[Max[y /. R0[σ, α1]],
 {σ, 0.01, .99}, {α1, 300, 350},
 PlotLegends -> Automatic,
 LabelStyle -> {Black, 14},
 FrameLabel -> {"σ", "α1"},
 ColorFunction -> "RedBlueTones"]

EDIT: If you want to both see contours and have a smooth color gradient use

Show[
 DensityPlot[Max[y /. R0[σ, α1]],
  {σ, 0.01, .99}, {α1, 300, 350},
  PlotLegends -> Automatic,
  LabelStyle -> {Black, 14},
  FrameLabel -> {"σ", "α1"},
  ColorFunction -> "RedBlueTones"],
 ContourPlot[Max[y /. R0[σ, α1]],
  {σ, 0.01, .99}, {α1, 300, 350},
  ContourShading -> None,
  PlotLegends -> Automatic,
  LabelStyle -> {Black, 14},
  FrameLabel -> {"σ", "α1"},
  ColorFunction -> "RedBlueTones"]]