Pattern formation

Pattern formation: amplitude equations and rolls.

BrusselatorElement(label, visual=None, *, u=1.0, v=3.0, a=1.0, b=3.0, Du=1.0, Dv=8.0)

Bases: BrusselatorReactionElement

Site of the Brusselator, the reactor of the Brusselator with the diffusion coefficients of its intermediates.

The four reactions of brusselator run at every site, and its diffusions read Du and Dv.

Parameters:

Name	Type	Description	Default
label	str	The label of the site or the field.	required
visual	VisualSpec | None	How the element is drawn, or None for default_visual_.	None
u	float	The initial concentration of u.	1.0
v	float	The initial concentration of v.	3.0
a	float	The feed of u.	1.0
b	float	The conversion of u into v.	3.0
Du	Annotated[object, ParameterSpec()]	The diffusion of u.	1.0
Dv	Annotated[object, ParameterSpec()]	The diffusion of v.	8.0

Examples:

>>> from skcomplex.systems.chemistry.pattern_formation import BrusselatorElement
>>> BrusselatorElement('cell').Dv
8.0

Source code in src/skcomplex/systems/chemistry/pattern_formation/_reaction_diffusion.py

def __init__(
    self,
    label: str,
    visual: VisualSpec | None = None,
    *,
    u: float = 1.0,
    v: float = 3.0,
    a: float = 1.0,
    b: float = 3.0,
    Du: Annotated[object, ParameterSpec()] = 1.0,
    Dv: Annotated[object, ParameterSpec()] = 8.0,
) -> None:
    super().__init__(label, visual=visual, u=u, v=v, a=a, b=b)
    self.Du = Du
    self.Dv = Dv

GrayScottElement(label, visual=None, *, u=1.0, v=0.0, Du=0.16, Dv=0.08, F=0.035, k=0.065)

Bases: Element

Site of the Gray-Scott system, the substrate u and the activator v.

The reactions of gray_scott at every site, the cubic autocatalysis of the activator on the substrate, the feed F of the substrate, and the removal F + k of the activator, read F and k, and its diffusions read Du and Dv.

Parameters:

Name	Type	Description	Default
label	str	The label of the site or the field.	required
visual	VisualSpec | None	How the element is drawn, or None for default_visual_.	None
u	Annotated[object, StateSpec()]	The initial substrate.	1.0
v	Annotated[object, StateSpec()]	The initial activator.	0.0
Du	Annotated[object, ParameterSpec()]	The diffusion of the substrate.	0.16
Dv	Annotated[object, ParameterSpec()]	The diffusion of the activator.	0.08
F	Annotated[object, ParameterSpec()]	The feed rate.	0.035
k	Annotated[object, ParameterSpec()]	The kill rate.	0.065

Examples:

>>> from skcomplex.systems.chemistry.pattern_formation import GrayScottElement
>>> GrayScottElement('cell').k
0.065

Source code in src/skcomplex/systems/chemistry/pattern_formation/_reaction_diffusion.py

def __init__(
    self,
    label: str,
    visual: VisualSpec | None = None,
    *,
    u: Annotated[object, StateSpec()] = 1.0,
    v: Annotated[object, StateSpec()] = 0.0,
    Du: Annotated[object, ParameterSpec()] = 0.16,
    Dv: Annotated[object, ParameterSpec()] = 0.08,
    F: Annotated[object, ParameterSpec()] = 0.035,
    k: Annotated[object, ParameterSpec()] = 0.065,
) -> None:
    super().__init__(label, visual=visual)
    self.u = u
    self.v = v
    self.Du = Du
    self.Dv = Dv
    self.F = F
    self.k = k

SchnakenbergElement(label, visual=None, *, u=1.0, v=0.9, a=0.1, b=0.9, Du=1.0, Dv=40.0)

Bases: Element

Site of the Schnakenberg system, the activator u and the inhibitor v.

The reactions of schnakenberg at every site, the productions a of u and b of v, the decay of u, and the autocatalysis 2U + V -> 3U, read a and b, and its diffusions read Du and Dv.

Parameters:

Name	Type	Description	Default
label	str	The label of the site or the field.	required
visual	VisualSpec | None	How the element is drawn, or None for default_visual_.	None
u	Annotated[object, StateSpec()]	The initial activator.	1.0
v	Annotated[object, StateSpec()]	The initial inhibitor.	0.9
a	Annotated[object, ParameterSpec()]	The production of u.	0.1
b	Annotated[object, ParameterSpec()]	The production of v.	0.9
Du	Annotated[object, ParameterSpec()]	The diffusion of the activator.	1.0
Dv	Annotated[object, ParameterSpec()]	The diffusion of the inhibitor.	40.0

Examples:

>>> from skcomplex.systems.chemistry.pattern_formation import SchnakenbergElement
>>> SchnakenbergElement('cell').Dv
40.0

