Title: Ordinary Differential Equation Systems in Ecology
Version: 0.1.0
Description: A framework to simulate ecosystem dynamics through ordinary differential equations (ODEs). You create an ODE model, tells 'ecode' to explore its behaviour, and perform numerical simulations on the model. 'ecode' also allows you to fit model parameters by machine learning algorithms. Potential users include researchers who are interested in the dynamics of ecological community and biogeochemical cycles.
Imports: ggplot2, cowplot, rlang, stringr
License: MIT + file LICENSE
Encoding: UTF-8
RoxygenNote: 7.2.3
BugReports: https://github.com/HaoranPopEvo/ecode/issues
URL: https://github.com/HaoranPopEvo/ecode
NeedsCompilation: no
Packaged: 2024-06-30 14:55:42 UTC; asa
Author: Haoran Wu ORCID iD [aut, cre], Chen Peng ORCID iD [aut]
Maintainer: Haoran Wu <haoran.wu@wolfson.ox.ac.uk>
Repository: CRAN
Date/Publication: 2024-07-02 06:20:02 UTC

ecode: Ordinary Differential Equation Systems in Ecology

Description

A framework to simulate ecosystem dynamics through ordinary differential equations (ODEs). You create an ODE model, tells 'ecode' to explore its behaviour, and perform numerical simulations on the model. 'ecode' also allows you to fit model parameters by machine learning algorithms. Potential users include researchers who are interested in the dynamics of ecological community and biogeochemical cycles.

Author(s)

Maintainer: Haoran Wu haoran.wu@wolfson.ox.ac.uk (ORCID)

Authors:

See Also

Useful links:

Multiply Operator for Phase Points

Description

The multiply operator used to perform arithmetic on phase points.

Usage

## S3 method for class 'pp'
x * y

Arguments

x

an object of "pp" class representing a phase point.

y

an object of "pp" class representing another phase point.

Value

an object of "pp" class representing the calculated result.

Examples

pp(list(x = 1, y = 1)) * pp(list(x = 3, y = 4))

Add Operator for Phase Points

Description

The add operator used to perform arithmetic on phase points.

Usage

## S3 method for class 'pp'
x + y

Arguments

x

an object of "pp" class representing a phase point.

y

an object of "pp" class representing another phase point.

Value

an object of "pp" class representing the calculated result.

Examples

pp(list(x = 1, y = 1)) + pp(list(x = 3, y = 4))

Subtract Operator for Phase Points

Description

The subtract operator used to perform arithmetic on phase points.

Usage

## S3 method for class 'pp'
x - y

Arguments

x

an object of "pp" class representing a phase point.

y

an object of "pp" class representing another phase point.

Value

an object of "pp" class representing the calculated result.

Examples

pp(list(x = 1, y = 1)) - pp(list(x = 3, y = 4))

Divide Operator for Phase Points

Description

The devide operator used to perform arithmetic on phase points.

Usage

## S3 method for class 'pp'
x / y

Arguments

x

an object of "pp" class representing a phase point.

y

an object of "pp" class representing another phase point.

Value

an object of "pp" class representing the calculated result.

Examples

pp(list(x = 1, y = 1)) / pp(list(x = 3, y = 4))

Extract Parts of an Population Dynamics Data

Description

Operators acting on population dynamics data.

Usage

## S3 method for class 'pdata'
x[i]

Arguments

x

Object of class "pdata".

i

indice specifying elements to extract.

Value

pdata object

Examples

eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
pdat <- pdata(x,
  init = data.frame(x = c(10, 20), y = c(5, 15)),
  t = c(3, 3),
  lambda = data.frame(z = c(15, 30)),
  formula = "z = x + y"
)
pdat[1]

Create an ODE System

Description

Creates an object of class "eode" representing an ODE system that describes population or community dynamics.

Usage

eode(..., constraint = NULL)

Arguments

...

Objects of class "function", each representing a single differential equation.

constraint

Character strings that must have a format of "variable>value" or "variable<value", such as "x>1", "y>3", etc. If not specified, then all model variables are considered as positive real numbers by default.

Value

An object of class "eode" describing population or community dynamics.

Examples

## Example1: Lotka-Volterra competition model
eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
x

## Example2: Susceptible-infected model
dX_Cdt <- function(X_C, Y_C, X_A, Y_A, nu = 0.15, beta = 0.1, mu = 0.15, g = 0.04) {
  nu * (X_A + Y_A) - beta * X_C * (Y_C + Y_A) - (mu + g) * X_C
}

dY_Cdt <- function(X_C, Y_C, Y_A, beta = 0.1, mu = 0.15, g = 0.04, rho = 0.2) {
  beta * X_C * (Y_C + Y_A) - (mu + g + rho) * Y_C
}

dX_Adt <- function(X_C, Y_C, X_A, Y_A, beta = 0.1, g = 0.04) {
  g * X_C - beta * X_A * (Y_C + Y_A)
}

dY_Adt <- function(X_A, Y_C, Y_A, beta = 0.1, g = 0.04, rho = 0.2) {
  beta * X_A * (Y_C + Y_A) + g * Y_C - rho * Y_A
}

x <- eode(
  dX_Cdt = dX_Cdt, dY_Cdt = dY_Cdt, dX_Adt = dX_Adt, dY_Adt = dY_Adt,
  constraint = c("X_C>=0", "Y_C>=0", "X_A>=0", "Y_A>=0")
)
x

Find Equilibrium Point

Description

Finds an equilibrium point (or critical point) of an ODE system based on Newton iteration method.

