datai={{54.902`,-0.369`},{19.182`,-0.275`},{25.617`,-0.255`},{31.64`,-0.228`},{39.396`,-0.21`},{129.025`,-0.258`},{165.196`,-0.187`},{83.659`,-0.136`},{26.429`,-0.171`},{24.319`,-0.139`},{23.271`,-0.119`},{48.166`,-0.116`},{78.27`,-0.035`},{134.42`,0.028`},{204.963`,0.166`},{218.885`,0.166`},{204.812`,0.134`},{160.102`,-0.03`},{165.744`,-0.008`},{181.919`,0.004`},{191.553`,-0.005`},{199.164`,-0.018`},{199.138`,-0.032`},{187.609`,-0.055`},{180.212`,-0.057`},{192.799`,0.059`},{197.906`,0.045`},{204.54`,0.042`},{208.569`,0.029`},{217.127`,0.051`},{281.713`,0.091`},{299.678`,0.06`},{322.048`,0.067`},{265.609`,0.004`},{276.833`,-0.009`},{257.675`,-0.064`},{283.112`,-0.051`},{319.601`,-0.062`},{340.118`,-0.063`}};

ListPlot[datai,PlotRange->{-.4,.2},AspectRatio->1,Frame->True,FrameLabel->{"Population size","Per capita growth rate"},LabelStyle->Directive[FontSize->14],PlotStyle->{Red,PointSize[.015]}]

This given Log[N(t+1)/N(t)] as a function of N(t). What we want is N(t+1)-N(t) as a function of N(t). So we need to transform the y variable to N(t) (Exp[y]-1).

Length[datai]

data=Table[{datai[[i,1]],(Exp[datai[[i,2]]]-1)*datai[[i,1]]},{i,1,Length[datai]}]

Append[data,{0,0}]

Fit[Append[Append[data,{0,0}],{0,0}],{1,x,(x^2),(x^3)},x]

Plot[4.4 -0.5 x+0.00425 (x^2)-8.67*^-6 (x^3),{x,19,350}]
ListPlot[data,PlotRange->{-40,50},AspectRatio->1,Frame->True,FrameLabel->{"Population size (individuals)","Population growth rate (individuals per year)"},LabelStyle->Directive[FontSize->14],PlotStyle->{Red,PointSize[.015]}]
Show[%,%%]

Export["1.eps",%72,"EPS"]

Solve[4.4 -0.5 x+0.00425 (x^2)-8.67*^-6 (x^3)==0,x]

D[4.4 -0.5 x+0.00425 (x^2)-8.67*^-6 (x^3),x]

Solve[-0.5`+0.0085` x-0.00002601` (x^2)==0,x]

4.4 -0.5 x+0.00425 (x^2)-8.67*^-6 (x^3)/.x->249.8611539563592`

-0.28+0.0024 x-4.6*(10^(-6)) (x^2)/.x->260.8695652173913`

Exp[0.03304347826086951`]