PredatorPreyPlant — Predator-Prey Dynamics including Vegetation

This is an expansion of the Predator-Prey model in which plants (the resource the prey feeds on) are modelled explicitly. The main additions/changes are the following:

• The movement mechanism is different, and the movement rule now involves a movement radius so that each entity can move further than one cell per turn.

• The presence of plants which can grow, based on different growth models

Implementation Details

Movement

In contrast to the PredatorPrey model, the movement rule is no longer called once per cell each turn. Instead, it is called for num_moves random cells each turn. This makes it possible for one cell to be selected multiple times to carry out the movement rule and other cells might not be selected at all within that time step.

The internal num_moves parameter can be set in the configuration via the num_moves_fraction key, which allows specifying it in units of the total number of grid cells.

Three possible actions take place on a selected cell:

• In case both a predator and a prey are on a cell, the prey will flee in the same fashion as in the standard Predator-Prey model.

• If there is a predator but no prey on the cell, the predator will move until it finds a prey or until it reaches the move_limit.

• If there is only a prey but no resource, the prey will move to find resources until it reaches it or arrives at the move_limit. In this phase the prey will move to a cell even if there is a predator on it.

Vegetation Growth Model

There are three options for the growth models of the vegatation:

• none: deactivates explicit modelling of vegetation. Grass regrows every turn, thus prey can also eat every turn.

• deterministic: vegetation will regrow after a specific number of time-steps.

• stochastic: vegetation will regrow with a certain probability, evaluated each time step.

Default configuration parameters

Below are the default configuration parameters for the PredatorPreyPlant model:

# --- Space -------------------------------------------------------------------
space:
  periodic: true


# --- CellManager -------------------------------------------------------------
cell_manager:
  grid:
    structure: square
    resolution: 64

  neighborhood:
    mode: Moore

  cell_params:
    # Initial prey and predator resources are drawn from a uniform integer (!)
    # distribution with the limits specified below.
    predator:
      min_init_resources: !is-positive-or-zero 2
      max_init_resources: !is-positive-or-zero 8
    prey:
      min_init_resources: !is-positive-or-zero 2
      max_init_resources: !is-positive-or-zero 8
 
    # The probabilities to have a predator, plant, and/or prey on a cell.
    # These probabilities are evaluated independently and for each cell.
    p_predator: !is-probability 0.2
    p_prey: !is-probability 0.3
    p_plant: !is-probability 1.

# Load species positions from datasets of a HDF5 file.
# If enabled, the presence of an entity is set using data from the respective
# dataset, i.e. `predator`, `prey`, and `plant`.
# Values can only be 0 or 1, specifying whether the respective entity is
# present on the corresponding cell.
cell_states_from_file:
  hdf5_file: !is-string /abs/path/to/data.hdf5  # TODO Set this in your run.yml
  load_predator: !is-bool false
  load_prey: !is-bool false
  load_plant: !is-bool false


# --- Model dynamics ----------------------------------------------------------
# How many cells to randomly select each time step and apply the movement rule
# to (in units of the number of available cells). Can also be > 1.
num_moves_fraction: !is-positive-or-zero 1.


# Species-specific parameters; the model dynamics arise from these
predator:
  # Resource intake from eating and maximum resource value
  resource_intake: !is-positive-or-zero 15.

  resource_max: !is-positive-or-zero 30.

  # Cost of living (per time step)
  cost_of_living: !is-positive-or-zero 3.5

  # Movement limit; how many single steps a predator can make when moving.
  move_limit: !is-positive-int 2
    
  # Reproduction parameters: minimum resources required, probability for
  # the reproduction taking place, and cost of reproduction.
  repro_resource_requ: !is-positive-or-zero 2.

  repro_prob: !is-probability 0.2

  repro_cost: !is-positive-or-zero 1.
    


prey:
  resource_intake: !is-positive-or-zero 3.
    
  resource_max: !is-positive-or-zero 8.

  cost_of_living: !is-positive-or-zero 1.

  move_limit: !is-positive-int 1

  # Fleeing probability when on the same cell together with a predator
  p_flee: !is-probability 0.3

  repro_resource_requ: !is-positive-or-zero 1.5
    
  repro_prob: !is-probability 0.4

  repro_cost: !is-positive-or-zero 1.2
    


plant:
  # Plants' growth model
  growth_model: !param
    dtype: str
    default: stochastic
    is_any_of: [none, deterministic, stochastic]
  # Available models:
  #  - none:           As in the standard Predator-Prey model, the presence of
  #                    plants is ignored. Thus, available resources for prey
  #                    are unlimited.
  #
  #  - deterministic   The system will regrow plants deterministically after a
  #                    constant regeneration time given by ``regen_time``.
  # 
  #  - stochastic      The system will regrow plants with a probability
  #                    given by parameter ``regen_prob``, evaluated each time
  #                    step for each cell without a plant on it.