Usage

eode_get_cripoi(
  x,
  init_value,
  eps = 0.001,
  max_step = 0.01,
  method = c("Newton", "GradDesc"),
  verbose = TRUE
)

Arguments

x

Object of class "eode" representing an ODE system.

init_value

An object of class "pp" representing a phase point giving start estimates.

eps

Precision for the stopping criterion. Iteration will stop after the movement of the phase point in a single step is smaller than eps.

max_step

Maximum number

method

one of "Newton", "GradDesc"

verbose

Logical, whether to print the iteration process.

Value

An object of class "pp" representing an equilibrium point found.

Examples

## Example 1: Lotka-Volterra competition model
eq1 <- function(x, y, r1 = 1, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
eode_get_cripoi(x, init_value = pp(list(x = 0.5, y = 0.5)))

## Example 2: Susceptible-infected model
dX_Cdt <- function(X_C, Y_C, X_A, Y_A, nu = 0.15, beta = 0.1, mu = 0.15, g = 0.04) {
  nu * (X_A + Y_A) - beta * X_C * (Y_C + Y_A) - (mu + g) * X_C
}

dY_Cdt <- function(X_C, Y_C, Y_A, beta = 0.1, mu = 0.15, g = 0.04, rho = 0.2) {
  beta * X_C * (Y_C + Y_A) - (mu + g + rho) * Y_C
}

dX_Adt <- function(X_C, Y_C, X_A, Y_A, beta = 0.1, g = 0.04) {
  g * X_C - beta * X_A * (Y_C + Y_A)
}

dY_Adt <- function(X_A, Y_C, Y_A, beta = 0.1, g = 0.04, rho = 0.2) {
  beta * X_A * (Y_C + Y_A) + g * Y_C - rho * Y_A
}

x <- eode(
  dX_Cdt = dX_Cdt, dY_Cdt = dY_Cdt, dX_Adt = dX_Adt, dY_Adt = dY_Adt,
  constraint = c("X_C>=0", "Y_C>=0", "X_A>=0", "Y_A>=0")
)
eode_get_cripoi(x, init_value = pp(list(X_C = 1, Y_C = 1, X_A = 1, Y_A = 1)))

System Matrix

Description

Linearlise an ODE system and calculate its system matrix.

Usage

eode_get_sysmat(x, value, delta = 0.00001)

Arguments

x

object of class "eode" representing an ODE system.

value

an object of class "pp" representing a phase point in the ODE system under consideration.

delta

Spacing or step size of a numerical grid used for calculating numerical differentiation.

Value

An object of class "matrix". Each matrix entry, (i,j), represents the partial derivative of the i-th equation of the system with respect to the j-th variable taking other variables as a constant.

Examples

## Example1: Lotka-Volterra competition model
eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
eode_get_sysmat(x, value = pp(list(x = 1, y = 1)), delta = 10e-6)

## Example2: Susceptible-infected model
dX_Cdt <- function(X_C, Y_C, X_A, Y_A, nu = 0.15, beta = 0.1, mu = 0.15, g = 0.04) {
  nu * (X_A + Y_A) - beta * X_C * (Y_C + Y_A) - (mu + g) * X_C
}

dY_Cdt <- function(X_C, Y_C, Y_A, beta = 0.1, mu = 0.15, g = 0.04, rho = 0.2) {
  beta * X_C * (Y_C + Y_A) - (mu + g + rho) * Y_C
}

dX_Adt <- function(X_C, Y_C, X_A, Y_A, beta = 0.1, g = 0.04) {
  g * X_C - beta * X_A * (Y_C + Y_A)
}

dY_Adt <- function(X_A, Y_C, Y_A, beta = 0.1, g = 0.04, rho = 0.2) {
  beta * X_A * (Y_C + Y_A) + g * Y_C - rho * Y_A
}

x <- eode(
  dX_Cdt = dX_Cdt, dY_Cdt = dY_Cdt, dX_Adt = dX_Adt, dY_Adt = dY_Adt,
  constraint = c("X_C>=0", "Y_C>=0", "X_A>=0", "Y_A>=0")
)
eode_get_sysmat(x, value = pp(list(X_A = 4, Y_A = 4, X_C = 4, Y_C = 4)), delta = 10e-6)

Velocity Vector

Description

Calculates the velocity vector at a given phase point.

Usage

eode_get_velocity(x, value, phase_curve = NULL)

Arguments

x

The ODE system under consideration. An object of class "eode".

value

an object of "pp" classs representing a phase point in the ODE system under consideration.

phase_curve

an object of "pc" class representing a phase curve. Only used when there is delayed items in the ODE system.

Value

an object of class "velocity" representing a velocity vector at a given phase point.

