Logistic Models

These models and simulations have been tagged “Logistic”.

This (simplest!) model demonstrates logistic growth.The original differential equation looks like  y'(t) = b y(t) (1 - y(t)/K)   where K is the carrying capacity of the quantity y.       But if we divide each side of the equation by K, we obtain      d(y/K)/dt = b (y/K) (1-y/K)     Defining a new va
This (simplest!) model demonstrates logistic growth.The original differential equation looks like

y'(t) = b y(t) (1 - y(t)/K)

where K is the carrying capacity of the quantity y.

But if we divide each side of the equation by K, we obtain

d(y/K)/dt = b (y/K) (1-y/K)

Defining a new variable w, the population relative to its carrying capacity, we obtain

dw/dt = b w (1 - w)

Finally we divide both sides by b, to write

dw/d(bt) = w (1 - w)

So if we work in dimensionless time units of bt, we have

w' = w (1 - w)

where the derivative is with respect to the variable bt=τ. .
τ=τ
This
       This equation, as simple as possible, contains all the dynamics (all the ways the population can behave), while masking the "trivialities"; but it kind of hides the physical aspects of the problem. So it's easy to study, but harder to interpret: alas, you can't have it all!:) 

τ=1 when t=1b: so if b=.5/year, then τ=1 when t=2.

So the larger b (the greater the birthrate), the shorter the real time t to give τ=1.
τ=τ=

τ=

This model shows the growth of one type of organism as a function of the carrying capacity (i.e., logistic growth).
This model shows the growth of one type of organism as a function of the carrying capacity (i.e., logistic growth).
logistic model    uses vector to plot multiple solutions at once
logistic model

uses vector to plot multiple solutions at once
11 months ago
This model shows the growth of two organisms competing for a limiting resource (space) .
This model shows the growth of two organisms competing for a limiting resource (space) .
This (simplest!) model demonstrates logistic growth.The original differential equation looks like  y'(t) = b y(t) (1 - y(t)/K)   where K is the carrying capacity of the quantity y.       But if we divide each side of the equation by K, we obtain      d(y/K)/dt = b (y/K) (1-y/K)     Defining a new va
This (simplest!) model demonstrates logistic growth.The original differential equation looks like

y'(t) = b y(t) (1 - y(t)/K)

where K is the carrying capacity of the quantity y.

But if we divide each side of the equation by K, we obtain

d(y/K)/dt = b (y/K) (1-y/K)

Defining a new variable w, the population relative to its carrying capacity, we obtain

dw/dt = b w (1 - w)

Finally we divide both sides by b, to write

dw/d(bt) = w (1 - w)

So if we work in dimensionless time units of bt, we have

w' = w (1 - w)

where the derivative is with respect to the variable bt=τ. .
τ=τ
This
       This equation, as simple as possible, contains all the dynamics (all the ways the population can behave), while masking the "trivialities"; but it kind of hides the physical aspects of the problem. So it's easy to study, but harder to interpret: alas, you can't have it all!:) 

τ=1 when t=1b: so if b=.5/year, then τ=1 when t=2.

So the larger b (the greater the birthrate), the shorter the real time t to give τ=1.
τ=τ=

τ=

This model shows the growth of one type of organism as a function of the carrying capacity (i.e., logistic growth).
This model shows the growth of one type of organism as a function of the carrying capacity (i.e., logistic growth).
This (simplest!) model demonstrates logistic growth.The original differential equation looks like  y'(t) = b y(t) (1 - y(t)/K)   where K is the carrying capacity of the quantity y.       But if we divide each side of the equation by K, we obtain      d(y/K)/dt = b (y/K) (1-y/K)     Defining a new va
This (simplest!) model demonstrates logistic growth.The original differential equation looks like

y'(t) = b y(t) (1 - y(t)/K)

where K is the carrying capacity of the quantity y.

But if we divide each side of the equation by K, we obtain

d(y/K)/dt = b (y/K) (1-y/K)

Defining a new variable w, the population relative to its carrying capacity, we obtain

dw/dt = b w (1 - w)

Finally we divide both sides by b, to write

dw/d(bt) = w (1 - w)

So if we work in dimensionless time units of bt, we have

w' = w (1 - w)

where the derivative is with respect to the variable bt=τ. .
τ=τ
This
       This equation, as simple as possible, contains all the dynamics (all the ways the population can behave), while masking the "trivialities"; but it kind of hides the physical aspects of the problem. So it's easy to study, but harder to interpret: alas, you can't have it all!:) 

τ=1 when t=1b: so if b=.5/year, then τ=1 when t=2.

So the larger b (the greater the birthrate), the shorter the real time t to give τ=1.
τ=τ=

τ=

Richards model    uses vector to plot multiple solutions at once
Richards model

uses vector to plot multiple solutions at once
10 months ago
This (simplest!) model demonstrates logistic growth.The original differential equation looks like  y'(t) = b y(t) (1 - y(t)/K)   where K is the carrying capacity of the quantity y.       But if we divide each side of the equation by K, we obtain      d(y/K)/dt = b (y/K) (1-y/K)     Defining a new va
This (simplest!) model demonstrates logistic growth.The original differential equation looks like

y'(t) = b y(t) (1 - y(t)/K)

where K is the carrying capacity of the quantity y.

But if we divide each side of the equation by K, we obtain

d(y/K)/dt = b (y/K) (1-y/K)

Defining a new variable w, the population relative to its carrying capacity, we obtain

dw/dt = b w (1 - w)

Finally we divide both sides by b, to write

dw/d(bt) = w (1 - w)

So if we work in dimensionless time units of bt, we have

w' = w (1 - w)

where the derivative is with respect to the variable bt=τ. .
τ=τ
This
       This equation, as simple as possible, contains all the dynamics (all the ways the population can behave), while masking the "trivialities"; but it kind of hides the physical aspects of the problem. So it's easy to study, but harder to interpret: alas, you can't have it all!:) 

τ=1 when t=1b: so if b=.5/year, then τ=1 when t=2.

So the larger b (the greater the birthrate), the shorter the real time t to give τ=1.
τ=τ=

τ=

2 months ago