Intrinsic-instability populations
Jensen's coupled populations, whose steady state becomes unstable without a change of the surroundings.
- Name:
intrinsic_instability - Field:
biology/ecology - Formalism: discrete map
- References: Jensen 2023
Factory
@register_model(
field='biology/ecology',
tags=('tipping point', 'stochastic', 'discrete map', 'early warning'),
references=('Jensen 2023',),
defaults={'until': 2000, 'dt': 10},
)
def intrinsic_instability(
*,
x: float = 1.1,
eta: float = 0.1,
alpha: float = 0.01,
beta: float = 0.01,
nu: float = 0.01,
p: float = 0.001,
sigma: float = 0.01,
) -> System:
"""Return Jensen's coupled populations, whose steady state becomes unstable without a change of the surroundings.
The resident population fluctuates about its level `X = 1` until a rare mutation seeds the invader. The invader
then grows by the factor `1 + beta` per step and suppresses the resident, so the trajectory leaves the fixed point
`(X, 0)` along its unstable direction, with eigenvalue `beta`, and the resident declines although no parameter has
changed.
Args:
x:
The initial resident population.
eta:
The restoring rate of the resident population.
alpha:
The suppression of the resident population by the invader.
beta:
The growth rate of the invader.
nu:
The share of the resident population that a mutation moves to the invader.
p:
The probability of a mutation per step.
sigma:
The amplitude of the noise.
Returns:
system:
The model, with the populations `populations.x` and `populations.y`.
Examples:
>>> from skcomplex.models.biology.ecology import intrinsic_instability
>>> intrinsic_instability().elements[0].beta
0.01
"""
populations = UnstablePopulationsElement(
'populations',
x=x,
eta=eta,
alpha=alpha,
beta=beta,
nu=nu,
p=p,
sigma=sigma,
)
return System(
elements=[populations],
interactions=[
MapInteraction('resident', target='populations', state='x', function=_get_resident),
MapInteraction('invader', target='populations', state='y', function=_get_invader),
],
)
Components
class UnstablePopulationsElement(Element):
"""Resident population `x` and invading population `y`, after equations (13.3) and (13.4) of Jensen.
The maps of `intrinsic_instability` step both sizes. At every step `x ← [x - eta (x - X)] exp(-alpha y) + √x sigma
chi`, an Ornstein-Uhlenbeck step damped by the invader. With probability `p` a mutation from `x` adds `nu x` to `y`,
and otherwise `y ← y (1 + beta) + √y sigma chi'`. The noises `chi` and `chi'` are independent standard normal
values, scaled by the square root of the population sizes as in birth and death processes, and both sizes are cut
at 0.
Args:
label:
The label of the populations.
visual:
How the element is drawn, or `None` for `default_visual_`.
x:
The initial resident population.
y:
The initial invading population.
eta:
The restoring rate `eta` of the resident population.
mean:
The level `X` the resident population returns to.
alpha:
The suppression `alpha` of the resident population by the invader.
beta:
The growth rate `beta` of the invader.
nu:
The share `nu` of the resident population that a mutation moves to the invader.
p:
The probability of a mutation per step.
sigma:
The amplitude `sigma` of the noise.
Examples:
>>> from skcomplex.models.biology.ecology import UnstablePopulationsElement
>>> UnstablePopulationsElement('populations').p
0.001
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = None,
*,
x: Annotated[float, StateSpec(domain=RealSet.closed_open(0, oo))] = 1.1,
y: Annotated[float, StateSpec(domain=RealSet.closed_open(0, oo))] = 0.0,
eta: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.1,
mean: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 1.0,
alpha: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.01,
beta: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.01,
nu: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.01,
p: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.001,
sigma: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.01,
) -> None:
super().__init__(label, visual=visual)
self.x = x
self.y = y
self.eta = eta
self.mean = mean
self.alpha = alpha
self.beta = beta
self.nu = nu
self.p = p
self.sigma = sigma
class MapInteraction(Interaction):
"""Map of a state of discrete time, whose next value is a function, or the state plus it.
In the mode `'replace'` the next value of the state is `f`, and in the mode `'increment'` it is the state plus `f`.
Under the `'synchronous'` schedule every member of a bound group updates at once, from the states at the start of
the step. Under the `'sequential'` schedule the members update one after the other, in their order, and under the
`'random-sequential'` schedule in a new random order every step, each member reading the values the members before
it set. Under the `'member'` schedule of the agent form, every member updates in its own activation, together with
the rules of its element and the other member interactions, in the order the system lists them.