Source code in src/skcomplex/systems/chemistry/pattern_formation/_reaction_diffusion.py

def __init__(
    self,
    label: str,
    visual: VisualSpec | None = None,
    *,
    u: Annotated[object, StateSpec()] = 1.0,
    v: Annotated[object, StateSpec()] = 0.9,
    a: Annotated[object, ParameterSpec()] = 0.1,
    b: Annotated[object, ParameterSpec()] = 0.9,
    Du: Annotated[object, ParameterSpec()] = 1.0,
    Dv: Annotated[object, ParameterSpec()] = 40.0,
) -> None:
    super().__init__(label, visual=visual)
    self.u = u
    self.v = v
    self.a = a
    self.b = b
    self.Du = Du
    self.Dv = Dv

brusselator(shape=(64, 64), *, a=1.0, b=3.0, Du=1.0, Dv=8.0, random_state=0)

Return the Brusselator reaction-diffusion system from small noise around its steady state (a, b / a).

Parameters:

Name	Type	Description	Default
shape	Sequence[int]	The number of rows and columns of the periodic lattice.	(64, 64)
a	float	The feed of u.	1.0
b	float	The conversion of u into v.	3.0
Du	float	The diffusion of u.	1.0
Dv	float	The diffusion of v.	8.0
random_state	int | Generator | None	The seed or generator of the noise.	0

Returns:

Name	Type	Description
system	System	The system, with the concentrations cells.u and cells.v.

Examples:

>>> from skcomplex.systems.chemistry.pattern_formation import brusselator
>>> brusselator((8, 8), b=2.5).elements[0].values['b']
2.5

Source code in src/skcomplex/systems/chemistry/pattern_formation/_reaction_diffusion.py

@register_system(
    field='chemistry/pattern_formation',
    tags=('field', 'reaction-diffusion', 'Turing', 'oscillation', 'pde'),
    references=('Prigogine and Lefever 1968', 'Gros 2015'),
    defaults={'until': 100.0, 'dt': 1.0, 'solver': RK4Solver(dt=0.01)},
)
def brusselator(
    shape: Sequence[int] = (64, 64),
    *,
    a: float = 1.0,
    b: float = 3.0,
    Du: float = 1.0,
    Dv: float = 8.0,
    random_state: int | np.random.Generator | None = 0,
) -> System:
    """Return the Brusselator reaction-diffusion system from small noise around its steady state `(a, b / a)`.

    Args:
        shape:
            The number of rows and columns of the periodic lattice.

        a:
            The feed of `u`.

        b:
            The conversion of `u` into `v`.

        Du:
            The diffusion of `u`.

        Dv:
            The diffusion of `v`.

        random_state:
            The seed or generator of the noise.

    Returns:
        system:
            The system, with the concentrations `cells.u` and `cells.v`.

    Examples:
        >>> from skcomplex.systems.chemistry.pattern_formation import brusselator
        >>> brusselator((8, 8), b=2.5).elements[0].values['b']
        2.5
    """
    shape = tuple(shape)
    values = {
        'u': a + _noise(shape, 0.01, random_state),
        'v': np.full(shape, b / a),
        'a': a,
        'b': b,
        'Du': Du,
        'Dv': Dv,
    }
    return System(
        elements=[FieldElement('cells', element=BrusselatorElement, space=LatticeSpace(shape), values=values)],
        interactions=[*_get_brusselator_reactions(), *_get_diffusions()],
    )

gray_scott(shape=(128, 128), *, Du=0.16, Dv=0.08, F=0.035, k=0.065, random_state=0)

Return the Gray-Scott reaction-diffusion system, seeded in its middle, which grows spots that replicate.

Parameters:

Name	Type	Description	Default
shape	Sequence[int]	The number of rows and columns of the periodic lattice.	(128, 128)
Du	float	The diffusion of the substrate u.	0.16
Dv	float	The diffusion of the activator v.	0.08
F	float	The feed rate.	0.035
k	float	The kill rate.	0.065
random_state	int | Generator | None	The seed or generator of the noise on the seed.	0

Returns:

Name	Type	Description
system	System	The system, with the substrate cells.u and the activator cells.v.

Examples:

>>> from skcomplex.systems.chemistry.pattern_formation import gray_scott
>>> gray_scott((32, 32), F=0.04).elements[0].values['F']
0.04

Source code in src/skcomplex/systems/chemistry/pattern_formation/_reaction_diffusion.py

@register_system(
    field='chemistry/pattern_formation',
    tags=('field', 'reaction-diffusion', 'self-replication', 'pde'),
    references=('Gray and Scott 1984', 'Pearson 1993'),
    defaults={'until': 2000.0, 'dt': 20.0, 'solver': RK4Solver(dt=1.0)},
)
def gray_scott(
    shape: Sequence[int] = (128, 128),
    *,
    Du: float = 0.16,
    Dv: float = 0.08,
    F: float = 0.035,
    k: float = 0.065,
    random_state: int | np.random.Generator | None = 0,
) -> System:
    """Return the Gray-Scott reaction-diffusion system, seeded in its middle, which grows spots that replicate.