  # The deterministic regeneration time
  regen_time: !is-unsigned 7

  # The regeneration probability (for stochastic regrowth)
  regen_prob: !is-probability 0.25

Available plots

The following plot configurations are available for the PredatorPreyPlant model:

Default Plot Configuration

phase_space/predator_prey:
  based_on: phase_space

  x: predator_density
  y: prey_density
  hue: time

  helpers:
    set_labels:
      x: Predator Density $[1/A]$
      y: Prey Density $[1/A]$

phase_space/prey_plant:
  based_on: phase_space

  x: prey_density
  y: plant_density
  hue: time

  helpers:
    set_labels:
      x: Prey Density $[1/A]$
      y: Plant Density $[1/A]$


# --- Time series of mean values ----------------------------------------------
species_densities:
  based_on: species_densities

mean_resources:
  based_on: mean_resources

# --- Animation of the spatial resource development of prey and predators -----
ca/resources:
  based_on: resources

# --- Animation of the spatial development of prey and predator populations ---
ca/population:
  based_on: population


# --- More detailed animation: prey only, combined, predator only -------------
ca/population_detailed:
  based_on: population_detailed

Base Plot Configuration

.variables:
  model_name: &model_name PredatorPreyPlant
  base_path: &base_path data/PredatorPreyPlant

  # Colors used throughout these plots
  cmap: &cmap
    empty: &color_empty white
    predator: &color_predator '#FFCC66'
    prey: &color_prey '#006666'
    plant: &color_plant '#339999'
    both: &color_both '#CC3333'



# =============================================================================
#  ╔╦╗╔═╗╔╦╗╔═╗╦  ╔═╗╔╦╗╔═╗╔═╗
#   ║ ║╣ ║║║╠═╝║  ╠═╣ ║ ║╣ ╚═╗
#   ╩ ╚═╝╩ ╩╩  ╩═╝╩ ╩ ╩ ╚═╝╚═╝
# =============================================================================
# -- Overloads ----------------------------------------------------------------
# Overload some configs to insert model-specific settings

# Model-specific defaults
.defaults:
  based_on: .defaults

  # Can define something here ...


# .. Creators .................................................................
.creator.universe:
  based_on:
    - .creator.universe
    - .defaults

  dag_options:
    select_path_prefix: *base_path

.creator.multiverse:
  based_on:
    - .creator.multiverse
    - .defaults

  select_and_combine:
    base_path: *base_path





# =============================================================================
#  ╔═╗╦  ╔═╗╔╦╗╔═╗
#  ╠═╝║  ║ ║ ║ ╚═╗
#  ╩  ╩═╝╚═╝ ╩ ╚═╝
# =============================================================================
# -- Phase space plots --------------------------------------------------------
phase_space:
  based_on:
    - .creator.universe
    - .plot.facet_grid.scatter

  select: &pred_prey_plant_density
    predator_density:
      path: predator
      transform:
        - .mean: [ !dag_prev , [ 'x', 'y' ] ]
    prey_density:
      path: prey
      transform:
        - .mean: [ !dag_prev , [ 'x', 'y' ] ]
    plant_density:
      path: plant
      transform:
        - .mean: [ !dag_prev , [ 'x', 'y' ] ]

  transform:
    - operation: xr.Dataset
      kwargs:
        data_vars:
          predator_density: !dag_tag predator_density
          prey_density: !dag_tag prey_density
          plant_density: !dag_tag plant_density
      tag: data

  # How to color-code the time axis
  cmap: viridis_r

  # All below: passed on to plt.scatter
  s: 3.5

# -- Time series of the predator and prey densities ---------------------------
species_densities:
  based_on:
    - .creator.universe
    - .plot.facet_grid.line
    - .hlpr.kind.time_series

  select:
    <<: *pred_prey_plant_density

  transform:
    - xr.Dataset:
        data_vars:
          predator: !dag_tag predator_density
          prey: !dag_tag prey_density
    - .to_array: [!dag_prev ]
      kwargs:
        dim: kind
      tag: data

  hue: kind

  helpers:
    set_labels:
      y: Density $[1/A]$
    set_title:
      title: Predator and Prey Densities
    set_legend:
      title: Species

  style: &color_cycler
    axes.prop_cycle: !format
      fstr: "cycler('color', ['{cmap[predator]:}', '{cmap[prey]:}'])"
      cmap: *cmap

