f[n_,p_]:=n(1-(n/6))-((n*p)/((4+1.2n+p)));
g[n_,p_]:=((n*p)/((4+1.2n+p)))-p;
Solve[n(1-(n/6))-((n*p)/((4+1.2n+p)))==0&&((n*p)/((4+1.2n+p)))-p==0,{n,p},Reals]

jacobianMatrix[n_,p_]={{D[f[n,p],n],D[f[n,p],p]},{D[g[n,p],n],D[g[n,p],p]}};
MatrixForm[jacobianMatrix[n,p]]

ss1=jacobianMatrix[0,0]

Eigenvalues[ss1]

ss2=jacobianMatrix[2.479473661428269`,-3.5041052677143463`]

Eigenvalues[ss2]

ss3=jacobianMatrix[6,0]

Eigenvalues[ss3]

ss4=jacobianMatrix[9.67947366142827`,5.935894732285654`]

Eigenvalues[ss4]

u1=NDSolve[{f[n[t],p[t]]==n'[t]&&g[n[t],p[t]]==p'[t]&&n[0]==1&&p[0]==1},{n[t],p[t]},{t,0,10}];
plot1=ParametricPlot[Evaluate[{n[t],p[t]}/.u1],{t,0,10},PlotRange->All,PlotStyle->{Blue,Thick},AxesOrigin->{0,0},AxesLabel->{"N","P"}]

u2=NDSolve[{f[n[t],p[t]]==n'[t]&&g[n[t],p[t]]==p'[t]&&n[0]==3&&p[0]==4},{n[t],p[t]},{t,0,10}];
plot2=ParametricPlot[Evaluate[{n[t],p[t]}/.u2],{t,0,10},PlotRange->All,PlotStyle->{Red,Dashed},AxesOrigin->{0,0},AxesLabel->{"N","P"}]

u3=NDSolve[{f[n[t],p[t]]==n'[t]&&g[n[t],p[t]]==p'[t]&&n[0]==5&&p[0]==6},{n[t],p[t]},{t,0,10}];
plot3=ParametricPlot[Evaluate[{n[t],p[t]}/.u3],{t,0,10},PlotRange->All,PlotStyle->Green,AxesOrigin->{0,0},AxesLabel->{"N","P"}]

Show[plot1,plot2,plot3]