Examples


## Example1: Lotka-Volterra competition model
eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
eode_get_velocity(x, value = pp(list(x = 1, y = 1)))

## Example2: Susceptible-infected model
dX_Cdt <- function(X_C, Y_C, X_A, Y_A, nu = 0.15, beta = 0.1, mu = 0.15, g = 0.04) {
  nu * (X_A + Y_A) - beta * X_C * (Y_C + Y_A) - (mu + g) * X_C
}

dY_Cdt <- function(X_C, Y_C, Y_A, beta = 0.1, mu = 0.15, g = 0.04, rho = 0.2) {
  beta * X_C * (Y_C + Y_A) - (mu + g + rho) * Y_C
}

dX_Adt <- function(X_C, Y_C, X_A, Y_A, beta = 0.1, g = 0.04) {
  g * X_C - beta * X_A * (Y_C + Y_A)
}

dY_Adt <- function(X_A, Y_C, Y_A, beta = 0.1, g = 0.04, rho = 0.2) {
  beta * X_A * (Y_C + Y_A) + g * Y_C - rho * Y_A
}

x <- eode(
  dX_Cdt = dX_Cdt, dY_Cdt = dY_Cdt, dX_Adt = dX_Adt, dY_Adt = dY_Adt,
  constraint = c("X_C>=0", "Y_C>=0", "X_A>=0", "Y_A>=0")
)
eode_get_velocity(x, value = pp(list(X_A = 5, Y_A = 5, X_C = 3, Y_C = 2)))

Grid Search For Optimal Parameters

Description

Find optimal parameters in the ODE system using grid search method.

Usage

eode_gridSearch(x, pdat, space, step = 0.01)

Arguments

x

the ODE system under consideration. An object of "eode" class.

pdat

observed population dynamics. An object of "pdata" class.

space

space

step

interval of time for running simulations. Parameter of the function "eode_proj()".

Value

a data frame showing attempted parameters along with the corresponding values of loss function.

Examples


dX_Cdt <- function(X_C, Y_C, X_A, Y_A, nu = 0.15, beta = 0.1, mu = 0.15, g = 0.04) {
  nu * (X_A + Y_A) - beta * X_C * (Y_C + Y_A) - (mu + g) * X_C
}

dY_Cdt <- function(X_C, Y_C, Y_A, beta = 0.1, mu = 0.15, g = 0.04, rho = 0.2) {
  beta * X_C * (Y_C + Y_A) - (mu + g + rho) * Y_C
}

dX_Adt <- function(X_C, Y_C, X_A, Y_A, beta = 0.1, g = 0.04) {
  g * X_C - beta * X_A * (Y_C + Y_A)
}

dY_Adt <- function(X_A, Y_C, Y_A, beta = 0.1, g = 0.04, rho = 0.2) {
  beta * X_A * (Y_C + Y_A) + g * Y_C - rho * Y_A
}

x <- eode(
  dX_Cdt = dX_Cdt, dY_Cdt = dY_Cdt, dX_Adt = dX_Adt, dY_Adt = dY_Adt,
  constraint = c("X_C>=0", "Y_C>=0", "X_A>=0", "Y_A>=0")
)
training_data <- pdata(x,
  init = data.frame(
    X_A = c(9, 19, 29, 39),
    Y_A = c(1, 1, 1, 1),
    X_C = c(5, 5, 5, 5),
    Y_C = c(0, 0, 0, 0)
  ),
  t = c(3, 3, 3, 3),
  lambda = data.frame(incidence = c(0.4, 0.8, 0.9, 0.95)),
  formula = "incidence = (Y_A + Y_C)/(X_A + X_C + Y_A + Y_C)"
)
res <- eode_gridSearch(x, pdat = training_data, space = list(beta = seq(0.05, 0.15, 0.01)))
res

Centre

Description

Check whether an equilibrium point is a neutral centre or not.

Usage

eode_is_centre(x, value, eps = 0.001)

Arguments

x

Object of class "eode" representing an ODE system.

value

an object of class "pp" representing a phase point in the ODE system under consideration.

eps

Precision used to check whether the input phase point is an equilibrium point. If the absolute value of any component of the phase velocity vector at a phase point is lower than eps, an error would be thrown out.

Value

a TRUE or FALSE.

Examples

eq1 <- function(x, y, r1 = 1, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
eode_is_centre(x, value = pp(list(x = 0.3333, y = 0.3333)))

Saddle

Description

Check whether an equilibrium point is a saddle or not.

Usage

eode_is_saddle(x, value, eps = 0.001)

Arguments

x

Object of class "eode" representing an ODE system.

value

an object of class "pp" representing a phase point in the ODE system under consideration.

eps

Precision used to check whether the input phase point is an equilibrium point. If the absolute value of any component of the phase velocity vector at a phase point is lower than eps, an error would be thrown out.

Value

a TRUE or FALSE.

Examples

eq1 <- function(x, y, r1 = 1, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
eode_is_saddle(x, value = pp(list(x = 0.3333, y = 0.3333)))

Stable Focus

Description

Check whether an equilibrium point is a stable focus or not.

Usage

eode_is_stafoc(x, value, eps = 0.001)

Arguments

x

Object of class "eode" representing an ODE system.

value

an object of class "pp" representing a phase point in the ODE system under consideration.

eps

Precision used to check whether the input phase point is an equilibrium point. If the absolute value of any component of the phase velocity vector at a phase point is lower than eps, an error would be thrown out.

Value

a TRUE or FALSE.

Examples

eq1 <- function(x, y, r1 = 1, a11 = 2, a12 = 1) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 1, a22 = 2) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
eode_is_stafoc(x, value = pp(list(x = 0.3333, y = 0.3333)))

Stable Node

Description

Check whether an equilibrium point is a stable node or not.

Usage

eode_is_stanod(x, value, eps = 0.001)

Arguments

x

Object of class "eode" representing an ODE system.

value

an object of class "pp" representing a phase point in the ODE system under consideration.

eps

Precision used to check whether the input phase point is an equilibrium point. If the absolute value of any component of the phase velocity vector at a phase point is lower than eps, an error would be thrown out.

