Leslie Model

The population consists of 7 age classes.

leslie = {{f1,f2,f3,f4,f5,f6,f7},
		  {P1,0,0,0,0,0,0},
		  {0,P2,0,0,0,0,0},
		  {0,0,P3,0,0,0,0},
		  {0,0,0,P4,0,0,0},
		  {0,0,0,0,P5,0,0},
		  {0,0,0,0,0,P6,0}}

MatrixForm[leslie]

Data of grey squirrels in North Carolina

para = {f1->0.32, f2->0.57, f3->0.57, f4->0.57,
	    f5->0.57, f6->0.57,f7->0.57,
	    P1->0.46, P2->0.77, P3->0.65, P4->0.67,
	    P5->0.64, P6->0.88};

leslie2 = leslie/.para

The initial distribution

init = {0.1,0,0,0,0,0,0};

leslie2.init

leslie2.leslie2.init

MatrixPower[leslie2,2].init

MatrixPower[leslie2,3].init

MatrixPower[leslie2,4].init

dist[t_]:=MatrixPower[leslie2,t].init

dist[10]

dist[20]

dist[100]

<<Graphics`Graphics3D`

data = Table[dist[t],{t,0,20}]

BarChart3D[data,BoxRatios->{1,1,0.5},
	PlotRange->All,ViewPoint->{-3.059, 0.285, 1.418}]

data2 = Table[dist[t],{t,0,50}];

BarChart3D[data2,BoxRatios->{1,1,0.5},ViewPoint->{-3.059, 0.285, 1.418}]

data = Table[dist[t],{t,0,50}];

totalDensity = Apply[Plus,Transpose[data]]

ListPlot[totalDensity,PlotJoined->True,PlotRange->{0,0.7}]

ListPlot[ Log[totalDensity] ,PlotJoined->True]

Fit[ Log[totalDensity],{1,t},t]

Exp[0.0377887]

Eigenvalues[leslie2]

Eigenvectors[leslie2]

ratios = %[[1]]

ratios/Apply[Plus,ratios]

Last[data]

Last[data]/Apply[Plus, Last[data] ]

fertility = f1 + f2 P1 + f3 P1 P2 + f4 P1 P2 P3 + f5 P1 P2 P3 P4 +
	f6 P1 P2 P3 P4 P5 + f7 P1 P2 P3 P4 P5 P6

fertility/.para