Skip to main content
Mathematica

Return to Question

deleted 46 characters in body
Source Link
pyring
pyring
  • 103
  • 3

I have a system of differential equations that I have reduced to this:

\begin{equation} \small \frac{db}{dt}= \beta_1 (1-b(t)-c(t)) b(t) + (1-\beta_1) \beta_3 (1-b(t)-c(t)) d(t) - \phi_{w1} b(t) c(t) - \phi_{b1} b(t) e(t) - \mu_B b(t) - \gamma_1 b(t) \end{equation} \begin{equation} \small \frac{dc}{dt}= \beta_2 (1-b(t)-c(t)) c(t) + (1-\beta_2) \beta_4 (1-b(t)-c(t)) e(t) + \phi_{w1} b(t) c(t) + \phi_{b1} b(t) e(t) - \mu_C c(t) - \gamma_2 c(t) \end{equation} \begin{equation} \small \frac{dd}{dt}= \beta_3 (1-d(t)-e(t)) d(t) + (1-\beta_3) \beta_1 (1-d(t)-e(t)) b(t) - \phi_{w2} d(t) e(t) - \phi_{b2} d(t) c(t) - \mu_D d(t) - \gamma_3 d(t) \end{equation} \begin{equation} \small \frac{de}{dt}= \beta_4 (1-d(t)-e(t)) e(t) + (1-\beta_4) \beta_2 (1-d(t)-e(t)) c(t) + \phi_{w2} d(t) e(t) + \phi_{b2} d(t) c(t) - \mu_E e(t) - \gamma_4 e(t) \end{equation}

Here the code for the equations as input in mathematica:

 ode1 = b'[t] == Subscript[\[Beta], 1]*(1 \[Minus] b[t] \[Minus] c[t])* b[t] + (1 \[Minus] Subscript[\[Beta], 1])* Subscript[\[Beta], 3]*(1 \[Minus] b[t] \[Minus] c[t])* d[t] \[Minus] Subscript[\[Phi], w1]*b[t]*c[t] \[Minus] Subscript[\[Phi], b1]*b[t]*e[t] \[Minus] Subscript[\[Mu], B]*b[t] \[Minus] Subscript[\[Gamma], 1]*b[t]; ode2 = c'[t] == Subscript[\[Beta], 2]*(1 \[Minus] b[t] \[Minus] c[t])* c[t] + (1 \[Minus] Subscript[\[Beta], 2])* Subscript[\[Beta], 4]*(1 \[Minus] b[t] \[Minus] c[t])*e[t] + Subscript[\[Phi], w1]*b[t]*c[t] + Subscript[\[Phi], b1]*b[t]*e[t] \[Minus] Subscript[\[Mu], C]*c[t] \[Minus] Subscript[\[Gamma], 2]*c[t]; ode3 = d'[t] == Subscript[\[Beta], 3] *(1 \[Minus] d[t] \[Minus] e[t])* d[t] + (1 \[Minus] Subscript[\[Beta], 3])* Subscript[\[Beta], 1]*(1 \[Minus] d[t] \[Minus] e[t])* b[t] \[Minus] Subscript[\[Phi], w2]*d[t]*e[t] \[Minus] Subscript[\[Phi], b2]*d[t]*c[t] \[Minus] Subscript[\[Mu], D]*d[t] \[Minus] Subscript[\[Gamma], 3]*d[t]; ode4 = e'[t] == Subscript[\[Beta], 4] *(1 \[Minus] d[t] \[Minus] e[t])* e[t] + (1 \[Minus] Subscript[\[Beta], 4])* Subscript[\[Beta], 2]*(1 \[Minus] d[t] \[Minus] e[t])*c[t] + Subscript[\[Phi], w2]*d[t]*e[t] + Subscript[\[Phi], b2]*d[t]*c[t] \[Minus] Subscript[\[Mu], E]*e[t] \[Minus] Subscript[\[Gamma], 4]*e[t];

Would the code to solve the system in mathematica be something like this?

 b[0] = 100 c[0] = 500 d[0] = 100 e[0] = 400 (fb, fc, fd, fe} = NDSolveValue[{ode1, ode2, ode3, ode4, b[0] == k1, c[0] == k2, d[0] == k3, e[0] == k4}, {b, c, d, e}, {t, 0, 1000}].

I was trying to generate a visualisation of the Phase portrait of this system in mathematica.

Here b, c, d and e are positive real variables (population) (in general taking values between 0 and 1000). Variable t goes from 0 to 1000. All the rest of parameters are all decimal numbers taking values from 0 to 1. We can assume constant prameters $\mu$ = 0.06 and constant parameters $\gamma$=0.01 in all the cases.

How can I a make a good visualisation of phase portraits of this system. For example, a 3D plot.

Can anyone help me?

