The impact of additive noise on the spatial patterns emerging in a predator-prey model with a fear effect and Holling-type I function system is explored in this paper. The amplitude equation is obtained when the system undergoes Hopf bifurcation via the weakly nonlinear analysis around the Hopf bifurcation parameters. Through theoretical analysis, a threshold value of the noise intensity is obtained. This implies that the additive noise can govern the shift of the Hopf bifurcation point. We also investigated the influence of noise on Turing instability. In addition, numerical simulations were performed to validate the accuracy of the theory.
The main focus of this paper is to investigate the dynamical behaviors of a predator–prey model incorporating cooperative hunting among predators, which is widely observed in real ecological communities. Supposing identical temporal evolution for prey and predator populations, the present paper examines saddle-node, transcritical, Hopf, cusp, and codimension-2 Bogdanov–Takens bifurcations. In view of the realistic ecological fact that the growth rate of prey populations is considerably larger than the mortality rate of predator populations, the model is further extended to a slow–fast system. With the aid of geometric singular perturbation theory and asymptotic expansion techniques, the singular Hopf bifurcation and canard explosion are systematically analyzed. Finally, we investigate the transient dynamics. Theoretical results are then illustrated and verified by numerical simulations. The current paper improves the understanding of sophisticated predator–prey dynamical interactions within single and multi-time-scale frameworks, and it provides valuable insights for revealing population persistence and community stability in natural ecosystems.
In this work, a reduced model is proposed for the description of combustion of rich hydrogen–air mixtures. The detailed reaction mechanism is reduced to a system, described by partial differential equations governing the evolution of the profiles of H, O$_2$, and temperature. For all the radicals except H, a quasi-steady-state assumption is introduced, so that their profiles are determined from algebraic relations without the need to solve additional differential equations. The model is tested on two classical configurations: freely propagating one-dimensional flames and flames stabilized on a flat porous burner. Both steady solutions and diffusive–thermal pulsations are investigated. It is shown that, for sufficiently high equivalence ratios, the model reproduces flame structure and agrees with detailed kinetic models in terms of propagation speed and unsteady dynamics. The approach combines accuracy of detailed kinetic models with computational efficiency and is suitable for analyzing combustion instabilities.
In this paper, we study an incompressible viscous fluid flowing through a periodically constricted pipe with thickness Omicron(epsilon). Starting from the Navier-Stokes system with moderately large Reynolds number Re, we propose a new mathematical model in a form of the non-linear cubic law describing the relation between the effective velocity and the pressure drop. To accomplish that, we use homogenization and boundary-layer analysis with respect to the local Reynolds number epsilon Re. The derived higher-order asymptotic solution, acknowledging the effects of the flow inertia and pipe's roughness, is justified via error estimates.
In this paper, we study a model about non-toxic phytoplankton (NTP), toxin-producing phytoplankton (TPP), and zooplankton defined on a two-dimensional circular domain with predator-taxis and time-delay. The relationship between NTP, TPP and zooplankton is explored by combining selective grazing, cannibalism, Allee effect and additional food. For the proposed model, a rigorous proof of global solution existence with uniform boundedness is provided. By applying the Brouwer fixed-point theorem, we demonstrate the existence of a positive constant steady-state solution. Within the functional analytic framework, we establish conditions for both standard and equivariant Hopf bifurcations occurring in the neighborhood of the spatially homogeneous equilibrium. Next we obtain the existence condition and specific properties of periodic solutions by bifurcation theory and normal form method. Numerically, we select appropriate parameters with practical significance and validate analytical predictions via computational experiments. Specifically, we show different types of rotating waves and standing waves, which are the rather complex dynamic behaviors in the two-dimensional circular domain. And then we reveal the periodic phenomena of NTP, TPP and zooplankton, as well as biological phenomena caused by selective grazing, cannibalism, Allee effect and additional food, which provides valuable results for monitoring and controlling the distribution of plankton populations and protecting aquatic ecosystems.
