Model of predator-prey dynamics

The predator-prey model emulates the population dynamics of predator and prey animals who live in a common ecosystem and compete over limited resources. This model is an agent-based analog to the classic Lotka-Volterra differential equation model. This example illustrates how to develop models with heterogeneous agents (sometimes referred to as a mixed agent based model).

The environment is a two dimensional grid containing sheep, wolves and grass. In the model, wolves eat sheep and sheep eat grass. Their populations will oscillate over time if the correct balance of resources is achieved. Without this balance however, a population may become extinct. For example, if wolf population becomes too large, they will deplete the sheep and subsequently die of starvation.

We will begin by loading the required packages and defining three subtypes of AbstractAgent: Sheep, Wolf, and Grass. All three agent types have id and pos properties, which is a requirement for all subtypes of AbstractAgent when they exist upon a GridSpace. Sheep and wolves have identical properties, but different behaviors as explained below. The property energy represents an animals current energy level. If the level drops below zero, the agent will die. Sheep and wolves reproduce asexually in this model, with a probability given by reproduction_prob. The property Δenergy controls how much energy is acquired after consuming a food source.

Grass is a replenishing resource that occupies every cell in the grid space. Grass can be consumed only if it is fully_grown. Once the grass has been consumed, it replenishes after a delay specified by the property regrowth_time. The property countdown tracks the delay between being consumed and the regrowth time.

It is also available from the Models module as Models.predator_prey.

using Agents, AgentsPlots, StatsPlots

mutable struct Sheep <: AbstractAgent
    id::Int
    pos::Tuple{Int,Int}
    energy::Float64
    reproduction_prob::Float64
    Δenergy::Float64
end

mutable struct Wolf <: AbstractAgent
    id::Int
    pos::Tuple{Int,Int}
    energy::Float64
    reproduction_prob::Float64
    Δenergy::Float64
end

mutable struct Grass <: AbstractAgent
    id::Int
    pos::Tuple{Int,Int}
    fully_grown::Bool
    regrowth_time::Int
    countdown::Int
end

WARNING: Method definition dot(Any, Any, Any) in module LinearAlgebra at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.5/LinearAlgebra/src/generic.jl:924 overwritten in module Optim at /home/travis/.julia/packages/Optim/Agd3B/src/multivariate/precon.jl:23.
  ** incremental compilation may be fatally broken for this module **

WARNING: Method definition dot(Any, Any, Any) in module LinearAlgebra at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.5/LinearAlgebra/src/generic.jl:924 overwritten in module Optim at /home/travis/.julia/packages/Optim/Agd3B/src/multivariate/precon.jl:23.
  ** incremental compilation may be fatally broken for this module **

WARNING: Method definition dot(Any, Any, Any) in module LinearAlgebra at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.5/LinearAlgebra/src/generic.jl:924 overwritten in module Optim at /home/travis/.julia/packages/Optim/Agd3B/src/multivariate/precon.jl:23.
  ** incremental compilation may be fatally broken for this module **

The function initialize_model returns a new model containing sheep, wolves, and grass using a set of pre-defined values (which can be overwritten). The environment is a two dimensional grid space with moore = true, which enables animals to walk in all directions. Heterogeneous agents are specified in the model as a Union. Agents are scheduled by_type, which randomizes the order of agents with the constraint that agents of a particular type are scheduled consecutively.