I have a system of differential equations that I have reduced to this:

\begin{equation} \small \frac{db}{dt}= \beta_1 (1-b(t)-c(t)) b(t) + (1-\beta_1) \beta_3 (1-b(t)-c(t)) d(t) - \phi_{w1} b(t) c(t) - \phi_{b1} b(t) e(t) - \mu_B b(t) - \gamma_1 b(t) \end{equation} \begin{equation} \small \frac{dc}{dt}= \beta_2 (1-b(t)-c(t)) c(t) + (1-\beta_2) \beta_4 (1-b(t)-c(t)) e(t) + \phi_{w1} b(t) c(t) + \phi_{b1} b(t) e(t) - \mu_C c(t) - \gamma_2 c(t) \end{equation} \begin{equation} \small \frac{dd}{dt}= \beta_3 (1-d(t)-e(t)) d(t) + (1-\beta_3) \beta_1 (1-d(t)-e(t)) b(t) - \phi_{w2} d(t) e(t) - \phi_{b2} d(t) c(t) - \mu_D d(t) - \gamma_3 d(t) \end{equation} \begin{equation} \small \frac{de}{dt}= \beta_4 (1-d(t)-e(t)) e(t) + (1-\beta_4) \beta_2 (1-d(t)-e(t)) c(t) + \phi_{w2} d(t) e(t) + \phi_{b2} d(t) c(t) - \mu_E e(t) - \gamma_4 e(t) \end{equation}

Here the code for the equations as input in mathematica:

 ode1 = b'[t] == Subscript[\[Beta], 1]*(1 \[Minus] b[t] \[Minus] c[t])* b[t] + (1 \[Minus] Subscript[\[Beta], 1])* Subscript[\[Beta], 3]*(1 \[Minus] b[t] \[Minus] c[t])* d[t] \[Minus] Subscript[\[Phi], w1]*b[t]*c[t] \[Minus] Subscript[\[Phi], b1]*b[t]*e[t] \[Minus] Subscript[\[Mu], B]*b[t] \[Minus] Subscript[\[Gamma], 1]*b[t]; ode2 = c'[t] == Subscript[\[Beta], 2]*(1 \[Minus] b[t] \[Minus] c[t])* c[t] + (1 \[Minus] Subscript[\[Beta], 2])* Subscript[\[Beta], 4]*(1 \[Minus] b[t] \[Minus] c[t])*e[t] + Subscript[\[Phi], w1]*b[t]*c[t] + Subscript[\[Phi], b1]*b[t]*e[t] \[Minus] Subscript[\[Mu], C]*c[t] \[Minus] Subscript[\[Gamma], 2]*c[t]; ode3 = d'[t] == Subscript[\[Beta], 3] *(1 \[Minus] d[t] \[Minus] e[t])* d[t] + (1 \[Minus] Subscript[\[Beta], 3])* Subscript[\[Beta], 1]*(1 \[Minus] d[t] \[Minus] e[t])* b[t] \[Minus] Subscript[\[Phi], w2]*d[t]*e[t] \[Minus] Subscript[\[Phi], b2]*d[t]*c[t] \[Minus] Subscript[\[Mu], D]*d[t] \[Minus] Subscript[\[Gamma], 3]*d[t]; ode4 = e'[t] == Subscript[\[Beta], 4] *(1 \[Minus] d[t] \[Minus] e[t])* e[t] + (1 \[Minus] Subscript[\[Beta], 4])* Subscript[\[Beta], 2]*(1 \[Minus] d[t] \[Minus] e[t])*c[t] + Subscript[\[Phi], w2]*d[t]*e[t] + Subscript[\[Phi], b2]*d[t]*c[t] \[Minus] Subscript[\[Mu], E]*e[t] \[Minus] Subscript[\[Gamma], 4]*e[t];

Would the code to solve the system in mathematica be something like this?

 b[0] = 100 c[0] = 500 d[0] = 100 e[0] = 400 (fb, fc, fd, fe} = NDSolveValue[{ode1, ode2, ode3, ode4, b[0] == k1, c[0] == k2, d[0] == k3, e[0] == k4}, {b, c, d, e}, {t, 0, 1000}].

I was trying to generate a visualisation of the Phase portrait of this system in mathematica.

Here b, c, d and e are positive real variables (population) (in general taking values between 0 and 1000). Variable t goes from 0 to 1000. All the rest of parameters are all decimal numbers taking values from 0 to 1.

How can I a make a good visualisation of phase portraits of this system. For example, a 3D plot.

Can anyone help me?

I have a system of differential equations that I have reduced to this:

\begin{equation} \small \frac{db}{dt}= \beta_1 (1-b(t)-c(t)) b(t) + (1-\beta_1) \beta_3 (1-b(t)-c(t)) d(t) - \phi_{w1} b(t) c(t) - \phi_{b1} b(t) e(t) - \mu_B b(t) - \gamma_1 b(t) \end{equation} \begin{equation} \small \frac{dc}{dt}= \beta_2 (1-b(t)-c(t)) c(t) + (1-\beta_2) \beta_4 (1-b(t)-c(t)) e(t) + \phi_{w1} b(t) c(t) + \phi_{b1} b(t) e(t) - \mu_C c(t) - \gamma_2 c(t) \end{equation} \begin{equation} \small \frac{dd}{dt}= \beta_3 (1-d(t)-e(t)) d(t) + (1-\beta_3) \beta_1 (1-d(t)-e(t)) b(t) - \phi_{w2} d(t) e(t) - \phi_{b2} d(t) c(t) - \mu_D d(t) - \gamma_3 d(t) \end{equation} \begin{equation} \small \frac{de}{dt}= \beta_4 (1-d(t)-e(t)) e(t) + (1-\beta_4) \beta_2 (1-d(t)-e(t)) c(t) + \phi_{w2} d(t) e(t) + \phi_{b2} d(t) c(t) - \mu_E e(t) - \gamma_4 e(t) \end{equation}

