CoevolvingPredatorPrey

Systems / ODEs / Population dynamics

An eco-evolutionary predator-prey model where a fast-evolving prey trait feeds back on the populations, generating chaos without external forcing.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= x \left(\frac{a_{1} \alpha \left(- k_{1} x \left(\alpha \delta - \alpha\right) + 1\right)}{\alpha b_{1} + 1} - \frac{a_{3} y}{b_{2} x + 1} - d_{1} \left(- k_{2} \left(\alpha^{2} \delta^{2} - \alpha^{2}\right) + k_{4} \left(\alpha^{4} \delta^{4} - \alpha^{4}\right) + 1\right)\right) \\ \dot{y} &= y \left(\frac{a_{2} x}{b_{2} x + 1} - d_{2}\right) \\ \dot{alpha} &= vv \left(- \frac{a_{1} \alpha \delta k_{1} x}{\alpha b_{1} + 1} - d_{1} \left(4 \alpha^{3} \delta^{4} k_{4} - 2 \alpha \delta^{2} k_{2}\right)\right) \end{aligned} \]

Parameters

Symbol	Default	Role
a1	2.5	predation / conversion rate coefficient
a2	0.05	predation / conversion rate coefficient
a3	0.4	predation / conversion rate coefficient
b1	6	saturation constant of the functional response
b2	1.333	saturation constant of the functional response
d1	0.16	prey intrinsic death rate
d2	0.004	predator intrinsic death rate
delta	1	trait-asymmetry parameter of the fitness landscape
k1	6	coupling strength of the fitness landscape
k2	9	coupling strength of the fitness landscape
k4	9	coupling strength of the fitness landscape
vv	0.33333	relative timescale of trait evolution vs population dynamics

State variables: x, y, alpha

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = 2 d_{1} \delta^{2} k_{2} vv - d_{1} - d_{2}$
constant part shown — state-dependent remainder omitted (too large)

Equilibria

8 equilibria
0 stable · 8 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.CoevolvingPredatorPrey()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference

Gilpin & Feldman (2017), PLoS Comput. Biol. 13, e1005644
doi:10.1371/journal.pcbi.1005644

BibTeX

@misc{coevolvingpredatorprey,
  title = {CoevolvingPredatorPrey system},
  note = {Gilpin & Feldman (2017), PLoS Comput. Biol. 13, e1005644},
  doi = {10.1371/journal.pcbi.1005644}
}