Exponential growth

A population that grows or declines at a constant per-capita rate.

• Name: exponential_growth
• Field: biology/ecology
• Formalism: ordinary differential equations
• References: Malthus 1798

Factory

@register_model(
    field='biology/ecology',
    tags=('population', 'ode'),
    references=('Malthus 1798',),
    defaults={'until': 20.0, 'dt': 0.1},
)
def exponential_growth(n: float = 10.0, r: float = 0.1) -> System:
    """Return a population that grows or declines at a constant per-capita rate.

    Args:
        n:
            The initial size.

        r:
            The per-capita growth rate.

    Returns:
        system:
            The model, with the population `population`.

    Examples:
        >>> from skcomplex.models.biology.ecology import exponential_growth
        >>> exponential_growth(r=0.2).interactions[0].r
        0.2
    """
    return System(
        elements=[PopulationElement('population', n=n)],
        interactions=[ExponentialGrowthInteraction('growth', target='population', r=r)],
    )

Components

class PopulationElement(CompartmentElement):
    """Population of one species, a compartment whose state `n` holds its size.

    Args:
        label:
            The label of the population.

        visual:
            How the element is drawn, or `None` for `default_visual_`.

        n:
            The initial size.

    Examples:
        >>> from skcomplex.models.biology.common.populations import PopulationElement
        >>> PopulationElement('rabbits', n=40).n
        40
    """

    def __init__(
        self,
        label: str,
        visual: VisualSpec | None = None,
        *,
        n: Annotated[float, StateSpec(domain=RealSet.closed_open(0, oo))] = 1.0,
    ) -> None:
        super().__init__(label, visual=visual, n=n)

class ExponentialGrowthInteraction(GrowthInteraction):
    """Growth at a constant per capita rate, `dn/dt = r n`, the exponential form of a growth without harvest.

    Args:
        label:
            The label of the interaction.

        target:
            The path of the element or group whose population grows.

        r:
            The per capita growth rate, negative for decay.

        state:
            The name of the state of the target that holds the population.

    Examples:
        >>> from skcomplex.core.elements import CompartmentElement
        >>> from skcomplex.core.system import System
        >>> from skcomplex.engine.runtime import Simulation
        >>> from skcomplex.models.biology.common.populations import ExponentialGrowthInteraction
        >>> system = System(
        ...     elements=[CompartmentElement('rabbits', n=1.0)],
        ...     interactions=[ExponentialGrowthInteraction('growth', target='rabbits', r=0.5)],
        ... )
        >>> trajectory = Simulation(system, dt=0.01).check().prepare().run(until=2.0).trajectory_
        >>> round(float(trajectory.values('rabbits.n')[-1]), 3)
        2.718
    """

    def __init__(
        self,
        label: str,
        *,
        target: Annotated[str, RoleSpec()],
        r: Annotated[float, ParameterSpec()] = 1.0,
        state: str = 'n',
    ) -> None:
        super().__init__(label, target=target, r=r, form='exponential', state=state)

    def _get_rate(self, namespace: SimpleNamespace) -> Expression:
        """Return the growth `r n`, with no harvest.

        Args:
            namespace:
                The namespace of the symbols the rate can use, with the namespace of the target.

        Returns:
            rate:
                The rate of the population.
        """
        return self._get_growth(namespace)

Equations

population.n: diff(n_population(t), t) == r_growth*n_population(t)