Here the code for the equations as input in mathematica:

 ode1 = b'[t] == Subscript[\[Beta], 1]*(1 \[Minus] b[t] \[Minus] c[t])* b[t] + (1 \[Minus] Subscript[\[Beta], 1])* Subscript[\[Beta], 3]*(1 \[Minus] b[t] \[Minus] c[t])* d[t] \[Minus] Subscript[\[Phi], w1]*b[t]*c[t] \[Minus] Subscript[\[Phi], b1]*b[t]*e[t] \[Minus] Subscript[\[Mu], B]*b[t] \[Minus] Subscript[\[Gamma], 1]*b[t]; ode2 = c'[t] == Subscript[\[Beta], 2]*(1 \[Minus] b[t] \[Minus] c[t])* c[t] + (1 \[Minus] Subscript[\[Beta], 2])* Subscript[\[Beta], 4]*(1 \[Minus] b[t] \[Minus] c[t])*e[t] + Subscript[\[Phi], w1]*b[t]*c[t] + Subscript[\[Phi], b1]*b[t]*e[t] \[Minus] Subscript[\[Mu], C]*c[t] \[Minus] Subscript[\[Gamma], 2]*c[t]; ode3 = d'[t] == Subscript[\[Beta], 3] *(1 \[Minus] d[t] \[Minus] e[t])* d[t] + (1 \[Minus] Subscript[\[Beta], 3])* Subscript[\[Beta], 1]*(1 \[Minus] d[t] \[Minus] e[t])* b[t] \[Minus] Subscript[\[Phi], w2]*d[t]*e[t] \[Minus] Subscript[\[Phi], b2]*d[t]*c[t] \[Minus] Subscript[\[Mu], D]*d[t] \[Minus] Subscript[\[Gamma], 3]*d[t]; ode4 = e'[t] == Subscript[\[Beta], 4] *(1 \[Minus] d[t] \[Minus] e[t])* e[t] + (1 \[Minus] Subscript[\[Beta], 4])* Subscript[\[Beta], 2]*(1 \[Minus] d[t] \[Minus] e[t])*c[t] + Subscript[\[Phi], w2]*d[t]*e[t] + Subscript[\[Phi], b2]*d[t]*c[t] \[Minus] Subscript[\[Mu], E]*e[t] \[Minus] Subscript[\[Gamma], 4]*e[t];

Would the code to solve the system in mathematica be something like this?

 b[0] = 100 c[0] = 500 d[0] = 100 e[0] = 400 (fb, fc, fd, fe} = NDSolveValue[{ode1, ode2, ode3, ode4, b[0] == k1, c[0] == k2, d[0] == k3, e[0] == k4}, {b, c, d, e}, {t, 0, 1000}].

I was trying to generate a visualisation of the Phase portrait of this system in mathematica.

Here b, c, d and e are positive real variables (population). Variable t goes from 0 to 1000. All the rest of parameters are all decimal numbers taking values from 0 to 1. We can assume constant prameters $\mu$ = 0.06 and constant parameters $\gamma$=0.01 in all the cases.

How can I a make a good visualisation of phase portraits of this system. For example, a 3D plot.

Can anyone help me?

added 3 characters in body
Source Link
pyring
pyring
  • 103
  • 3

I have a system of differential equations that I have reduced to this:

\begin{equation} \small \frac{db}{dt}= \beta_1 (1-b(t)-c(t)) b(t) + (1-\beta_1) \beta_3 (1-b(t)-c(t)) d(t) - \phi_{w1} b(t) c(t) - \phi_{b1} b(t) e(t) - \mu_B b(t) - \gamma_1 b(t) \end{equation} \begin{equation} \small \frac{dc}{dt}= \beta_2 (1-b(t)-c(t)) c(t) + (1-\beta_2) \beta_4 (1-b(t)-c(t)) e(t) + \phi_{w1} b(t) c(t) + \phi_{b1} b(t) e(t) - \mu_C c(t) - \gamma_2 c(t) \end{equation} \begin{equation} \small \frac{dd}{dt}= \beta_3 (1-d(t)-e(t)) d(t) + (1-\beta_3) \beta_1 (1-d(t)-e(t)) b(t) - \phi_{w2} d(t) e(t) - \phi_{b2} d(t) c(t) - \mu_D d(t) - \gamma_3 d(t) \end{equation} \begin{equation} \small \frac{de}{dt}= \beta_4 (1-d(t)-e(t)) e(t) + (1-\beta_4) \beta_2 (1-d(t)-e(t)) c(t) + \phi_{w2} d(t) e(t) + \phi_{b2} d(t) c(t) - \mu_E e(t) - \gamma_4 e(t) \end{equation}

