23 research outputs found
Multiple wave solutions in a diffusive predator-prey model with strong Allee effect on prey and ratio-dependent functional response
A thorough analysis is performed in a predator-prey reaction-diffusion model
which includes three relevant complex dynamical ingredients: (a) a strong Allee
effect; (b) ratio-dependent functional responses; and (c) transport attributes
given by a diffusion process. As is well-known in the specialized literature,
these aspects capture adverse survival conditions for the prey, predation
search features and non-homogeneous spatial dynamical distribution of both
populations. We look for traveling-wave solutions and provide rigorous results
coming from a standard local analysis, numerical bifurcation analysis, and
relevant computations of invariant manifolds to exhibit homoclinic and
heteroclinic connections and periodic orbits in the associated dynamical system
in . In so doing, we present and describe a diverse zoo of traveling wave
solutions; and we relate their occurrence to the Allee effect, the spreading
rates and propagation speed. In addition, homoclinic chaos is manifested via
both saddle-focus and focus-focus bifurcations as well as a Belyakov point. An
actual computation of global invariant manifolds near a focus-focus homoclinic
bifurcation is also presented to enravel a multiplicity of wave solutions in
the model. A deep understanding of such ecological dynamics is therefore
highlighted.Comment: 35 pages, 22 figure
Dynamic Analysis of a Phytoplankton-Fish Model with Biological and Artificial Control
We investigate a nonlinear model of the interaction between phytoplankton and fish, which uses a pair of semicontinuous systems with biological and artificial control. First, the existence of an order-1 periodic solution to the system is analyzed using a Poincaré map and a geometric method. The stability conditions of the order-1 periodic solution are obtained by a theoretical mathematical analysis. Furthermore, based on previous analysis, we investigate the bifurcation in the order-1 periodic solution and prove that the order-1 periodic solution breaks up an order-1 periodic solution at least. In addition, the transcritical bifurcation of the system is described. Finally, we provide a series of numerical results that illustrate the feasibility of the theoretical results. Based on the theoretical and numerical results, we analyzed the feasibility of biological and artificial control, which showed that biological and artificial methods can control phytoplankton blooms. These results are expected to be useful for the study of phytoplankton dynamics in aquatic ecosystems
Impulsive perturbations to differential equations: stable/unstable pseudo-manifolds, heteroclinic connections, and flux
State-dependent time-impulsive perturbations to a two-dimensional autonomous
flow with stable and unstable manifolds are analysed by posing in terms of an
integral equation which is valid in both forwards- and backwards-time. The
impulses destroy the smooth invariant manifolds, necessitating new definitions
for stable and unstable pseudo-manifolds. Their time-evolution is characterised
by solving a Volterra integral equation of the second kind with discontinuous
inhomogeniety. A criteria for heteroclinic trajectory persistence in this
impulsive context is developed, as is a quantification of an instantaneous flux
across broken heteroclinic manifolds. Several examples, including a kicked
Duffing oscillator and an underwater explosion in the vicinity of an eddy, are
used to illustrate the theory