# 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 position 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
mutable struct Sheep <: AbstractAgent
id::Int
pos::Dims{2}
energy::Float64
reproduction_prob::Float64
Δenergy::Float64
end
mutable struct Wolf <: AbstractAgent
id::Int
pos::Dims{2}
energy::Float64
reproduction_prob::Float64
Δenergy::Float64
end
mutable struct Grass <: AbstractAgent
id::Int
pos::Dims{2}
fully_grown::Bool
regrowth_time::Int
countdown::Int
end
```

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, 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, periodic = false)
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 p in positions(model)
id += 1
fully_grown = rand(model.rng, Bool)
countdown = fully_grown ? regrowth_time : rand(model.rng, 1:regrowth_time) - 1
grass = Grass(id, (0, 0), fully_grown, regrowth_time, countdown)
add_agent!(grass, p, 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 position 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 = collect(agents_in_position(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(model.rng) <= sheep.reproduction_prob
reproduce!(sheep, model)
end
end
function agent_step!(wolf::Wolf, model)
move!(wolf, model)
wolf.energy -= 1
agents = collect(agents_in_position(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(model.rng) <= 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 position with the `move!`

function.

```
function move!(agent, model)
neighbors = nearby_positions(agent, model)
position = rand(model.rng, collect(neighbors))
move_agent!(agent, position, 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(model.rng, 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.

```
using InteractiveDynamics
using AbstractPlotting
import CairoMakie
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 size (20, 20), metric=chebyshev and periodic=false 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) = '⚫'
mshape(a::Wolf) = '▲'
mshape(a::Grass) = '■'
mcolor(a::Sheep) = RGBAf0(1.0, 1.0, 1.0, 0.8)
mcolor(a::Wolf) = RGBAf0(0.2, 0.2, 0.2, 0.8)
mcolor(a::Grass) = cgrad([:brown, :green])[a.countdown/a.regrowth_time]
figure, = abm_plot(
model;
resolution = (800, 600),
offset = offset,
am = mshape,
as = 22,
ac = mcolor,
scheduler = by_type((Grass, Sheep, Wolf), false),
equalaspect = true,
)
figure
```

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)
```

(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 8 │ 7 41 71 136 ⋮ │ ⋮ ⋮ ⋮ ⋮ 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 0 400 486 rows omitted, 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.

```
figure = Figure(resolution = (600, 400))
ax = figure[1, 1] = Axis(figure; xlabel = "Step", ylabel = "Population")
sheepl = lines!(ax, results.step, results.count_sheep, color = :blue)
wolfl = lines!(ax, results.step, results.count_wolves, color = :orange)
grassl = lines!(ax, results.step, results.count_grass, color = :green)
figure[1, 2] = Legend(figure, [sheepl, wolfl, grassl], ["Sheep", "Wolves", "Grass"])
figure
```

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)
figure = Figure(resolution = (600, 400))
ax = figure[1, 1] = Axis(figure, xlabel = "Step", ylabel = "Population")
sheepl = lines!(ax, results.step, results.count_sheep, color = :blue)
wolfl = lines!(ax, results.step, results.count_wolves, color = :orange)
grassl = lines!(ax, results.step, results.count_grass, color = :green)
figure[1, 2] = Legend(figure, [sheepl, wolfl, grassl], ["Sheep", "Wolves", "Grass"])
figure
```