Here the code for the equations as input in mathematica:

 ode1 = b'[t] == Subscript[\[Beta], 1]*(1 \[Minus] b[t] \[Minus] c[t])* b[t] + (1 \[Minus] Subscript[\[Beta], 1])* Subscript[\[Beta], 3]*(1 \[Minus] b[t] \[Minus] c[t])* d[t] \[Minus] Subscript[\[Phi], w1]*b[t]*c[t] \[Minus] Subscript[\[Phi], b1]*b[t]*e[t] \[Minus] Subscript[\[Mu], B]*b[t] \[Minus] Subscript[\[Gamma], 1]*b[t]; ode2 = c'[t] == Subscript[\[Beta], 2]*(1 \[Minus] b[t] \[Minus] c[t])* c[t] + (1 \[Minus] Subscript[\[Beta], 2])* Subscript[\[Beta], 4]*(1 \[Minus] b[t] \[Minus] c[t])*e[t] + Subscript[\[Phi], w1]*b[t]*c[t] + Subscript[\[Phi], b1]*b[t]*e[t] \[Minus] Subscript[\[Mu], C]*c[t] \[Minus] Subscript[\[Gamma], 2]*c[t]; ode3 = d'[t] == Subscript[\[Beta], 3] *(1 \[Minus] d[t] \[Minus] e[t])* d[t] + (1 \[Minus] Subscript[\[Beta], 3])* Subscript[\[Beta], 1]*(1 \[Minus] d[t] \[Minus] e[t])* b[t] \[Minus] Subscript[\[Phi], w2]*d[t]*e[t] \[Minus] Subscript[\[Phi], b2]*d[t]*c[t] \[Minus] Subscript[\[Mu], D]*d[t] \[Minus] Subscript[\[Gamma], 3]*d[t]; ode4 = e'[t] == Subscript[\[Beta], 4] *(1 \[Minus] d[t] \[Minus] e[t])* e[t] + (1 \[Minus] Subscript[\[Beta], 4])* Subscript[\[Beta], 2]*(1 \[Minus] d[t] \[Minus] e[t])*c[t] + Subscript[\[Phi], w2]*d[t]*e[t] + Subscript[\[Phi], b2]*d[t]*c[t] \[Minus] Subscript[\[Mu], E]*e[t] \[Minus] Subscript[\[Gamma], 4]*e[t];

Would the code to colvesolve the system in mathematica be something like this?

 b[0] = 100 c[0] = 500 d[0] = 100 e[0] = 400 (fb, fc, fd, fe} = NDSolveValue[{ode1, ode2, ode3, ode4, b[0] == k1, c[0] == k2, d[0] == k3, e[0] == k4}, {b, c, d, e}, {t, 0, 1000}].

I was trying to generate a visualisation of the Phase portrait of this system in mathematica.

Here b, c, d and e are positive real variables (population) (in general taking values between 0 and 1000). Variable t goes from 0 to 1000. All the rest of parameters are all decimal numbers taking values from 0 to 1.

How can I a make a good visualisation of phase portraits of this system. For example, a 3D plot.

Can anyone help me?

I have a system of differential equations that I have reduced to this:

\begin{equation} \small \frac{db}{dt}= \beta_1 (1-b(t)-c(t)) b(t) + (1-\beta_1) \beta_3 (1-b(t)-c(t)) d(t) - \phi_{w1} b(t) c(t) - \phi_{b1} b(t) e(t) - \mu_B b(t) - \gamma_1 b(t) \end{equation} \begin{equation} \small \frac{dc}{dt}= \beta_2 (1-b(t)-c(t)) c(t) + (1-\beta_2) \beta_4 (1-b(t)-c(t)) e(t) + \phi_{w1} b(t) c(t) + \phi_{b1} b(t) e(t) - \mu_C c(t) - \gamma_2 c(t) \end{equation} \begin{equation} \small \frac{dd}{dt}= \beta_3 (1-d(t)-e(t)) d(t) + (1-\beta_3) \beta_1 (1-d(t)-e(t)) b(t) - \phi_{w2} d(t) e(t) - \phi_{b2} d(t) c(t) - \mu_D d(t) - \gamma_3 d(t) \end{equation} \begin{equation} \small \frac{de}{dt}= \beta_4 (1-d(t)-e(t)) e(t) + (1-\beta_4) \beta_2 (1-d(t)-e(t)) c(t) + \phi_{w2} d(t) e(t) + \phi_{b2} d(t) c(t) - \mu_E e(t) - \gamma_4 e(t) \end{equation}

