8 research outputs found
Zero-hopf bifurcation in the volterra-gause system of predator-prey type
We prove that the Volterra-Gause system of predator-prey type exhibits 2 kinds of zero-Hopf bifurcations for convenient values of their parameters. In the first, 1 periodic solution bifurcates from a zero-Hopf equilibrium, and in the second, 4 periodic solutions bifurcate from another zero-Hopf equilibrium. This study is done using the averaging theory of second order
Canard Trajectories in 3D piecewise linear systems
We present some results on singularly perturbed piecewise linear systems, similar to those obtained by the Geometric Singular Perturbation Theory. Unlike the differentiable case, in the piecewise linear case we obtain the global expression of the slow manifold Sε. As a result, we characterize the existence of canard orbits in such systems. Finally, we apply the above theory to a specific case where we show numerical evidences of the existence of a canard cycle
Chaotic dynamics in a simple predator-prey model with discrete delay
A discrete delay is included to model the time between the capture of the
prey and its conversion to viable biomass in the simplest classical Gause type
predator-prey model that has equilibrium dynamics without delay. As the delay
increases from zero, the coexistence equilibrium undergoes a supercritical Hopf
bifurcation, two saddle-node bifurcations of limit cycles, and a cascade of
period doublings, eventu1ally leading to chaos. The resulting periodic orbits
and the strange attractor resemble their counterparts for the Mackey-Glass
equation. Due to the global stability of the system without delay, these
complicated dynamics can be solely attributed to the introduction of the delay.
Since many models include predator-prey like interactions as submodels, this
study emphasizes the importance of understanding the implications of
overlooking delay in such models on the reliability of the model-based
predictions, especially since the temperature is known to have an effect on the
length of certain delays.Comment: This paper has 28 pages, 12 figures and has been accepted to DCDS-B.
Please cite the journal version once it is published in DCDS-B. Appreciate
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Robust computational methods to simulate slow-fast dynamical systems governed by predator-prey models
Philosophiae Doctor - PhDNumerical approximations of multiscale problems of important applications in ecology
are investigated. One of the class of models considered in this work are singularly perturbed
(slow-fast) predator-prey systems which are characterized by the presence of a
very small positive parameter representing the separation of time-scales between the
fast and slow dynamics. Solution of such problems involve multiple scale phenomenon
characterized by repeated switching of slow and fast motions, referred to as relaxationoscillations,
which are typically challenging to approximate numerically. Granted with
a priori knowledge, various time-stepping methods are developed within the framework
of partitioning the full problem into fast and slow components, and then numerically
treating each component differently according to their time-scales. Nonlinearities that
arise as a result of the application of the implicit parts of such schemes are treated by
using iterative algorithms, which are known for their superlinear convergence, such as
the Jacobian-Free Newton-Krylov (JFNK) and the Anderson’s Acceleration (AA) fixed
point methods