The Logistic Growth Model

Description

The derivative function of the logistic growth model, an example of a two dimensional autonomous ODE system.

Usage

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logistic(t, y, parameters)

Arguments

t

Value of t, the independent variable, to evaluate the derivative at. Should be a single number.

y

Value of y, the dependent variable, to evaluate the derivative at. Should be a single number.

parameters

Values of the parameters of the system. Should be a vector with parameters specified in the following order: beta, K.

Details

Evaluates the derivative of the following ODE at the point (t, y):

dy/dt = beta*y*(1 - y/K).

Format is designed to be compatible with ode from the deSolve package.

Value

Returns a list dy containing the value of the derivative at (t, y).

Author(s)

Michael J. Grayling

See Also

ode

Examples

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# Plot the velocity field, nullclines and several trajectories.
logistic.flowField  <- flowField(logistic, x.lim = c(0, 5), y.lim = c(-1, 3),
                                 parameters = c(1, 2), points = 21, system = "one.dim",
								 add = FALSE)
logistic.nullclines <- nullclines(logistic, x.lim = c(0, 5), y.lim = c(-1, 3),
                                  parameters = c(1, 2), system = "one.dim")
logistic.trajectory <- trajectory(logistic, y0 = c(-0.5, 0.5, 1.5, 2.5), t.end = 5,
                                  parameters = c(1, 2), system = "one.dim")

# Plot the phase portrait.
logistic.phasePortrait <- phasePortrait(logistic, y.lim = c(-0.5, 2.5),
                                        parameters = c(1, 2), points = 10, frac = 0.5)

# Determine the stability of the equilibrium points.
logistic.stability.1 <- stability(logistic, y.star = 0, parameters = c(1, 2),
                                  system = "one.dim")
logistic.stability.2 <- stability(logistic, y.star = 2, parameters = c(1, 2),
                                  system = "one.dim")