Modeling

We start by defining the dynamical variables

vars = {x[t], y[t]} ;

Define the equation of the model

f1 = ro (1 - x[t]/xo) x[t] - (k x[t] y[t])/(1 + x[t]) ;

f2 = (k x[t] y[t])/(1 + x[t]) - y[t] ;

Set the differential equations

odes = {x '[t] == f1, y '[t] == f2} ;

Steady states satisfy  the following equation:
                eq1 = 0
                eq2 = 0

Steady states satisfy  the following equation:
                f1 = 0
                f2 = 0

Now we first solve for the srteady states

solSS = Solve[{f1 == 0, f2 == 0}, vars]

{{x[t] →0, y[t] →0}, {y[t] →0, x[t] →xo}, {y[t] → (ro (-1 - xo + k xo))/((-1 + k)^2 xo), x[t] →1/(-1 + k)}}

jac = Outer[D, {f1, f2}, vars] ;

MatrixForm[Simplify[jac]]

( {{ro - (k y[t])/(1 + x[t]) + x[t] (-(2 ro)/xo + (k y[t])/(1 + x[t])^2), -(k x[t])/(1 + x[t])}, {(k y[t])/(1 + x[t])^2, -1 + (k x[t])/(1 + x[t])}} )

eigenV = Map[Eigenvalues[jac/.#] &, solSS] ;

par1 = {ro→4.8, xo→8, k→1.3} ;

MatrixForm[solSS/.par1]

( {{x[t] →0, y[t] →0}, {y[t] →0, x[t] →8}, {y[t] →9.33333, x[t] →3.33333}} )

eigenV1 = eigenV/.par1 ;

MatrixForm[eigenV1]

( {{-1, 4.8}, {-4.8, 0.155556}, {0.0769231 - 0.800148 , 0.0769231 + 0.800148 }} )

Created by Mathematica  (September 7, 2006) Valid XHTML 1.1!