Value

a TRUE or FALSE.

Examples

eq1 <- function(x, y, r1 = 1, a11 = 2, a12 = 1) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 1, a22 = 2) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
eode_is_stanod(x, value = pp(list(x = 0.3333, y = 0.3333)))

Stable Equilibrium Point

Description

Check whether an equilibrium point is stable or not.

Usage

eode_is_stapoi(x, value, eps = 0.001)

Arguments

x

Object of class "eode" representing an ODE system.

value

an object of class "pp" representing a phase point in the ODE system under consideration.

eps

Precision used to check whether the input phase point is an equilibrium point. If the absolute value of any component of the phase velocity vector at a phase point is lower than eps, an error would be thrown out.

Value

a TRUE or FALSE.

Examples

eq1 <- function(x, y, r1 = 1, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
eode_is_stapoi(x, value = pp(list(x = 0.3333, y = 0.3333)))

## Example2: Susceptible-infected model
dX_Cdt <- function(X_C, Y_C, X_A, Y_A, nu = 0.15, beta = 0.1, mu = 0.15, g = 0.04) {
  nu * (X_A + Y_A) - beta * X_C * (Y_C + Y_A) - (mu + g) * X_C
}

dY_Cdt <- function(X_C, Y_C, Y_A, beta = 0.1, mu = 0.15, g = 0.04, rho = 0.2) {
  beta * X_C * (Y_C + Y_A) - (mu + g + rho) * Y_C
}

dX_Adt <- function(X_C, Y_C, X_A, Y_A, beta = 0.1, g = 0.04) {
  g * X_C - beta * X_A * (Y_C + Y_A)
}

dY_Adt <- function(X_A, Y_C, Y_A, beta = 0.1, g = 0.04, rho = 0.2) {
  beta * X_A * (Y_C + Y_A) + g * Y_C - rho * Y_A
}

x <- eode(
  dX_Cdt = dX_Cdt, dY_Cdt = dY_Cdt, dX_Adt = dX_Adt, dY_Adt = dY_Adt,
  constraint = c(
    "X_C>=0", "Y_C>=0", "X_A>=0", "Y_A>=0",
    "X_C<5", "Y_C<5", "X_A<5", "Y_A<5"
  )
)
eode_is_stapoi(x, value = pp(list(X_A = 1.3443, Y_A = 0.2304, X_C = 1.0655, Y_C = 0.0866)))

Unstable Focus

Description

Check whether an equilibrium point is an unstable focus or not.

Usage

eode_is_unsfoc(x, value, eps = 0.001)

Arguments

x

Object of class "eode" representing an ODE system.

value

an object of class "pp" representing a phase point in the ODE system under consideration.

eps

Precision used to check whether the input phase point is an equilibrium point. If the absolute value of any component of the phase velocity vector at a phase point is lower than eps, an error would be thrown out.

Value

a TRUE or FALSE.

Examples

eq1 <- function(x, y, r1 = 1, a11 = 2, a12 = 1) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 1, a22 = 2) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
eode_is_unsfoc(x, value = pp(list(x = 0.3333, y = 0.3333)))

Unstable Node

Description

Check whether an equilibrium point is an unstable node or not.

Usage

eode_is_unsnod(x, value, eps = 0.001)

Arguments

x

Object of class "eode" representing an ODE system.

value

an object of class "pp" representing a phase point in the ODE system under consideration.

eps

Precision used to check whether the input phase point is an equilibrium point. If the absolute value of any component of the phase velocity vector at a phase point is lower than eps, an error would be thrown out.

Value

a TRUE or FALSE.

Examples

eq1 <- function(x, y, r1 = 1, a11 = 2, a12 = 1) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 1, a22 = 2) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
eode_is_unsnod(x, value = pp(list(x = 0.3333, y = 0.3333)))

Test the Validity of a Phase Point

Description

Test whether a phase point sits out of the boundary of an ODE system.

Usage

eode_is_validval(x, value)

Arguments

x

object of class "eode" representing an ODE system.

value

an object of class "pp" representing a phase point in the ODE system under consideration.

Value

returns TRUE or FALSE depending on whether the values of the system variables are within the boundary or not.

Examples

eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
eode_is_validval(x, value = pp(list(x = 1, y = 1)))

Calculate Loss Function

Description

Calculate the quadratic loss function given an ODE system and observed population dynamics.

Usage

eode_lossf(x, pdat, step = 0.01)

Arguments

x

the ODE system under consideration. An object of "eode" class.

pdat

observed population dynamics. An object of "pdata" class.

step

interval of time for each step. Parameter of the function "eode_proj()".

