Fig 1.
Description of model and fitness landscape.
(A) A schematic depiction of the “gradient dynamics” model, illustrating the model’s assumptions regarding the role of selection in the evolutionary dynamics. (B) The two distinct topologies of the density-dependent fitness landscape, for typical values of the chosen parameters and observed predator and prey densities in the numerical work described below. Color scale ranges between −1 (blue) and 1 (red).
Fig 2.
The onset of chaos with increasing prey evolutionary rate in a predator-prey system.
(A) As the timescale of the evolutionary dynamics, V, increases relative to the timescales of predator-prey interactions, the system undergoes a Hopf bifurcation from stable limit cycles (upper panel, ) to chaotic cycling (bottom panel,
). (B) The “teacup” strange attractor for the system in its chaotic state (
). Projections of the dynamics onto pairs of dynamical variables are shown in the inset. For this figure, a1 = 2.5, a2 = 0.05, d1 = 0.16, d2 = 0.004, b1 = 6,
, k1 = 6, k2 = 9, k4 = 9, ya = 8, Simulation time: 4 × 104.
Fig 3.
A gradual decrease in predator density remodels the fitness landscape, triggering transient cycling.
(A) A closeup of the dynamics for a typical entry into cycling. A high predator density y (magenta) slowly decays while the prey density x (turquoise) remains nearly constant. Once the predator density becomes small enough, the prey density abruptly increases, causing a decrease in the mean trait value (black) that provokes cycling. The right vertical axis ticks (x*, c*, etc.) correspond to analytical predictions for critical points, as described in the next section. (B) The fitness function r (from Eq (9)) computed for the values of (x, y,
) at the timepoints shown in (A), plotted with c as the vertical axis. Local minima (white) and maxima (black) in c are overlaid for the portion of the dynamics shown in (A). The color gradient is centered with white at 0 and the positive (red) and negative(blue) fitness values scaled by the log transform modulus. (C) The fitness function as a function of c at two representative timepoints (indicated by white dashed lines in (B)). All parameters as given in Fig 2B.
Fig 4.
Hysteresis and discontinuity induced by a first-order fitness phase transition.
(A) Under the fast equilibration approximation , an analytic phase diagram for the real parts of the locations of the local extrema of the fitness function as a function of the prey density, x. The red “forward” overlay indicates the apparent jump in the dynamics as x increases; the blue “backward” overlay indicates the apparent jump as x decreases. Dashed line indicates unstable equilibria. All parameters are as given in Fig 2B.
Fig 5.
Lyapunov exponents associated with the chaotic dynamics.
(A) The eigenvalues of the Jacobian evaluated during a portion of the dynamics in which the system enters a series of chaotic cycles. The global Lyapunov exponent is underlaid as a dashed black line. (B) The distribution of the lengths of periods in the dynamics during which the largest local Lyapunov exponent is greater than the threshold value 0.2, chosen to correspond to periods of strongly disruptive selection in which the exponent takes values an order of magnitude greater than its median across the time series. Simulation time 2 × 105, other parameters are as given in Fig 2B.