Here the code for the equations as input in mathematica:

 ode1 = b'[t] == Subscript[\[Beta], 1]*(1 \[Minus] b[t] \[Minus] c[t])* b[t] + (1 \[Minus] Subscript[\[Beta], 1])* Subscript[\[Beta], 3]*(1 \[Minus] b[t] \[Minus] c[t])* d[t] \[Minus] Subscript[\[Phi], w1]*b[t]*c[t] \[Minus] Subscript[\[Phi], b1]*b[t]*e[t] \[Minus] Subscript[\[Mu], B]*b[t] \[Minus] Subscript[\[Gamma], 1]*b[t]; ode2 = c'[t] == Subscript[\[Beta], 2]*(1 \[Minus] b[t] \[Minus] c[t])* c[t] + (1 \[Minus] Subscript[\[Beta], 2])* Subscript[\[Beta], 4]*(1 \[Minus] b[t] \[Minus] c[t])*e[t] + Subscript[\[Phi], w1]*b[t]*c[t] + Subscript[\[Phi], b1]*b[t]*e[t] \[Minus] Subscript[\[Mu], C]*c[t] \[Minus] Subscript[\[Gamma], 2]*c[t]; ode3 = d'[t] == Subscript[\[Beta], 3] *(1 \[Minus] d[t] \[Minus] e[t])* d[t] + (1 \[Minus] Subscript[\[Beta], 3])* Subscript[\[Beta], 1]*(1 \[Minus] d[t] \[Minus] e[t])* b[t] \[Minus] Subscript[\[Phi], w2]*d[t]*e[t] \[Minus] Subscript[\[Phi], b2]*d[t]*c[t] \[Minus] Subscript[\[Mu], D]*d[t] \[Minus] Subscript[\[Gamma], 3]*d[t]; ode4 = e'[t] == Subscript[\[Beta], 4] *(1 \[Minus] d[t] \[Minus] e[t])* e[t] + (1 \[Minus] Subscript[\[Beta], 4])* Subscript[\[Beta], 2]*(1 \[Minus] d[t] \[Minus] e[t])*c[t] + Subscript[\[Phi], w2]*d[t]*e[t] + Subscript[\[Phi], b2]*d[t]*c[t] \[Minus] Subscript[\[Mu], E]*e[t] \[Minus] Subscript[\[Gamma], 4]*e[t];

Would the code to colve the system in mathematica something like this?

 b[0] = 100 c[0] = 500 d[0] = 100 e[0] = 400 (fb, fc, fd, fe} = NDSolveValue[{ode1, ode2, ode3, ode4, b[0] == k1, c[0] == k2, d[0] == k3, e[0] == k4}, {b, c, d, e}, {t, 0, 1000}].

I was trying to generate a visualisation of the Phase portrait of this system in mathematica.

Here b, c, d and e are positive real variables (population) (in general taking values between 0 and 1000). Variable t goes from 0 to 1000. All the rest of parameters are all decimal numbers taking values from 0 to 1.

How can I a make a good visualisation of phase portraits of this system. For example, a 3D plot.

Can anyone help me?

I have a system of differential equations that I have reduced to this:

\begin{equation} \small \frac{db}{dt}= \beta_1 (1-b(t)-c(t)) b(t) + (1-\beta_1) \beta_3 (1-b(t)-c(t)) d(t) - \phi_{w1} b(t) c(t) - \phi_{b1} b(t) e(t) - \mu_B b(t) - \gamma_1 b(t) \end{equation} \begin{equation} \small \frac{dc}{dt}= \beta_2 (1-b(t)-c(t)) c(t) + (1-\beta_2) \beta_4 (1-b(t)-c(t)) e(t) + \phi_{w1} b(t) c(t) + \phi_{b1} b(t) e(t) - \mu_C c(t) - \gamma_2 c(t) \end{equation} \begin{equation} \small \frac{dd}{dt}= \beta_3 (1-d(t)-e(t)) d(t) + (1-\beta_3) \beta_1 (1-d(t)-e(t)) b(t) - \phi_{w2} d(t) e(t) - \phi_{b2} d(t) c(t) - \mu_D d(t) - \gamma_3 d(t) \end{equation} \begin{equation} \small \frac{de}{dt}= \beta_4 (1-d(t)-e(t)) e(t) + (1-\beta_4) \beta_2 (1-d(t)-e(t)) c(t) + \phi_{w2} d(t) e(t) + \phi_{b2} d(t) c(t) - \mu_E e(t) - \gamma_4 e(t) \end{equation}

Here the code for the equations as input in mathematica:

