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I have Lotka Volterra Equations where the prey equation is modified. The Equations are

x'[t]=a*x[t]*(1-x[t]/k)-b*x[t]*y[t]
y'[t]=-c*y[t]+d*x[t]*y[t]

To plot the limit cycles, i used NDSolve to solve the coupled nonlinear differential equations for x[t] and y[t]. I then used ParametricPlot to plot for x[t] and y[t] with respect to t, but I'm not getting appropriate limit cycles.

I did a steady state analysis of the equations and got certain conditions on the value of a, b, c, d and k. The equation has a non-trivial steady state, at $(x,y)=(c/d,(a/b)-(ac/bdk))$. The commands are:

Equations = {x'[t] == ax[t]*(1 - x[t]/k) - bx[t]*y[t], x[0] == 2, y[0] == 1, y'[t] == -cy[t] + dx[t]*y[t]}; 
s = NDSolve[{Equations}, {x, y}, {t, 0, time}];
ParametricPlot[Evaluate[{x[t], y[t]} /. s], {t, 0, 30}] 

For the value of a = 2; b = 1; c = 2.5; d = 1.2; k = 2.8, I'm supposed to get a unstable spiral. But instead, the trajectory approaches the steady state of $(x,y) = (2.08333 , 0.511905)$.

Any help will be greatly appreciated.

I have Lotka Volterra Equations where the prey equation is modified. The Equations are

x'[t]=a*x[t]*(1-x[t]/k)-b*x[t]*y[t]
y'[t]=-c*y[t]+d*x[t]*y[t]

To plot the limit cycles, i used NDSolve to solve the coupled nonlinear differential equations for x[t] and y[t]. I then used ParametricPlot to plot for x[t] and y[t] with respect to t, but I'm not getting appropriate limit cycles.

I did a steady state analysis of the equations and got certain conditions on the value of a, b, c, d and k. The equation has a non-trivial steady state, at $(x,y)=(c/d,(a/b)-(ac/bdk))$. The commands are:

Equations = {x'[t] == a*x[t]*(1 - x[t]/k) - b*x[t]*y[t], x[0] == 2, y[0] == 1, y'[t] == -c*y[t] + d*x[t]*y[t]}; 
s = NDSolve[Equations, {x, y}, {t, 0, time}];
ParametricPlot[Evaluate[{x[t], y[t]} /. s], {t, 0, 30}] 

For the value of a = 2; b = 1; c = 2.5; d = 1.2; k = 2.8, I'm supposed to get a unstable spiral. But instead, the trajectory approaches the steady state of $(x,y) = (2.08333 , 0.511905)$.

Any help will be greatly appreciated.

I have Lotka Volterra Equations where the prey equation is modified. The Equations are

x'[t]=ax[t](1-x[t]/k)-b*x[t]*y[t]

y'[t]=-cy[t]+dx[t]*y[t]

To plot the limit cycles, i used NDsolve to solve the coupled nonlinear differential equations to solve for x[t] and y[t]. I then used ParametricPlot to plot for x[t] and y[t] with respect to t, but I'm not getting appropriate limit cycles.

I did a steady state analysis of the equations and got certain conditions on the value of a,b,c,d and k.

Now, the equation has a non trivial steady state, at (x,y)=(c/d,(a/b)-(ac/bdk)).

  The commands are -

Equations = {x'[t] == ax[t]*(1 - x[t]/k) - bx[t]*y[t], x[0] == 2, y[0] == 1, y'[t] == -cy[t] + dx[t]*y[t]};

s = NDSolve[{Equations}, {x, y}, {t, 0, time}];

ParametricPlot[Evaluate[{x[t], y[t]} /. s], {t, 0, 30}]

For the value of a = 2; b = 1; c = 2.5; d = 1.2; k = 2.8, I'm supposed to get a unstable spiral. But instead, the Parametric Plot goes to the steady state of (x,y) = (2.08333 , 0.511905)

Any help will be greatly appreciated.

I have Lotka Volterra Equations where the prey equation is modified. The Equations are

x'[t]=a*x[t]*(1-x[t]/k)-b*x[t]*y[t]
y'[t]=-c*y[t]+d*x[t]*y[t]

To plot the limit cycles, i used NDSolve to solve the coupled nonlinear differential equations for x[t] and y[t]. I then used ParametricPlot to plot for x[t] and y[t] with respect to t, but I'm not getting appropriate limit cycles.

I did a steady state analysis of the equations and got certain conditions on the value of a, b, c, d and k. The equation has a non-trivial steady state, at $(x,y)=(c/d,(a/b)-(ac/bdk))$. The commands are:

Equations = {x'[t] == ax[t]*(1 - x[t]/k) - bx[t]*y[t], x[0] == 2, y[0] == 1, y'[t] == -cy[t] + dx[t]*y[t]}; 
s = NDSolve[{Equations}, {x, y}, {t, 0, time}];
ParametricPlot[Evaluate[{x[t], y[t]} /. s], {t, 0, 30}] 

For the value of a = 2; b = 1; c = 2.5; d = 1.2; k = 2.8, I'm supposed to get a unstable spiral. But instead, the trajectory approaches the steady state of $(x,y) = (2.08333 , 0.511905)$.

Any help will be greatly appreciated.

I have Lotka Volterra Equations where the prey equation is modified. The Equations are

x'[t]=ax[t](1-x[t]/k)-b*x[t]*y[t]

y'[t]=-cy[t]+dx[t]*y[t]

To plot the limit cycles, i used NDsolve to solve the coupled nonlinear differential equations to solve for x[t] and y[t]. I then used ParametricPlot to plot for x[t] and y[t] with respect to t, but I'm not getting appropriate limit cycles.

Any help will be greatly appreciated.

I have Lotka Volterra Equations where the prey equation is modified. The Equations are

x'[t]=ax[t](1-x[t]/k)-b*x[t]*y[t]

y'[t]=-cy[t]+dx[t]*y[t]

To plot the limit cycles, i used NDsolve to solve the coupled nonlinear differential equations to solve for x[t] and y[t]. I then used ParametricPlot to plot for x[t] and y[t] with respect to t, but I'm not getting appropriate limit cycles.

I did a steady state analysis of the equations and got certain conditions on the value of a,b,c,d and k.

Now, the equation has a non trivial steady state, at (x,y)=(c/d,(a/b)-(ac/bdk)).

The commands are -

Equations = {x'[t] == ax[t]*(1 - x[t]/k) - bx[t]*y[t], x[0] == 2, y[0] == 1, y'[t] == -cy[t] + dx[t]*y[t]};

s = NDSolve[{Equations}, {x, y}, {t, 0, time}];

ParametricPlot[Evaluate[{x[t], y[t]} /. s], {t, 0, 30}]

For the value of a = 2; b = 1; c = 2.5; d = 1.2; k = 2.8, I'm supposed to get a unstable spiral. But instead, the Parametric Plot goes to the steady state of (x,y) = (2.08333 , 0.511905)

Any help will be greatly appreciated.