Value

a value of loss function

Examples

dX_Cdt <- function(X_C, Y_C, X_A, Y_A, nu = 0.15, beta = 0.1, mu = 0.15, g = 0.04) {
  nu * (X_A + Y_A) - beta * X_C * (Y_C + Y_A) - (mu + g) * X_C
}

dY_Cdt <- function(X_C, Y_C, Y_A, beta = 0.1, mu = 0.15, g = 0.04, rho = 0.2) {
  beta * X_C * (Y_C + Y_A) - (mu + g + rho) * Y_C
}

dX_Adt <- function(X_C, Y_C, X_A, Y_A, beta = 0.1, g = 0.04) {
  g * X_C - beta * X_A * (Y_C + Y_A)
}

dY_Adt <- function(X_A, Y_C, Y_A, beta = 0.1, g = 0.04, rho = 0.2) {
  beta * X_A * (Y_C + Y_A) + g * Y_C - rho * Y_A
}

x <- eode(
  dX_Cdt = dX_Cdt, dY_Cdt = dY_Cdt, dX_Adt = dX_Adt, dY_Adt = dY_Adt,
  constraint = c("X_C>=0", "Y_C>=0", "X_A>=0", "Y_A>=0")
)
training_data <- pdata(x,
  init = data.frame(
    X_A = c(9, 19, 29, 39),
    Y_A = c(1, 1, 1, 1),
    X_C = c(5, 5, 5, 5),
    Y_C = c(0, 0, 0, 0)
  ),
  t = c(3, 3, 3, 3),
  lambda = data.frame(incidence = c(0.4, 0.8, 0.9, 0.95)),
  formula = "incidence = (Y_A + Y_C)/(X_A + X_C + Y_A + Y_C)"
)
eode_lossf(x, pdat = training_data)

Solve ODEs

Description

Provides the numerical solution of an ODE under consideration given an initial condition.

Usage

eode_proj(x, value0, N, step = 0.01)

Arguments

x

the ODE system under consideration. An object of class "eode".

value0

value an object of class "pp" representing the phase point at the initial state.

N

number of iterations

step

interval of time for each step

Value

an object of class "pc" that represents a phase curve

Examples

## Example1: Lotka-Volterra competition model
eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2, constraint = c("x<1", "y<1"))
eode_proj(x, value0 = pp(list(x = 0.2, y = 0.1)), N = 100)

## Example2: Susceptible-infected model
dX_Cdt <- function(X_C, Y_C, X_A, Y_A, nu = 0.15, beta = 0.1, mu = 0.15, g = 0.04) {
  nu * (X_A + Y_A) - beta * X_C * (Y_C + Y_A) - (mu + g) * X_C
}

dY_Cdt <- function(X_C, Y_C, Y_A, beta = 0.1, mu = 0.15, g = 0.04, rho = 0.2) {
  beta * X_C * (Y_C + Y_A) - (mu + g + rho) * Y_C
}

dX_Adt <- function(X_C, Y_C, X_A, Y_A, beta = 0.1, g = 0.04) {
  g * X_C - beta * X_A * (Y_C + Y_A)
}

dY_Adt <- function(X_A, Y_C, Y_A, beta = 0.1, g = 0.04, rho = 0.2) {
  beta * X_A * (Y_C + Y_A) + g * Y_C - rho * Y_A
}

x <- eode(
  dX_Cdt = dX_Cdt, dY_Cdt = dY_Cdt, dX_Adt = dX_Adt, dY_Adt = dY_Adt,
  constraint = c("X_C>=0", "Y_C>=0", "X_A>=0", "Y_A>=0")
)
eode_proj(x, value0 = pp(list(X_A = 5, Y_A = 5, X_C = 3, Y_C = 2)), N = 100)

Sensitivity Analysis

Description

Run a sensitivity analysis on an ODE system.

Usage

eode_sensitivity_proj(x, valueSpace, N, step = 0.01)

Arguments

x

object of class "pc" that represents the ODE system under consideration.

valueSpace

a list indicating initial conditions and parameters. Model variables must be included to specify initial values of each variable. Values can be a vector, indicating all the potential values to be considered in the sensitivity analysis.

N

number of iterations

step

interval of time for running simulations. Parameter of the function "eode_proj()".

Value

an object of "pcfamily" class, each component having three sub-components: $grid_var: variables or parameters whose values are changed throughout the sensitivity analysis. $fixed_var: variables whose values are not changed. $pc: phase curve. An object of "pc" class.

Examples

eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2, constraint = c("x<100", "y<100"))
eode_sensitivity_proj(x, valueSpace = list(x = c(0.2, 0.3, 0.4), y = 0.1, a11 = 1:3), N = 100)

Set New Constraints

Description

Set new constraints for an ODE system.

Usage

eode_set_constraint(x, new_constraint)

Arguments

x

an object of class "eode".

new_constraint

a vector of characters indicating new constraints to be assigned to the ODE system.

Value

an object of "eode" class

Examples

eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
x
eode_set_constraint(x, c("x<5", "y<5"))

Set New Parameters

Description

Set new parameters for an ODE system.

Usage

eode_set_parameter(x, ParaList)

Arguments

x

an object of class "eode".

ParaList

a list of names of parameters with their corresponding values to be assigned to the ODE system.

Value

an object of "eode" class

Examples

eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
x
eode_set_parameter(x, ParaList = list(r1 = 3))

Simulated Annealing For Optimal Parameters

Description

Find optimal parameters in the ODE system using simulated annealing.

Usage

eode_simuAnnealing(
  x,
  pdat,
  paras = "ALL",
  max_disturb = 0.2,
  AnnN = 100,
  step = 0.01,
  prop.train = 1
)

Arguments

x

the ODE system under consideration. An object of "eode" class.

pdat

observed population dynamics. An object of "pdata" class.

paras

parameters to be optimised. A character vector. If multiple parameters are specified, the simulation annealing process will proceed by altering multiple parameters at the same time, and accept an alteration if it achieves a lower value of the loss function. Default is "ALL", which means to choose all the parameters.

max_disturb

maximum disturbance in proportion. The biggest disturbance acts on parameters at the beginning of the simulated annealing process.

