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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

\[ \begin{aligned} \dot{y_{0}} &= y_{0} \left(\frac{a_{1} y_{2} \left(- k_{1} y_{0} \left(\delta y_{2} - y_{2}\right) + 1\right)}{b_{1} y_{2} + 1} - \frac{a_{3} y_{1}}{b_{2} y_{0} + 1} - d_{1} \left(- k_{2} \left(\delta^{2} y_{2}^{2} - y_{2}^{2}\right) + k_{4} \left(\delta^{4} y_{2}^{4} - y_{2}^{4}\right) + 1\right)\right) \\ \dot{y_{1}} &= y_{1} \left(\frac{a_{2} y_{0}}{b_{2} y_{0} + 1} - d_{2}\right) \\ \dot{y_{2}} &= vv \left(- \frac{a_{1} \delta k_{1} y_{0} y_{2}}{b_{1} y_{2} + 1} - d_{1} \left(4 \delta^{4} k_{4} y_{2}^{3} - 2 \delta^{2} k_{2} y_{2}\right)\right) \end{aligned} \]

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

Properties

Lyapunov spectrum
$+0.001047,\; +0.00626,\; -0.9468$
computed at build
Kaplan–Yorke dimension
$D_{KY} = 2.008$
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
7 equilibria
0 stable · 7 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.CoevolvingPredatorPrey()
traj = sys.integrate(final_time=100.0, dt=0.01)

exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)

Reference

Gilpin & Feldman (2017), PLoS Comput. Biol. 13, e1005644

BibTeX 
@misc{coevolvingpredatorprey,
  title = {CoevolvingPredatorPrey system},
  note = {Gilpin & Feldman (2017), PLoS Comput. Biol. 13, e1005644}
}