In the `'symbolic'` form, `function(namespace)` returns a symbolic expression, or a vector of them for a vector
state, where `namespace.target` holds the states and parameters of the target, `namespace.sources` the namespaces
of the sources, and `namespace.t` the time, and may hold draws such as `normal('noise')`. In the `'agent'` form,
the target is an agent group and `function(agent, model)` returns the value as a number, or as an array of the
shape of a vector state, so it can read the agent, its neighbours, and the history and storage of the model view.
Args:
label:
The label of the interaction, unique among the parts of its system.
target:
The path of the element or group whose state the map sets.
state:
The name of the state of the target.
function:
The function `f(namespace)` in the symbolic form, or `f(agent, model)` in the agent form.
sources:
The paths of the elements, or members of groups such as `cells[0]`, that the function reads.
mode:
`'replace'` or `'increment'`.
schedule:
`'synchronous'`, `'sequential'`, `'random-sequential'`, or `'member'` for the agent form.
form:
`'symbolic'` for update laws of the states of the target, or `'agent'` for a rule over an agent group.
phase:
The phase of the step the agent form runs in, or `None` for the default phase.
Raises:
ValidationError:
The mode, the schedule, or the form is unknown, the symbolic form has the schedule `'member'`, the
function is not callable, or the agent form returns a value that does not match the state, raised by
`run` of the simulation.
Examples:
>>> from skcomplex.core.elements import Element
>>> from skcomplex.core.interactions import MapInteraction
>>> from skcomplex.core.spaces import DiscreteTime
>>> from skcomplex.core.system import System
>>> from skcomplex.engine.runtime import Simulation
>>> from skcomplex.specs import StateSpec
>>> from typing import Annotated, SupportsFloat
>>> class ValueElement(Element):
... def __init__(self, label, x: Annotated[float, StateSpec()] = 0.5):
... super().__init__(label)
... self.x = x
>>> logistic = MapInteraction(
... 'map', target='value', state='x', function=lambda s: 3 * s.target.x * (1 - s.target.x)
... )
>>> system = System(elements=[ValueElement('value')], interactions=[logistic], time=DiscreteTime())
>>> Simulation(system).check().prepare().run(until=2).trajectory_.values('value.x').tolist()
[0.5, 0.75, 0.5625]
>>> agents = MapInteraction('map', target='walkers', state='x', function=lambda a, m: a.x + 1, form='agent')
>>> type(agents.check().prepare().run(SimpleNamespace()).laws_[0]).__name__
'RuleLaw'
>>> from skcomplex.core.elements import GroupElement
>>> class WeightsElement(Element):
... def __init__(self, label, visual=None, *, w: Annotated[object, StateSpec(vector=True)] = 0.0):
... super().__init__(label, visual=visual)
... self.w = w
>>> group = GroupElement('weights', element=WeightsElement, size=1, values={'w': [1.0, 2.0]})
>>> doubling = MapInteraction(
... 'double', target='weights', state='w', function=lambda a, m: [2 * x for x in a.w], form='agent'
... )
>>> trajectory = Simulation(System(elements=[group], interactions=[doubling])).check().prepare().run(until=2)
>>> trajectory.trajectory_.agents('weights', 2).values['w'].tolist()
[[4.0, 8.0]]
"""
def __init__(
self,
label: str,
*,
target: Annotated[str, RoleSpec()],
state: str,
function: Callable[..., object],
sources: Annotated[Sequence[str], RoleSpec(variadic=True)] = (),
mode: str = 'replace',
schedule: str = 'synchronous',
form: str = 'symbolic',
phase: str | None = None,
) -> None:
super().__init__(label)
self.target = target
self.state = state
self.function = function
self.sources = sources
self.mode = mode
self.schedule = schedule
self.form = form
self.phase = phase
def _check(self) -> None:
"""Raise an error for an unknown mode, schedule, or form, or a function that cannot be called.
Raises:
ValidationError:
The mode, the schedule, or the form is unknown, the symbolic form has the schedule `'member'`, or the
function is not callable.
"""
super()._check()
self._check_variant('mode', self.mode, MODES, (BaseSpec,))
for name, value, variants in (
('schedule', self.schedule, SCHEDULES),
('form', self.form, FORMS),
):
if value not in variants:
error_msg = f'The {name} of map {self.label!r} is {value!r}. Use one of {", ".join(variants)}.'
raise ValidationError(error_msg)
if self.schedule == 'member' and self.form != 'agent':
error_msg = (
f'The map {self.label!r} updates every member in its activation, which only the agent form does.'