AnnN

steps of simulated annealing.

step

interval of time for running simulations. Parameter of the function "eode_proj()".

prop.train

proportion of training data set. In each step of annealing, a proportion of dataset will be randomly decided for model training, and the rest for model validation.

Value

a data frame showing attempted parameters along with the corresponding values of loss function.

Examples


dX_Cdt <- function(X_C, Y_C, X_A, Y_A, nu = 0.15, beta = 0.1, mu = 0.15, g = 0.04) {
  nu * (X_A + Y_A) - beta * X_C * (Y_C + Y_A) - (mu + g) * X_C
}

dY_Cdt <- function(X_C, Y_C, Y_A, beta = 0.1, mu = 0.15, g = 0.04, rho = 0.2) {
  beta * X_C * (Y_C + Y_A) - (mu + g + rho) * Y_C
}

dX_Adt <- function(X_C, Y_C, X_A, Y_A, beta = 0.1, g = 0.04) {
  g * X_C - beta * X_A * (Y_C + Y_A)
}

dY_Adt <- function(X_A, Y_C, Y_A, beta = 0.1, g = 0.04, rho = 0.2) {
  beta * X_A * (Y_C + Y_A) + g * Y_C - rho * Y_A
}

x <- eode(
  dX_Cdt = dX_Cdt, dY_Cdt = dY_Cdt, dX_Adt = dX_Adt, dY_Adt = dY_Adt,
  constraint = c("X_C>=0", "Y_C>=0", "X_A>=0", "Y_A>=0")
)
training_data <- pdata(x,
  init = data.frame(
    X_A = c(9, 19, 29, 39),
    Y_A = c(1, 1, 1, 1),
    X_C = c(5, 5, 5, 5),
    Y_C = c(0, 0, 0, 0)
  ),
  t = c(3, 3, 3, 3),
  lambda = data.frame(incidence = c(0.4, 0.8, 0.9, 0.95)),
  formula = "incidence = (Y_A + Y_C)/(X_A + X_C + Y_A + Y_C)"
)
res <- eode_simuAnnealing(x, pdat = training_data, paras = "beta", max_disturb = 0.05, AnnN = 20)
res

Stability Analysis

Description

Check whether an equilibrium point is stable or not, and give the specific type (i.e. stable node, stable focus, unstable node, unstable focus, saddle, or centre). Check whether an equilibrium point is stable or not.

Usage

eode_stability_type(x, value, eps = 0.001)

Arguments

x

Object of class "eode" representing an ODE system.

value

an object of class "pp" representing a phase point in the ODE system under consideration.

eps

Precision used to check whether the input phase point is an equilibrium point. If the absolute value of any component of the phase velocity vector at a phase point is lower than eps, an error would be thrown out.

Value

a string character indicate the type of the equilbrium point.

Examples

eq1 <- function(x, y, r1 = 1, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
eode_stability_type(x, value = pp(list(x = 0.3333, y = 0.3333)))

Length of an ODE System

Description

Get the number of equations in an ODE system.

Usage

## S3 method for class 'eode'
length(x, ...)

Arguments

x

Object of "eode" class representing an ODE system under consideration.

...

Ignored.

Value

The number of equations in the ODE system.

Examples

## Example 1: Lotka-Volterra competition model
eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
length(x)

## Example 2: Susceptible-infected model
dX_Cdt <- function(X_C, Y_C, X_A, Y_A, nu = 0.15, beta = 0.1, mu = 0.15, g = 0.04) {
  nu * (X_A + Y_A) - beta * X_C * (Y_C + Y_A) - (mu + g) * X_C
}

dY_Cdt <- function(X_C, Y_C, Y_A, beta = 0.1, mu = 0.15, g = 0.04, rho = 0.2) {
  beta * X_C * (Y_C + Y_A) - (mu + g + rho) * Y_C
}

dX_Adt <- function(X_C, Y_C, X_A, Y_A, beta = 0.1, g = 0.04) {
  g * X_C - beta * X_A * (Y_C + Y_A)
}

dY_Adt <- function(X_A, Y_C, Y_A, beta = 0.1, g = 0.04, rho = 0.2) {
  beta * X_A * (Y_C + Y_A) + g * Y_C - rho * Y_A
}

x <- eode(
  dX_Cdt = dX_Cdt, dY_Cdt = dY_Cdt, dX_Adt = dX_Adt, dY_Adt = dY_Adt,
  constraint = c("X_C>=0", "Y_C>=0", "X_A>=0", "Y_A>=0")
)
length(x)

Get Length Of Population Dynamics Data

Description

Get the number of observations of population dynamics data.

Usage

## S3 method for class 'pdata'
length(x, ...)

Arguments

x

Object of class "pdata".

...

Ignored.

Value

A scalar.

Examples

eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
pdata(x,
  init = data.frame(x = c(10, 20), y = c(5, 15)),
  t = c(3, 3),
  lambda = data.frame(z = c(15, 30)),
  formula = "z = x + y"
)

Variable Calculator In ODE Systems

Description

Calculate one or several variables during the simulation of an ODE system

Usage

pc_calculator(x, formula)

Arguments

x

the ODE system under consideration. An object of "eode" class.

formula

formula that specifies how to calculate the variable to be traced.

Value

an object of "pc" class with extra variables being attached.

