Complex oscillatory patterns in a three-timescale model of a generalist predator and a specialist predator competing for a common prey

Figure 1.  Two-parameter bifurcation diagram of (5) showing transitions between different dynamical regimes for all parameter values as in (8). See text for details. HB: Hopf bifurcation, TR: torus bifurcation

Figure 2.  Mixed-mode time series of (6) in intermediate time for $ \beta_2 = 0.01 $, $ \alpha = 0.75 $ and other parameters values as in (8). (A) For $ x $-component. (B) For $ y $ and $ z $ components

Figure 3.  Bursting patterns in system (6) for $ \beta_2 = 0.0245 $, $ \alpha = 0.8 $ and other parameters values as in (8). (A) For $ x $-component. (B) For $ y $ and $ z $ components

Figure 4.  Geometric structure and stable periodic orbits $ \Gamma $ of system (6) for different values of $ \beta_2 $ and $ \alpha $ and other parameters as in (8). Shown are the critical manifold $ \mathcal{M}_1 = \Pi \cup S $, the superslow manifold $ \mathcal{Z} $ (red) and the coexistence equilibrium state (blue dot). The plane $ \Pi $ is divided into an attracting component $ \Pi^a $ and a repelling component $ \Pi^r $ joined at the transcritical curve $ TC $. The surface $ S $ can consist of one or more attracting and repelling sheets joined at the folds $ \mathcal{F}^{0} $ and $ \mathcal{F}^{\pm} $ (also see table 5). Note that $ \Gamma $ displays three distinct timescales during the course of its cycle. (A) $ \beta_2 = 0.005 $ and $ \alpha = 0.6 $. (B) $ \beta_2 = 0.0245 $ and $ \alpha = 0.8 $

Figure 5.  Shape of the fold curve $ \mathcal{F} $ with respect to $ \alpha $ and $ \beta_2 $. Here $ {\mu}_1, {\mu}_2 $ are the roots of $ \mu(x) = 0 $. See Appendix for the details

Figure 6.  Geometric structure of the surface $ S $ of system (6) for different values of $ \beta_2 $ with $ \alpha = 0.6 $ and other parameters as in (8). The black dotted lines divide $ S $ into distinct regions, characterized by the number of attracting and repelling branches that the surface possesses. (A) The surface $ S $ is divided into four regions for $ \beta_2 = 0.048 $. Also shown are different cross-sections of $ S $ (in red) for constant $ y $ values chosen from each region. (B) The surface $ S $ is divided into three regions for $ \beta_2 = 0.025 $

Figure 7.  Two-parameter bifurcation structure of the desingularized system (14). The FSN Ⅰ (a) and FSN Ⅰ (b) curves represent folded saddle-node bifurcations of type Ⅰ, while FSN Ⅱ (a) and FSN Ⅱ (b) curves represent folded saddle-node bifurcations of type Ⅱ. See text for details

Figure 8.  Real parts of eigenvalues of the layer problem (15) for $ \beta_2 = 0.005 $ and $ \alpha = 0.6 $. The degenerate nodes are marked by black dots and the delayed Hopf bifurcation points by magenta dots

Figure 9.  Two parameter bifurcation diagram of (15) in $ (z, \beta_2) $ parameter plane with $ \alpha = 1 $. The HC curves intersect with each other at the "gluing bifurcation" denoted by a star and intersect with the $ SN_f $ curve at SNSL points denoted by dots. The inset shows a qualitative representation of the region around the gluing bifuraction. $ SN_f $ - saddle-node bifurcation, $ SN_p $- saddle-node of periodics, H - Hopf bifurcation, GH - generalized Hopf, HC - small homoclinic bifurcation, BHC - big homoclinic bifurcation, SNSL - saddle-node separatrix loop bifurcation

Figure 10.  One parameter bifurcation diagram of (6) with respect to $ \beta_2 $ with other parameters as in (8) and $ \alpha = 0.75 $. Filled green (open blue) circles represent maximum and minimum values of $ x $ in stable (unstable) limit cycles. H - Hopf bifurcation, PD- period-doubling bifurcation, TR - torus bifurcation