# -- Time series of the predator and prey resources ---------------------------
mean_resources:
  based_on:
    - .creator.universe
    - .plot.facet_grid.line
    - .hlpr.kind.time_series

  select:
    resource_predator:
      path: resource_predator
      transform:
        - .where: [ !dag_prev ge 1, !dag_prev , 0]  # TODO Syntax ok???
        - .mean: [!dag_prev , ['x', 'y']]
    resource_prey:
      path: resource_prey
      transform:
        - .where: [ !dag_prev ge 1, !dag_prev , 0]
        - .mean: [!dag_prev , ['x', 'y']]

  transform:
    - operation: xr.Dataset
      kwargs:
        data_vars:
          predator: !dag_tag resource_predator
          prey: !dag_tag resource_prey
    - .to_array: [ !dag_prev ]
      kwargs:
        dim: kind
      tag: data

  x: time
  hue: kind

  helpers:
    set_limits:
      y: [0, ~]
    set_labels:
      y: Resource Density $[1/A]$
    set_title:
      title: Mean Total Resources
    set_legend:
      title: Species

  style:
    <<: *color_cycler

# --- Animation of the spatial resource development of prey and predators -----
resources:
  based_on:
    - .creator.universe
    - .plot.ca

  select:
    resource_predator: resource_predator
    resource_prey: resource_prey

    resource_max_predator:
      path: "../../cfg"
      with_previous_result: true
      transform:
        - recursive_getitem: [[*model_name, predator, resource_max]]

    resource_max_prey:
      path: "../../cfg"
      with_previous_result: true
      transform:
        - recursive_getitem: [[*model_name, prey, resource_max]]

  # Select the properties to plot
  to_plot:
    resource_predator:
      title: Predator resources
      vmin: 0
      vmax: !dag_result resource_max_predator
      cmap: YlGn

    resource_prey:
      title: Prey resources
      vmin: 0
      vmax: !dag_result resource_max_prey
      cmap: YlGn


# --- Animation of the spatial development of prey and predator populations ---
population:
  based_on:
    - .creator.universe
    - .plot.ca

  select:
    predator:
      path: predator
      transform: [.data]  # resolve GridDC to have an xr-native object
    prey:
      path: prey
      transform: [.data]  # resolve GridDC to have an xr-native object

  transform:
    # Use base-2 encoding to denote whether a cell is empty, contains only a
    # prey, only a predator or both:
    - mul: [!dag_tag predator, 2]
    - add: [!dag_prev , !dag_tag prey]
      tag: combined

  to_plot:
    combined:
      title: Predator & Prey positions
      cmap:
        empty: *color_empty         # 0
        prey: *color_prey           # 1
        predator: *color_predator   # 2
        both: *color_both           # 3


# --- More detailed animation: prey only, combined, predator only -------------
population_detailed:
  based_on:
    - .creator.universe
    - .plot.ca
  select:
    predator:
      path: predator
      transform: [.data]  # resolve GridDC to have an xr-native object
    prey:
      path: prey
      transform: [.data]  # resolve GridDC to have an xr-native object
    plant:
      path: plant
      transform: [ .data ]  # resolve GridDC to have an xr-native object

  transform:
    # Use base-2 encoding to denote whether a cell is empty or contains both
    - mul: [ !dag_tag predator, !dag_tag prey ]
    - .assign_attrs: [!dag_prev , {'grid_structure': 'square'}]
      tag: combined

  to_plot:
    predator:
      title: Predator
      cmap:
        empty: *color_empty
        predator: *color_predator
      add_colorbar: false

    prey:
      title: Prey
      cmap:
        empty: *color_empty
        prey: *color_prey
      add_colorbar: false

    combined:
      title: Predator & Prey
      cmap:
        empty: *color_empty
        both: *color_both
      add_colorbar: false

    plant:
      title: Plant
      cmap:
        empty: *color_empty
        both: *color_plant
      add_colorbar: false

For available base plots, see Base Plot Configuration Pool.

References

Kurt Roth: Chaotic, Complex, and Evolving Environmental Systems, unpublished lecture notes, University of Heidelberg, 2019.