Examples

eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2, constraint = c("x<100", "y<100"))
pc <- eode_proj(x, value0 = pp(list(x = 0.2, y = 0.1)), N = 100)
pc_calculator(pc, formula = "z = x + y")

Plot a Phase Curve in Phase Plane

Description

Creates a plot of a phase curve in its phase plane

Usage

pc_plot(x)

Arguments

x

object of class "pc" that represents a phase curve.

Value

a graphic object

Examples

eq1 <- function(x, y, r1 = 1, a11 = 2, a12 = 1) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 1, a22 = 2) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2, constraint = c("x<100", "y<100"))
pc <- eode_proj(x, value0 = pp(list(x = 0.2, y = 0.1)), N = 50, step = 0.1)
pc_plot(pc)

Create Population Dynamics Data

Description

Create a new data set to describe population dynamics with initial conditions. The data is mainly used to train the ODE model described by an object of "eode" class.

Usage

pdata(x, init, t, lambda, formula)

Arguments

x

an object of class "eode" describing the ODE system in focus.

init

a data frame describing the initial conditions of the population. Each column should correspond to a variable in the ODE system, hence the name of column will be automatically checked to make sure it matches a variable in the ODE system. Each row represents an independent observation

t

a numerical vector that describes the time for each observation

lambda

a data frame describing the observation values of population dynamics. Each column is an variable and each row represents an independent observation.

formula

a character that specifies how the observed variable can be predicted by the ODE system.

Value

an object of "pdata" class

Examples

eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
pdata(x,
  init = data.frame(x = c(10, 20), y = c(5, 15)),
  t = c(3, 3),
  lambda = data.frame(z = c(15, 30)),
  formula = "z = x + y"
)

Distance between Phase Points

Description

Computes and returns the distance between two phase points.

Usage

pdist(x, y)

Arguments

x

an object of "pp" class representing a phase point.

y

an object of "pp" class representing another phase point.

Value

A scalar.

Examples

a <- pp(list(x = 1, y = 1))
b <- pp(list(x = 3, y = 4))
pdist(a, b)

Plot Phase Velocity Vector Field

Description

Creates a plot of phase velocity vector (or its field) of an ODE system. With two free variables the function creates a plot of the phase velocity vector field within the range defined by model constraints. With a single free variable the function creates a plot of one-dimensional phase velocity vector field. With more than two variables the function will automatically set a value (the middle of the lower and upper boundaries) for any extra variable to reduce axes. The values can also be set using parameter "set_covar". If all model variables have had their values, the function creates a plot to show the exact values of a single phase velocity vector.

Usage

## S3 method for class 'eode'
plot(
  x,
  n.grid = 20,
  scaleL = 1,
  scaleAH = 1,
  scaleS = 1,
  set_covar = NULL,
  ...
)

Arguments

x

The ODE system under consideration. An object of class "eode".

n.grid

number of grids per dimension. A larger number indicates a smaller grid size. Default 20.

scaleL

scale factor of the arrow length.

scaleAH

scale factor of the arrow head.

scaleS

scale factor of the arrow size.

set_covar

give certain values of model variables that are going to be fixed.

...

ignored arguments

Value

a graphic object

Examples

## Example 1: Lotka-Volterra competition model
eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
plot(x)

## Example 2: Susceptible-infected model
dX_Cdt <- function(X_C, Y_C, X_A, Y_A, nu = 0.15, beta = 0.1, mu = 0.15, g = 0.04) {
  nu * (X_A + Y_A) - beta * X_C * (Y_C + Y_A) - (mu + g) * X_C
}

dY_Cdt <- function(X_C, Y_C, Y_A, beta = 0.1, mu = 0.15, g = 0.04, rho = 0.2) {
  beta * X_C * (Y_C + Y_A) - (mu + g + rho) * Y_C
}

dX_Adt <- function(X_C, Y_C, X_A, Y_A, beta = 0.1, g = 0.04) {
  g * X_C - beta * X_A * (Y_C + Y_A)
}

dY_Adt <- function(X_A, Y_C, Y_A, beta = 0.1, g = 0.04, rho = 0.2) {
  beta * X_A * (Y_C + Y_A) + g * Y_C - rho * Y_A
}

x <- eode(
  dX_Cdt = dX_Cdt, dY_Cdt = dY_Cdt, dX_Adt = dX_Adt, dY_Adt = dY_Adt,
  constraint = c("X_C>=0", "Y_C>=0", "X_A>=0", "Y_A>=0")
)
plot(x)
plot(x, set_covar = list(X_A = 60, Y_A = 80))
plot(x, set_covar = list(Y_C = 80, X_A = 60, Y_A = 80))
plot(x, set_covar = list(X_C = 60, Y_C = 80, X_A = 60, Y_A = 80))

Plot a Phase Curve With Time

Description

Creates a plot of a phase curve

Usage

## S3 method for class 'pc'
plot(x, model_var_label = NULL, model_var_color = NULL, ...)

Arguments

x

object of class "pc" that represents a phase curve.

model_var_label

a list indicating labels of model variables. Name should be old variable names and values should be names of labels.

model_var_color

a list indicating colors of model variable. Name should be old variable names and values should be 16-bit color codes.

...

additional parameters accepted by the function "geom_line".

Value

a graphic object

Examples

eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2, constraint = c("x<1", "y<1"))
pc <- eode_proj(x, value0 = pp(list(x = 0.2, y = 0.1)), N = 100)
plot(pc)

Plot Phase Curve Family

Description

Creates a plot of a phase curve family.

