Example: Predator-Prey Model¶

This example replicates the great case study [1], which leverages the Lotka-Volterra equation [2] to describe the dynamics of Canada lynx (predator) and snowshoe hare (prey) populations. We will use the dataset obtained from [3] and run MCMC to get inferences about parameters of the differential equation governing the dynamics.

References:

```import argparse
import os

import matplotlib
import matplotlib.pyplot as plt

from jax.experimental.ode import odeint
import jax.numpy as jnp
from jax.random import PRNGKey

import numpyro
import numpyro.distributions as dist
from numpyro.examples.datasets import LYNXHARE, load_dataset
from numpyro.infer import MCMC, NUTS, Predictive

matplotlib.use("Agg")  # noqa: E402

def dz_dt(z, t, theta):
"""
Lotka–Volterra equations. Real positive parameters `alpha`, `beta`, `gamma`, `delta`
describes the interaction of two species.
"""
u = z[0]
v = z[1]
alpha, beta, gamma, delta = (
theta[..., 0],
theta[..., 1],
theta[..., 2],
theta[..., 3],
)
du_dt = (alpha - beta * v) * u
dv_dt = (-gamma + delta * u) * v
return jnp.stack([du_dt, dv_dt])

def model(N, y=None):
"""
:param int N: number of measurement times
:param numpy.ndarray y: measured populations with shape (N, 2)
"""
# initial population
z_init = numpyro.sample("z_init", dist.LogNormal(jnp.log(10), 1).expand([2]))
# measurement times
ts = jnp.arange(float(N))
# parameters alpha, beta, gamma, delta of dz_dt
theta = numpyro.sample(
"theta",
dist.TruncatedNormal(
low=0.0,
loc=jnp.array([1.0, 0.05, 1.0, 0.05]),
scale=jnp.array([0.5, 0.05, 0.5, 0.05]),
),
)
# integrate dz/dt, the result will have shape N x 2
z = odeint(dz_dt, z_init, ts, theta, rtol=1e-6, atol=1e-5, mxstep=1000)
# measurement errors
sigma = numpyro.sample("sigma", dist.LogNormal(-1, 1).expand([2]))
# measured populations
numpyro.sample("y", dist.LogNormal(jnp.log(z), sigma), obs=y)

def main(args):
_, fetch = load_dataset(LYNXHARE, shuffle=False)
year, data = fetch()  # data is in hare -> lynx order

# use dense_mass for better mixing rate
mcmc = MCMC(
NUTS(model, dense_mass=True),
num_warmup=args.num_warmup,
num_samples=args.num_samples,
num_chains=args.num_chains,
progress_bar=False if "NUMPYRO_SPHINXBUILD" in os.environ else True,
)
mcmc.run(PRNGKey(1), N=data.shape[0], y=data)
mcmc.print_summary()

# predict populations
pop_pred = Predictive(model, mcmc.get_samples())(PRNGKey(2), data.shape[0])["y"]
mu = jnp.mean(pop_pred, 0)
pi = jnp.percentile(pop_pred, jnp.array([10, 90]), 0)
plt.figure(figsize=(8, 6), constrained_layout=True)
plt.plot(year, data[:, 0], "ko", mfc="none", ms=4, label="true hare", alpha=0.67)
plt.plot(year, data[:, 1], "bx", label="true lynx")
plt.plot(year, mu[:, 0], "k-.", label="pred hare", lw=1, alpha=0.67)
plt.plot(year, mu[:, 1], "b--", label="pred lynx")
plt.fill_between(year, pi[0, :, 0], pi[1, :, 0], color="k", alpha=0.2)
plt.fill_between(year, pi[0, :, 1], pi[1, :, 1], color="b", alpha=0.3)
plt.gca().set(ylim=(0, 160), xlabel="year", ylabel="population (in thousands)")
plt.title("Posterior predictive (80% CI) with predator-prey pattern.")
plt.legend()

plt.savefig("ode_plot.pdf")

if __name__ == "__main__":
assert numpyro.__version__.startswith("0.13.2")
parser = argparse.ArgumentParser(description="Predator-Prey Model")
parser.add_argument("-n", "--num-samples", nargs="?", default=1000, type=int)
parser.add_argument("--num-warmup", nargs="?", default=1000, type=int)
parser.add_argument("--num-chains", nargs="?", default=1, type=int)
parser.add_argument("--device", default="cpu", type=str, help='use "cpu" or "gpu".')
args = parser.parse_args()

numpyro.set_platform(args.device)
numpyro.set_host_device_count(args.num_chains)

main(args)
```

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