function initialize_model(;
    n_sheep = 100,
    n_wolves = 50,
    dims = (20, 20),
    regrowth_time = 30,
    Δenergy_sheep = 4,
    Δenergy_wolf = 20,
    sheep_reproduce = 0.04,
    wolf_reproduce = 0.05,
)
    space = GridSpace(dims, moore = true)
    model =
        ABM(Union{Sheep,Wolf,Grass}, space, scheduler = by_type(true, true), warn = false)
    id = 0
    for _ in 1:n_sheep
        id += 1
        energy = rand(1:(Δenergy_sheep * 2)) - 1
        # Note that we must instantiate agents before adding them in a mixed-ABM
        # to confirm their type.
        sheep = Sheep(id, (0, 0), energy, sheep_reproduce, Δenergy_sheep)
        add_agent!(sheep, model)
    end
    for _ in 1:n_wolves
        id += 1
        energy = rand(1:(Δenergy_wolf * 2)) - 1
        wolf = Wolf(id, (0, 0), energy, wolf_reproduce, Δenergy_wolf)
        add_agent!(wolf, model)
    end
    for n in nodes(model)
        id += 1
        fully_grown = rand(Bool)
        countdown = fully_grown ? regrowth_time : rand(1:regrowth_time) - 1
        grass = Grass(id, (0, 0), fully_grown, regrowth_time, countdown)
        add_agent!(grass, n, model)
    end
    return model
end

The function agent_step! is dispatched on each subtype in order to produce type-specific behavior. The agent_step! is similar for sheep and wolves: both lose 1 energy unit by moving to an adjacent cell and both consume a food source if available. If their energy level is below zero, an agent dies. Otherwise, the agent lives and reproduces with some probability.

function agent_step!(sheep::Sheep, model)
    move!(sheep, model)
    sheep.energy -= 1
    agents = get_node_agents(sheep.pos, model)
    dinner = filter!(x -> isa(x, Grass), agents)
    eat!(sheep, dinner, model)
    if sheep.energy < 0
        kill_agent!(sheep, model)
        return
    end
    if rand() <= sheep.reproduction_prob
        reproduce!(sheep, model)
    end
end

function agent_step!(wolf::Wolf, model)
    move!(wolf, model)
    wolf.energy -= 1
    agents = get_node_agents(wolf.pos, model)
    dinner = filter!(x -> isa(x, Sheep), agents)
    eat!(wolf, dinner, model)
    if wolf.energy < 0
        kill_agent!(wolf, model)
        return
    end
    if rand() <= wolf.reproduction_prob
        reproduce!(wolf, model)
    end
end

The behavior of grass functions differently. If it is fully grown, it is consumable. Otherwise, it cannot be consumed until it regrows after a delay specified by regrowth_time.

function agent_step!(grass::Grass, model)
    if !grass.fully_grown
        if grass.countdown <= 0
            grass.fully_grown = true
            grass.countdown = grass.regrowth_time
        else
            grass.countdown -= 1
        end
    end
end

Sheep and wolves move to a random adjacent cell with the move! function.

function move!(agent, model)
    neighbors = node_neighbors(agent, model)
    cell = rand(neighbors)
    move_agent!(agent, cell, model)
end

Sheep and wolves have separate eat! functions. If a sheep eats grass, it will acquire additional energy and the grass will not be available for consumption until regrowth time has elapsed. If a wolf eats a sheep, the sheep dies and the wolf acquires more energy.

function eat!(sheep::Sheep, grass_array, model)
    isempty(grass_array) && return
    grass = grass_array[1]
    if grass.fully_grown
        sheep.energy += sheep.Δenergy
        grass.fully_grown = false
    end
end

function eat!(wolf::Wolf, sheep, model)
    if !isempty(sheep)
        dinner = rand(sheep)
        kill_agent!(dinner, model)
        wolf.energy += wolf.Δenergy
    end
end

Sheep and wolves share a common reproduction method. Reproduction has a cost of 1/2 the current energy level of the parent. The offspring is an exact copy of the parent, with exception of id.

function reproduce!(agent, model)
    agent.energy /= 2
    id = nextid(model)
    A = typeof(agent)
    offspring = A(id, agent.pos, agent.energy, agent.reproduction_prob, agent.Δenergy)
    add_agent_pos!(offspring, model)
    return
end

Running the model

We will run the model for 500 steps and record the number of sheep, wolves and consumable grass patches after each step. First: initialize the model.

n_steps = 500
model = initialize_model()