)
raise ValidationError(error_msg)
if not callable(self.function):
error_msg = f'The function of map {self.label!r} must be callable, but it is {self.function!r}.'
raise ValidationError(error_msg)
def _apply_mode(self, current: Value, value: Value) -> Value:
"""Return the next value of a state from its current value and the value of the function, by the mode."""
if self.mode == 'increment':
return current + value
if self.mode == 'replace':
return value
raise self._get_variant_error(self.mode, 'mode', '_apply_mode')
def _get_vector(self, agent: AgentViewProtocol, model: ModelViewProtocol) -> Value:
"""Return the next value of a vector state of an agent."""
current = np.asarray(getattr(agent, self.state), dtype=np.float64)
value = np.asarray(self.function(agent, model), dtype=np.float64)
if value.shape != current.shape:
error_msg = (
f'The function of map {self.label!r} returns the shape {value.shape} for the state {self.state!r} of '
f'the shape {current.shape}. Return an array of the shape of the state.'
)
raise ValidationError(error_msg)
return self._apply_mode(current, value)
def _get_value(self, agent: AgentViewProtocol, model: ModelViewProtocol) -> Value:
"""Return the next value of the state of an agent, a number or the array of a vector state."""
if np.ndim(getattr(agent, self.state)) > 0:
return self._get_vector(agent, model)
found = self.function(agent, model)
if not isinstance(found, SupportsFloat):
error_msg = f'The function of map {self.label!r} returns {found!r} for an agent. Return a number.'
raise ValidationError(error_msg)
value = float(found)
return self._apply_mode(float(getattr(agent, self.state)), value)
def _set_value(self, agent: AgentViewProtocol, value: Value) -> None:
"""Set the state of an agent to its next value."""
setattr(agent, self.state, value)
def _update_member(self, agent: AgentViewProtocol, model: ModelViewProtocol) -> None:
"""Set the state of one member to its next value."""
if agent.alive:
self._set_value(agent, self._get_value(agent, model))
def update(self, model: ModelViewProtocol) -> None:
"""Set the state of every agent of the target to its next value, by the schedule.
Args:
model:
The model view of the interaction.
"""
self._update_agents(model, self.schedule, self._get_value, self._set_value)
def _get_order(self) -> object:
"""Return the order of the members of the symbolic form: none, their own order, or a uniform draw."""
if self.schedule == 'synchronous':
return None
if self.schedule == 'sequential':
return SR(0)
return uniform(re.sub(r'\W', '_', f'order_{self.label}'))
def _get_laws(self, namespace: SimpleNamespace) -> list[BasePart]:
"""Return the update of the state to its next value, or the rule of the agent form.
Args:
namespace:
The namespace of the symbols the laws can use, with `namespace.target` and `namespace.sources`.
Returns:
laws:
The update law of the state, or the rule that runs the map of the agents.
"""
if self.form == 'agent' and self.schedule == 'member':
return [RuleLaw(self._update_member, phase=self.phase, members=self._get_role('target'))]
if self.form == 'agent':
return [RuleLaw(self.update, phase=self.phase)]
value = getattr(self._get_member(namespace, 'target'), self.state)
mapped = cast(Value, self.function(namespace))
next_value = self._apply_mode(value, mapped)
return [UpdateLaw(value, next_value, increment=self.mode == 'increment', order=self._get_order())]
Equations
populations.x: x_populations(t + 1) == where(normal_chi_x_resident*sigma_populations*sqrt(x_populations(t)) + (eta_populations*(mean_populations - x_populations(t)) + x_populations(t))*e^(-alpha_populations*y_populations(t)) < 0, 0, normal_chi_x_resident*sigma_populations*sqrt(x_populations(t)) + (eta_populations*(mean_populations - x_populations(t)) + x_populations(t))*e^(-alpha_populations*y_populations(t)))
populations.y: y_populations(t + 1) == where(where(where(bernoulli_mutation_invader < p_populations, 1, 0) > 0, nu_populations*x_populations(t) + y_populations(t), normal_chi_y_invader*sigma_populations*sqrt(y_populations(t)) + (beta_populations + 1)*y_populations(t)) < 0, 0, where(where(bernoulli_mutation_invader < p_populations, 1, 0) > 0, nu_populations*x_populations(t) + y_populations(t), normal_chi_y_invader*sigma_populations*sqrt(y_populations(t)) + (beta_populations + 1)*y_populations(t)))