 ode1 = b'[t] == Subscript[\[Beta], 1]*(1 \[Minus] b[t] \[Minus] c[t])* b[t] + (1 \[Minus] Subscript[\[Beta], 1])* Subscript[\[Beta], 3]*(1 \[Minus] b[t] \[Minus] c[t])* d[t] \[Minus] Subscript[\[Phi], w1]*b[t]*c[t] \[Minus] Subscript[\[Phi], b1]*b[t]*e[t] \[Minus] Subscript[\[Mu], B]*b[t] \[Minus] Subscript[\[Gamma], 1]*b[t]; ode2 = c'[t] == Subscript[\[Beta], 2]*(1 \[Minus] b[t] \[Minus] c[t])* c[t] + (1 \[Minus] Subscript[\[Beta], 2])* Subscript[\[Beta], 4]*(1 \[Minus] b[t] \[Minus] c[t])*e[t] + Subscript[\[Phi], w1]*b[t]*c[t] + Subscript[\[Phi], b1]*b[t]*e[t] \[Minus] Subscript[\[Mu], C]*c[t] \[Minus] Subscript[\[Gamma], 2]*c[t]; ode3 = d'[t] == Subscript[\[Beta], 3] *(1 \[Minus] d[t] \[Minus] e[t])* d[t] + (1 \[Minus] Subscript[\[Beta], 3])* Subscript[\[Beta], 1]*(1 \[Minus] d[t] \[Minus] e[t])* b[t] \[Minus] Subscript[\[Phi], w2]*d[t]*e[t] \[Minus] Subscript[\[Phi], b2]*d[t]*c[t] \[Minus] Subscript[\[Mu], D]*d[t] \[Minus] Subscript[\[Gamma], 3]*d[t]; ode4 = e'[t] == Subscript[\[Beta], 4] *(1 \[Minus] d[t] \[Minus] e[t])* e[t] + (1 \[Minus] Subscript[\[Beta], 4])* Subscript[\[Beta], 2]*(1 \[Minus] d[t] \[Minus] e[t])*c[t] + Subscript[\[Phi], w2]*d[t]*e[t] + Subscript[\[Phi], b2]*d[t]*c[t] \[Minus] Subscript[\[Mu], E]*e[t] \[Minus] Subscript[\[Gamma], 4]*e[t];

Would the code to solve the system in mathematica be something like this?

 b[0] = 100 c[0] = 500 d[0] = 100 e[0] = 400 (fb, fc, fd, fe} = NDSolveValue[{ode1, ode2, ode3, ode4, b[0] == k1, c[0] == k2, d[0] == k3, e[0] == k4}, {b, c, d, e}, {t, 0, 1000}].

I was trying to generate a visualisation of the Phase portrait of this system in mathematica.

Here b, c, d and e are positive real variables (population) (in general taking values between 0 and 1000). Variable t goes from 0 to 1000. All the rest of parameters are all decimal numbers taking values from 0 to 1.

How can I a make a good visualisation of phase portraits of this system. For example, a 3D plot.

Can anyone help me?

added 259 characters in body
Source Link
pyring
pyring
  • 103
  • 3

\begin{equation} \small \frac{db}{dt}= \beta_1 (1-b-c) b + (1-\beta_1) \beta_3 (1-b-c) d - \phi_{w1} b c - \phi_{b1} b e - \mu_B b - \gamma_1 b \end{equation}\begin{equation} \small \frac{db}{dt}= \beta_1 (1-b(t)-c(t)) b(t) + (1-\beta_1) \beta_3 (1-b(t)-c(t)) d(t) - \phi_{w1} b(t) c(t) - \phi_{b1} b(t) e(t) - \mu_B b(t) - \gamma_1 b(t) \end{equation} \begin{equation} \small \frac{dc}{dt}= \beta_2 (1-b-c) c + (1-\beta_2) \beta_4 (1-b-c) e + \phi_{w1} b c + \phi_{b1} b e - \mu_C c - \gamma_2 c \end{equation}\begin{equation} \small \frac{dc}{dt}= \beta_2 (1-b(t)-c(t)) c(t) + (1-\beta_2) \beta_4 (1-b(t)-c(t)) e(t) + \phi_{w1} b(t) c(t) + \phi_{b1} b(t) e(t) - \mu_C c(t) - \gamma_2 c(t) \end{equation} \begin{equation} \small \frac{dd}{dt}= \beta_3 (1-d-e) d + (1-\beta_3) \beta_1 (1-d-e) b - \phi_{w2} d e - \phi_{b2} d c - \mu_D d - \gamma_3 d \end{equation}\begin{equation} \small \frac{dd}{dt}= \beta_3 (1-d(t)-e(t)) d(t) + (1-\beta_3) \beta_1 (1-d(t)-e(t)) b(t) - \phi_{w2} d(t) e(t) - \phi_{b2} d(t) c(t) - \mu_D d(t) - \gamma_3 d(t) \end{equation} \begin{equation} \small \frac{de}{dt}= \beta_4 (1-d-e) e + (1-\beta_4) \beta_2 (1-d-e) c + \phi_{w2} d e + \phi_{b2} d c - \mu_E e - \gamma_4 e \end{equation}\begin{equation} \small \frac{de}{dt}= \beta_4 (1-d(t)-e(t)) e(t) + (1-\beta_4) \beta_2 (1-d(t)-e(t)) c(t) + \phi_{w2} d(t) e(t) + \phi_{b2} d(t) c(t) - \mu_E e(t) - \gamma_4 e(t) \end{equation}