Usage

## S3 method for class 'pcfamily'
plot(
  x,
  model_var_label = NULL,
  model_var_color = NULL,
  facet_grid_label = NULL,
  facet_paras = TRUE,
  ...
)

Arguments

x

object of class "pcfamily" that represents a phase curve family.

model_var_label

a list indicating labels of model variables. Name should be old variable names and values should be names of labels.

model_var_color

a list indicating colors of model variable. Name should be old variable names and values should be 16-bit color codes.

facet_grid_label

a list indicating facet labels. Name should be old variable names and values should be facet labels.

facet_paras

a logical vector indicating whether facet?

...

other parameters

Value

a graphic object

Examples

eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2, constraint = c("x<1", "y<1"))
value_space <- list(x = c(0.2, 0.3, 0.4), y = 0.1, a11 = 1:3)
plot(eode_sensitivity_proj(x, valueSpace = value_space, N = 100))

Create a Plot of a Phase Velocity Vector

Description

Plot a phase velocity vector

Usage

## S3 method for class 'velocity'
plot(x, ...)

Arguments

x

Object of class "velocity".

...

Ignored.

Value

ggplot object

Examples

eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
plot(eode_get_velocity(x, value = pp(list(x = 1, y = 1))))

Create a Phase Point

Description

Creates an object of class "pp" representing a phase point in the phase space.

Usage

pp(value)

Arguments

value

A list of several elements, each giving a value on one dimension.

Value

An object of class "pp" describing a phase point.

Examples

pp(list(x = 1, y = 1))

Print Brief Details of an ODE System

Description

Prints a very brief description of an ODE system.

Usage

## S3 method for class 'eode'
print(x, ...)

Arguments

x

Object of class "eode".

...

Ignored.

Value

No value

Examples

## Example 1: Lotka-Volterra competition model
eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
x

## Example 2: Susceptible-infected model
dX_Cdt <- function(X_C, Y_C, X_A, Y_A, nu = 0.15, beta = 0.1, mu = 0.15, g = 0.04) {
  nu * (X_A + Y_A) - beta * X_C * (Y_C + Y_A) - (mu + g) * X_C
}

dY_Cdt <- function(X_C, Y_C, Y_A, beta = 0.1, mu = 0.15, g = 0.04, rho = 0.2) {
  beta * X_C * (Y_C + Y_A) - (mu + g + rho) * Y_C
}

dX_Adt <- function(X_C, Y_C, X_A, Y_A, beta = 0.1, g = 0.04) {
  g * X_C - beta * X_A * (Y_C + Y_A)
}

dY_Adt <- function(X_A, Y_C, Y_A, beta = 0.1, g = 0.04, rho = 0.2) {
  beta * X_A * (Y_C + Y_A) + g * Y_C - rho * Y_A
}

x <- eode(
  dX_Cdt = dX_Cdt, dY_Cdt = dY_Cdt, dX_Adt = dX_Adt, dY_Adt = dY_Adt,
  constraint = c("X_C>=0", "Y_C>=0", "X_A>=0", "Y_A>=0")
)
x

Print Brief Details of a Phase Curve

Description

Prints a very brief description of a phase curve.

Usage

## S3 method for class 'pc'
print(x, ...)

Arguments

x

Object of class "pc".

...

Ignored.

Value

No value

Examples

eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2, constraint = c("x<1", "y<1"))
eode_proj(x, value0 = pp(list(x = 0.9, y = 0.9)), N = 8)

Print Brief Details of a Phase Curve Family

Description

Prints a very brief description of a phase curve family.

Usage

## S3 method for class 'pcfamily'
print(x, ...)

Arguments

x

Object of class "pcfamily".

...

Ignored.

Value

No value

Examples

eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2, constraint = c("x<100", "y<100"))
eode_sensitivity_proj(x, valueSpace = list(x = c(0.2, 0.3, 0.4), y = 0.1, a11 = 1:3), N = 100)

Print Brief Details of Population Dynamics Data

Description

Prints a very brief description of population dynamics data.

Usage

## S3 method for class 'pdata'
print(x, ...)

Arguments

x

Object of class "pdata".

...

Ignored.

Value

No value

Examples

eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
pdata(x,
  init = data.frame(x = c(10, 20), y = c(5, 15)),
  t = c(3, 3),
  lambda = data.frame(z = c(15, 30)),
  formula = "z = x + y"
)

Print Brief Details of a Phase Point

Description

Prints a very brief description of a phase point.

Usage

## S3 method for class 'pp'
print(x, ...)

Arguments

x

Object of class "pp".

...

Ignored.

Value

No value

Examples

pp(list(x = 1, y = 1))

Print Brief Details of a Phase Velocity Vector

Description

Prints a very brief description of a phase velocity vector.

Usage

## S3 method for class 'velocity'
print(x, ...)

Arguments

x

Object of class "velocity".

...

Ignored.

Value

No value

Examples

eq1 <- function(x, y, r1 = 4, a11 = 1, a12 = 2) (r1 - a11 * x - a12 * y) * x
eq2 <- function(x, y, r2 = 1, a21 = 2, a22 = 1) (r2 - a21 * x - a22 * y) * y
x <- eode(dxdt = eq1, dydt = eq2)
eode_get_velocity(x, value = pp(list(x = 1, y = 1)))