    Args:
        shape:
            The number of rows and columns of the periodic lattice.

        Du:
            The diffusion of the substrate `u`.

        Dv:
            The diffusion of the activator `v`.

        F:
            The feed rate.

        k:
            The kill rate.

        random_state:
            The seed or generator of the noise on the seed.

    Returns:
        system:
            The system, with the substrate `cells.u` and the activator `cells.v`.

    Examples:
        >>> from skcomplex.systems.chemistry.pattern_formation import gray_scott
        >>> gray_scott((32, 32), F=0.04).elements[0].values['F']
        0.04
    """
    shape = tuple(shape)
    u, v = np.ones(shape), np.zeros(shape)
    middle = tuple(slice(size // 2 - size // 10, size // 2 + size // 10) for size in shape)
    u[middle], v[middle] = 0.5, 0.25
    noise = _noise(shape, 0.01, random_state)
    values = {'u': u + noise, 'v': v + np.abs(noise), 'Du': Du, 'Dv': Dv, 'F': F, 'k': k}
    return System(
        elements=[FieldElement('cells', element=GrayScottElement, space=LatticeSpace(shape), values=values)],
        interactions=[*_get_gray_scott_reactions(), *_get_diffusions()],
    )

schnakenberg(shape=(64, 64), *, a=0.1, b=0.9, Du=1.0, Dv=40.0, spacing=1.0, random_state=0)

Return the Schnakenberg reaction-diffusion system, which forms Turing patterns from noise when Dv >> Du.

The steady state u = a + b, v = b / (a + b)^2 is stable without diffusion and unstable to perturbations of a band of wavelengths with it.

Parameters:

Name	Type	Description	Default
shape	Sequence[int]	The number of sites along every dimension of the periodic lattice.	(64, 64)
a	float	The production of u.	0.1
b	float	The production of v.	0.9
Du	float	The diffusion of the activator u.	1.0
Dv	float	The diffusion of the inhibitor v.	40.0
spacing	float	The distance between sites.	1.0
random_state	int | Generator | None	The seed or generator of the noise.	0

Returns:

Name	Type	Description
system	System	The system, with the concentrations cells.u and cells.v.

Examples:

>>> from skcomplex.systems.chemistry.pattern_formation import schnakenberg
>>> schnakenberg((8, 8)).elements[0].values['Dv']
40.0

Source code in src/skcomplex/systems/chemistry/pattern_formation/_reaction_diffusion.py

@register_system(
    field='chemistry/pattern_formation',
    tags=('field', 'reaction-diffusion', 'Turing', 'pde'),
    references=('Schnakenberg 1979', 'Turing 1952', 'Murray 2003'),
    defaults={'until': 200.0, 'dt': 5.0, 'solver': RK4Solver(dt=0.005)},
)
def schnakenberg(
    shape: Sequence[int] = (64, 64),
    *,
    a: float = 0.1,
    b: float = 0.9,
    Du: float = 1.0,
    Dv: float = 40.0,
    spacing: float = 1.0,
    random_state: int | np.random.Generator | None = 0,
) -> System:
    """Return the Schnakenberg reaction-diffusion system, which forms Turing patterns from noise when `Dv >> Du`.

    The steady state `u = a + b`, `v = b / (a + b)^2` is stable without diffusion and unstable to perturbations of a
    band of wavelengths with it.

    Args:
        shape:
            The number of sites along every dimension of the periodic lattice.

        a:
            The production of `u`.

        b:
            The production of `v`.

        Du:
            The diffusion of the activator `u`.

        Dv:
            The diffusion of the inhibitor `v`.

        spacing:
            The distance between sites.

        random_state:
            The seed or generator of the noise.

    Returns:
        system:
            The system, with the concentrations `cells.u` and `cells.v`.

    Examples:
        >>> from skcomplex.systems.chemistry.pattern_formation import schnakenberg
        >>> schnakenberg((8, 8)).elements[0].values['Dv']
        40.0
    """
    shape = tuple(shape)
    steady_u, steady_v = a + b, b / (a + b) ** 2
    values = {
        'u': steady_u * (1 + _noise(shape, 0.01, random_state)),
        'v': np.full(shape, steady_v),
        'a': a,
        'b': b,
        'Du': Du,
        'Dv': Dv,
    }
    return System(
        elements=[
            FieldElement(
                'cells',
                element=SchnakenbergElement,
                space=LatticeSpace(shape, spacing=spacing),
                values=values,
            ),
        ],
        interactions=[*_get_schnakenberg_reactions(), *_get_diffusions()],
    )