 ode1 = b'[t] == Subscript[\[Beta],  1] 1]*(1 \[Minus] bb[t] \[Minus] cc[t]) b + (1*  \[Minus]    b[t] + (1 \[Minus] Subscript[\[Beta], 1]) Subscript[\[Beta], *   3]Subscript[\[Beta], 3]*(1 \[Minus] bb[t] \[Minus] cc[t])*  d \[Minus]    d[t] \[Minus] Subscript[\[Phi], w1] bcw1]*b[t]*c[t] \[Minus] Subscript[\[Phi], b1] beb1]*b[t]*e[t] \[Minus] Subscript[\[Mu], B]   b \[Minus]    Subscript[\[Mu], B]*b[t] \[Minus] Subscript[\[Gamma], 1] b;1]*b[t]; ode2 = c'[t] == Subscript[\[Beta],  2] 2]*(1 \[Minus] bb[t] \[Minus] cc[t]) c*  + (1 \[Minus]   c[t] + (1 \[Minus] Subscript[\[Beta], 2]) Subscript[\[Beta], *   4]Subscript[\[Beta], 4]*(1 \[Minus] bb[t] \[Minus] cc[t]) e*e[t] +  Subscript[\[Phi], w1] bcw1]*b[t]*c[t] + Subscript[\[Phi], b1] beb1]*b[t]*e[t] \[Minus] Subscript[\[Mu],   C] c \[Minus]  Subscript[\[Mu], C]*c[t] \[Minus] Subscript[\[Gamma], 2] c;2]*c[t]; ode3 = d'[t] == Subscript[\[Beta],  3]  *(1 \[Minus] dd[t] \[Minus] ee[t]) d + (1*  \[Minus]    d[t] + (1 \[Minus] Subscript[\[Beta], 3]) Subscript[\[Beta], *   1]Subscript[\[Beta], 1]*(1 \[Minus] dd[t] \[Minus] ee[t])*  b \[Minus]    b[t] \[Minus] Subscript[\[Phi], w2] dew2]*d[t]*e[t] \[Minus] Subscript[\[Phi], b2] dcb2]*d[t]*c[t] \[Minus] Subscript[\[Mu], D]   d \[Minus]    Subscript[\[Mu], D]*d[t] \[Minus] Subscript[\[Gamma], 3] d;3]*d[t]; ode4 = e'[t] == Subscript[\[Beta], 4] *(1 \[Minus] d[t] \[Minus] e[t])*   e[t] + (1 \[Minus] Subscript[\[Beta], 4])*  Subscript[\[Beta], 2]*(1 \[Minus] dd[t] \[Minus] ee[t])*c[t] e+ Subscript[\[Phi], w2]*d[t]*e[t] + (1  Subscript[\[Phi], b2]*d[t]*c[t] \[Minus] Subscript[\[Mu], E]*e[t] \[Minus] Subscript[\[Beta]Subscript[\[Gamma], 4])4]*e[t];

Would the code to colve the system in mathematica something like this?

 Subscript[\[Beta], b[0] = 100 c[0] = 500  2]d[0] (1= \[Minus]100  d \[Minus]e[0] e)= c400  + Subscript[\[Phi](fb, w2]fc, defd, +fe}   =    NDSolveValue[{ode1, ode2, Subscript[\[Phi]ode3, b2]ode4, dcb[0] \[Minus]== Subscript[\[Mu]k1, E]c[0] e== \[Minus]k2, Subscript[\[Gamma]d[0] == k3, 4]e[0] e;== k4}, {b, c, d, e}, {t, 0, 1000}].

\begin{equation} \small \frac{db}{dt}= \beta_1 (1-b-c) b + (1-\beta_1) \beta_3 (1-b-c) d - \phi_{w1} b c - \phi_{b1} b e - \mu_B b - \gamma_1 b \end{equation} \begin{equation} \small \frac{dc}{dt}= \beta_2 (1-b-c) c + (1-\beta_2) \beta_4 (1-b-c) e + \phi_{w1} b c + \phi_{b1} b e - \mu_C c - \gamma_2 c \end{equation} \begin{equation} \small \frac{dd}{dt}= \beta_3 (1-d-e) d + (1-\beta_3) \beta_1 (1-d-e) b - \phi_{w2} d e - \phi_{b2} d c - \mu_D d - \gamma_3 d \end{equation} \begin{equation} \small \frac{de}{dt}= \beta_4 (1-d-e) e + (1-\beta_4) \beta_2 (1-d-e) c + \phi_{w2} d e + \phi_{b2} d c - \mu_E e - \gamma_4 e \end{equation}

ode1 = b'[t] == Subscript[\[Beta],  1] (1 \[Minus] b \[Minus] c) b + (1 \[Minus]    Subscript[\[Beta], 1]) Subscript[\[Beta],    3] (1 \[Minus] b \[Minus] c) d \[Minus]    Subscript[\[Phi], w1] bc \[Minus] Subscript[\[Phi], b1] be \[Minus] Subscript[\[Mu], B] b \[Minus]    Subscript[\[Gamma], 1] b; ode2 = c'[t] == Subscript[\[Beta],  2] (1 \[Minus] b \[Minus] c) c + (1 \[Minus]   Subscript[\[Beta], 2]) Subscript[\[Beta],    4] (1 \[Minus] b \[Minus] c) e + Subscript[\[Phi], w1] bc + Subscript[\[Phi], b1] be \[Minus] Subscript[\[Mu], C] c \[Minus]  Subscript[\[Gamma], 2] c; ode3 = d'[t] == Subscript[\[Beta],  3]  (1 \[Minus] d \[Minus] e) d + (1 \[Minus]    Subscript[\[Beta], 3]) Subscript[\[Beta],    1] (1 \[Minus] d \[Minus] e) b \[Minus]    Subscript[\[Phi], w2] de \[Minus] Subscript[\[Phi], b2] dc \[Minus] Subscript[\[Mu], D] d \[Minus]    Subscript[\[Gamma], 3] d; ode4 = e'[t] == Subscript[\[Beta], 4] (1 \[Minus] d \[Minus] e) e + (1 \[Minus] Subscript[\[Beta], 4]) Subscript[\[Beta], 2] (1 \[Minus] d \[Minus] e) c + Subscript[\[Phi], w2] de +    Subscript[\[Phi], b2] dc \[Minus] Subscript[\[Mu], E] e \[Minus] Subscript[\[Gamma], 4] e;

