We consider an ecological model consisting of two species of predators competing for their common prey with explicit interference competition. With a proper rescaling, the model is portrayed as a singularly perturbed system with one fast (prey dynamics) and two slow variables (dynamics of the predators). The model exhibits a variety of rich and interesting dynamics, including, but not limited to mixed-mode oscillations (MMOs), featuring concatenation of small and large amplitude oscillations, relaxation oscillations and bistability between a semi-trivial equilibrium state and a coexistent oscillatory state. More interestingly, in a neighborhood of singular Hopf bifurcation, long lasting transient dynamics in the form of chaotic MMOs or relaxation oscillations are observed as the system approaches the periodic attractor born out of supercritical Hopf bifurcation or a semi-trivial equilibrium state respectively. The transient dynamics could persist for hundreds or thousands of generations before the ecosystem experiences a regime shift. The time series of population cycles with different types of irregular oscillations arising in this model stem from a biological realistic feature, namely, by the variation in the intraspecific competition amongst the predators. To explain these oscillations, we use bifurcation analysis and methods from geometric singular perturbation theory. The numerical continuation study reveals the rich bifurcation structure in the system, including the existence of codimension-two bifurcations such as fold-Hopf and generalized Hopf bifurcations.
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Figure 2.
(a) The critical manifold
Figure 3.
One-parameter bifurcation of the desingularized system (12) as
Figure 4.
A one-parameter bifurcation diagram in
Figure 5.
A one-parameter bifurcation structure of MMOs corresponding to the inset in Figure 4. Here
Figure 6.
Numerical continuation of periodic orbits of system (5) as
Figure 7.
(a) Phase portrait of a chaotic MMO orbit. (b) Corresponding time series in
Figure 8.
(a)-(b) Phase space and time series of a MMO orbit of signature
Figure 9.
(a) A chaotic MMO orbit rotating around the weak eigendirection (red) after leaving the equilibrium (magenta) along its unstable eigendirection. The folded node is shown in cyan. (b) Corresponding time series in
Figure 10.
Time series in
Figure 11.
Transient basins of attraction for
Figure 12.
(a) A trajectory exhibiting MMOs while approaching
Figure 13.
The Poincaré section
Figure 15.
(a)-(b) Phase portrait and time series of a trajectory starting out at
Figure 16.
A two-parameter bifurcation diagram in
Figure 17.
(a)-(b) Phase space and time series of a trajectory for
Figure 19.
A two-parameter bifurcation diagram in
Figure 20.
(a)-(b) Transient temporal dynamics of system (5) asymptotically approaching to the equilibrium
Figure 21.
Bistable dynamics exhibited by system (5) at
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