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Stochastic Logistic Growth Model

    
Define the stochastic differential equation describing the stochastic logistic growth model:
Wolfram Language code: slgm[r_, xmax_, x0_, σ_] := ItoProcess[ⅆx[t] == x[t](1 - (x[t]/xmax))(r ⅆt + σ ⅆw[t]), x[t], {x, x0}, t, wWienerProcess[]]
Deterministic solution of is well known:
Wolfram Language code: detsol[r_, xmax_, x0_] := Function[t, (E^r t x0 /(E^r t - 1) (x0/xmax) + 1)]
Simulate SDE using method of Kloeden–Platen–Schurz of strong order 1.5:
Wolfram Language code: stochsol[r_, xmax_, x0_, σ_] := RandomFunction[slgm[r, xmax, x0, σ], {0, 6, 0.005}, 10, Method -> "KloedenPlatenSchurz"]
Visualize stochastic and deterministic solutions:
Wolfram Language code: With[{r = 1, xmax = 4, x0 = 1, σ = 1 / 2}, Show[ListLinePlot[stochsol[r, xmax, x0, σ]], Plot[detsol[r, xmax, x0][t], {t, 0, 6}, PlotStyle -> Directive[Orange, Thick]]]]