\begin{equation} \small \frac{db}{dt}= \beta_1 (1-b(t)-c(t)) b(t) + (1-\beta_1) \beta_3 (1-b(t)-c(t)) d(t) - \phi_{w1} b(t) c(t) - \phi_{b1} b(t) e(t) - \mu_B b(t) - \gamma_1 b(t) \end{equation} \begin{equation} \small \frac{dc}{dt}= \beta_2 (1-b(t)-c(t)) c(t) + (1-\beta_2) \beta_4 (1-b(t)-c(t)) e(t) + \phi_{w1} b(t) c(t) + \phi_{b1} b(t) e(t) - \mu_C c(t) - \gamma_2 c(t) \end{equation} \begin{equation} \small \frac{dd}{dt}= \beta_3 (1-d(t)-e(t)) d(t) + (1-\beta_3) \beta_1 (1-d(t)-e(t)) b(t) - \phi_{w2} d(t) e(t) - \phi_{b2} d(t) c(t) - \mu_D d(t) - \gamma_3 d(t) \end{equation} \begin{equation} \small \frac{de}{dt}= \beta_4 (1-d(t)-e(t)) e(t) + (1-\beta_4) \beta_2 (1-d(t)-e(t)) c(t) + \phi_{w2} d(t) e(t) + \phi_{b2} d(t) c(t) - \mu_E e(t) - \gamma_4 e(t) \end{equation}

 ode1 = b'[t] == Subscript[\[Beta], 1]*(1 \[Minus] b[t] \[Minus] c[t])*  b[t] + (1 \[Minus] Subscript[\[Beta], 1])* Subscript[\[Beta], 3]*(1 \[Minus] b[t] \[Minus] c[t])*  d[t] \[Minus] Subscript[\[Phi], w1]*b[t]*c[t] \[Minus] Subscript[\[Phi], b1]*b[t]*e[t] \[Minus]    Subscript[\[Mu], B]*b[t] \[Minus] Subscript[\[Gamma], 1]*b[t]; ode2 = c'[t] == Subscript[\[Beta], 2]*(1 \[Minus] b[t] \[Minus] c[t])*  c[t] + (1 \[Minus] Subscript[\[Beta], 2])* Subscript[\[Beta], 4]*(1 \[Minus] b[t] \[Minus] c[t])*e[t] +  Subscript[\[Phi], w1]*b[t]*c[t] + Subscript[\[Phi], b1]*b[t]*e[t] \[Minus]    Subscript[\[Mu], C]*c[t] \[Minus] Subscript[\[Gamma], 2]*c[t]; ode3 = d'[t] == Subscript[\[Beta], 3] *(1 \[Minus] d[t] \[Minus] e[t])*  d[t] + (1 \[Minus] Subscript[\[Beta], 3])* Subscript[\[Beta], 1]*(1 \[Minus] d[t] \[Minus] e[t])*  b[t] \[Minus] Subscript[\[Phi], w2]*d[t]*e[t] \[Minus] Subscript[\[Phi], b2]*d[t]*c[t] \[Minus]    Subscript[\[Mu], D]*d[t] \[Minus] Subscript[\[Gamma], 3]*d[t]; ode4 = e'[t] == Subscript[\[Beta], 4] *(1 \[Minus] d[t] \[Minus] e[t])*   e[t] + (1 \[Minus] Subscript[\[Beta], 4])*  Subscript[\[Beta], 2]*(1 \[Minus] d[t] \[Minus] e[t])*c[t] + Subscript[\[Phi], w2]*d[t]*e[t] +   Subscript[\[Phi], b2]*d[t]*c[t] \[Minus] Subscript[\[Mu], E]*e[t] \[Minus] Subscript[\[Gamma], 4]*e[t];

Would the code to colve the system in mathematica something like this?

  b[0] = 100 c[0] = 500  d[0] = 100  e[0] = 400  (fb, fc, fd, fe} =    NDSolveValue[{ode1, ode2, ode3, ode4, b[0] == k1, c[0] == k2, d[0] == k3, e[0] == k4}, {b, c, d, e}, {t, 0, 1000}].
edited tags
Link
Michael E2
Michael E2
  • 258.7k
  • 21
  • 370
  • 830
added 1491 characters in body
Source Link
pyring
pyring
  • 103
  • 3
Source Link
pyring
pyring
  • 103
  • 3