AgentBasedModel with 550 agents of type Union{Main.ex-predator_prey.Grass, Main.ex-predator_prey.Sheep, Main.ex-predator_prey.Wolf}
 space: GridSpace with 400 nodes and 1482 edges
 scheduler: by_union

To view our starting population, we can build an overview plot:

offset(a::Sheep) = (0.2, 0.0)
offset(a::Wolf) = (-0.2, 0.0)
offset(a::Grass) = (0.0, 0.0)
mshape(a::Sheep) = :circle
mshape(a::Wolf) = :utriangle
mshape(a::Grass) = :square
mcolor(a::Sheep) = RGBA(1.0, 1.0, 1.0, 0.6)
mcolor(a::Wolf) = RGBA(0.6, 0.6, 0.6, 0.8)
mcolor(a::Grass) = cgrad([:brown, :green])[a.countdown / a.regrowth_time]
plotabm(
    model;
    offset = offset,
    am = mshape,
    as = 15,
    ac = mcolor,
    scheduler = by_type((Grass, Sheep, Wolf), false),
    grid = false,
    size = (800, 600),
    showaxis = false,
    aspect_ratio = :equal,
)

Now, lets run the simulation and collect some data.

sheep(a) = typeof(a) == Sheep
wolves(a) = typeof(a) == Wolf
grass(a) = typeof(a) == Grass && a.fully_grown
adata = [(sheep, count), (wolves, count), (grass, count)]
results, _ = run!(model, agent_step!, n_steps; adata = adata)

(501×4 DataFrame
│ Row │ step  │ count_sheep │ count_wolves │ count_grass │
│     │ Int64 │ Int64       │ Int64        │ Int64       │
├─────┼───────┼─────────────┼──────────────┼─────────────┤
│ 1   │ 0     │ 100         │ 50           │ 203         │
│ 2   │ 1     │ 89          │ 56           │ 171         │
│ 3   │ 2     │ 75          │ 60           │ 156         │
│ 4   │ 3     │ 68          │ 60           │ 152         │
│ 5   │ 4     │ 61          │ 66           │ 147         │
│ 6   │ 5     │ 55          │ 71           │ 139         │
│ 7   │ 6     │ 52          │ 70           │ 134         │
⋮
│ 494 │ 493   │ 0           │ 0            │ 400         │
│ 495 │ 494   │ 0           │ 0            │ 400         │
│ 496 │ 495   │ 0           │ 0            │ 400         │
│ 497 │ 496   │ 0           │ 0            │ 400         │
│ 498 │ 497   │ 0           │ 0            │ 400         │
│ 499 │ 498   │ 0           │ 0            │ 400         │
│ 500 │ 499   │ 0           │ 0            │ 400         │
│ 501 │ 500   │ 0           │ 0            │ 400         │, 0×0 DataFrame
)

The plot shows the population dynamics over time. Initially, wolves become extinct because they consume the sheep too quickly. The few remaining sheep reproduce and gradually reach an equilibrium that can be supported by the amount of available grass.

@df results plot(
    :step,
    :count_sheep,
    grid = false,
    xlabel = "Step",
    ylabel = "Population",
    label = "Sheep",
)
@df results plot!(:step, :count_wolves, label = "Wolves")
@df results plot!(:step, :count_grass, label = "Grass")

Altering the input conditions, we now see a landscape where all three agents find an equilibrium.

model = initialize_model(
    n_wolves = 20,
    dims = (25, 25),
    Δenergy_sheep = 5,
    sheep_reproduce = 0.2,
    wolf_reproduce = 0.08,
)
results, _ = run!(model, agent_step!, n_steps; adata = adata)
@df results plot(
    :step,
    :count_sheep,
    grid = false,
    xlabel = "Step",
    ylabel = "Population",
    label = "Sheep",
)
@df results plot!(:step, :count_wolves, label = "Wolves")
@df results plot!(:step, :count_grass, label = "Grass")