Figure 11.  Time profiles of the $ x $-components of solutions of system (6) for $ \alpha = 0.75 $ and other parameter values as in (8). (A) $ \beta_2 = 0.0053 $. (B) $ \beta_2 = 0.00536 $

Figure 12.  Time series for the $ x $-coordinate of the orbits of system (6) for $ \beta_2 = 0.005 $ and varying $ \alpha $. The other parameter values are as in (8). Note the change in profile of the MMO patterns. Top-left panel: MMOs spending prolonged time near $ \mathcal{F}^+ $ with SAOs near $ \mathcal{F}^0 $; top-right and bottom-left panels: MMOs spending prolonged time near $ \mathcal{F}^0 $ with SAOs near $ \mathcal{F}^0 $; bottom-right panel: mixed-type torus canard orbit

Figure 13.  (A) Overlay of the projection of trajectory $ \Gamma $ of (6) onto the $ (z, x) $ - plane with the bifurcation diagram of the fast subsystem (15) for $ \beta_2 = 0.005 $. (A) $ \alpha = 0.425 $ (B) $ \alpha = 0.6 $ (C) $ \alpha = 0.68 $ (D) $ \alpha = 0.742 $. Shown are the fold curves $ \mathcal{F}^{\pm}, \mathcal{F}^0 $ (yellow), the z-curves $ \mathcal{Z}^{\pm}, \mathcal{Z}^{0} $ (red), folded node (grey dot), equilibrium point $ E^* $ (cyan dot), delayed Hopf points (magenta dots) and degenerate nodes (black dots) of (15). Note that the trajectory exhibits delayed Hopf bifurcation on the lower branch of $ \mathcal{Z} $ in panels (B)-(D). FN - folded node, SH - subcritical Hopf, SNP - saddle-node bifurcation of periodic orbit, HC - homoclinic bifurcation, $ \mathcal{Z} $ - the superslow curve. Dashed lines denote instability. Open blue circles represent maximum and minimum values of $ x $ in unstable limit cycles

Figure 14.  (A) Mixed-mode oscillations exhibited by system (6) for parameter values $ \beta_2 = 0.01 $ and $ \alpha = 0.75 $. (B) Zoomed view of the dynamics near the lower fold $ \mathcal{F}^0 $ on the $ xz $-plane. The trajectory enters into the singular funnel on $ \mathcal{S}^{-a} $ shown by the shaded region and filters through the the folded-node singularity (FN) while passing close to the delayed-Hopf bifurcation point (DHB). The local vector field of the equilibrium $ E^* $ further influences its dynamics before it jumps to $ S^{+a} $

Figure 15.  Time series for the $ x $-coordinate of the orbits of system (6) for $ \beta_2 = 0.0245 $ and varying $ \alpha $. Note the transition from MMO patterns to bursting oscillations. Top-left panel: MMOs with SAOs near $ \mathcal{F}^+ $; top-right and bottom-left panels: large-amplitude oscillations featuring two-timescales; bottom-right panel: subcritical elliptic bursting

Figure 16.  (A) Overlay of the projection of the trajectory $ \Gamma $ of (6) onto the $ (z, x) $ - plane with the bifurcation diagram of the fast subsystem (15) for $ \beta_2 = 0.0245 $ and varying $ \alpha $. Note the transition from spiking to bursting patterns in the system. (A) $ \alpha = 0.4645 $ (B) $ \alpha = 0.526 $ (C) $ \alpha = 0.6 $ (D) $ \alpha = 0.8 $. $ SN_p $ - saddle-node of periodics, BHC - big homoclinic loop, remaining labels and curves are as in figure 13. Filled green (open blue) circles represent maximum and minimum values of $ x $ in stable (unstable) limit cycles

Figure 17.  (A) Mixed-mode oscillations exhibited by system (6) for parameter values $ \beta_2 = 0.0245 $ and $ \alpha = 0.4645 $. (B) Zoomed view of the dynamics near the upper fold $ \mathcal{F}^+ $. The trajectory enters into the singular funnel on $ \mathcal{S}^{+a} $ shown by the shaded region and filters through the the folded-node singularity (FN) while making its way to the delayed-Hopf bifurcation point (DHB). The local vector field of the equilibrium $ E^* $ further influences its dynamics before it jumps to $ S^{-a} $