A Case Study of Multiple Wave Solutions in a Reaction-Diffusion System Using Invariant Manifolds and Global Bifurcations

Villar-Sepúlveda, Edgardo; Aguirre, Pablo; Breña‐Medina, Víctor F.

doi:10.1137/22m1474709

SIAM J. Appl. Dyn. Syst.

2023

DOI: 10.1137/22m1474709

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A Case Study of Multiple Wave Solutions in a Reaction-Diffusion System Using Invariant Manifolds and Global Bifurcations

Edgardo Villar-Sepúlveda

Pablo Aguirre

Víctor F. Breña‐Medina

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Cited by 3 publications

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The effect of nonlocal interaction on chaotic dynamics, Turing patterns, and population invasion in a prey–predator model

Yadav,

Pal,

Sen

2023

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Pattern formation is a central process that helps to understand the individuals’ organizations according to different environmental conditions. This paper investigates a nonlocal spatiotemporal behavior of a prey–predator model with the Allee effect in the prey population and hunting cooperation in the predator population. The nonlocal interaction is considered in the intra-specific prey competition, and we find the analytical conditions for Turing and Hopf bifurcations for local and nonlocal models and the spatial-Hopf bifurcation for the nonlocal model. Different comparisons have been made between the local and nonlocal models through extensive numerical investigation to study the impact of nonlocal interaction. In particular, a legitimate range of nonlocal interaction coefficients causes the occurrence of spatial-Hopf bifurcation, which is the emergence of periodic patterns in both time and space from homogeneous periodic solutions. With an increase in the range of nonlocal interaction, the whole Turing pattern suppresses after a certain threshold, and no pure Turing pattern exists for such cases. Specifically, at low diffusion rates for the predators, nonlocal interaction in the prey population leads to the extinction of predators. As the diffusion rate of predators increases, impulsive wave solutions emerge in both prey and predator populations in a one-dimensional spatial domain. This study also includes the effect of nonlocal interaction on the invasion of populations in a two-dimensional spatial domain, and the nonlocal model produces a patchy structure behind the invasion where the local model predicts only the homogeneous structure for such cases.

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The effect of nonlocal interaction on chaotic dynamics, Turing patterns, and population invasion in a prey–predator model

Yadav,

Pal,

Sen

2023

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

show abstract

Bifurcation and onset of chaos in an eco-epidemiological system with the influence of time delay

Pandey,

Das,

Ghosh

et al. 2024

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In the present article, we investigated a delay-based eco-epidemic prey–predator system in the presence of environmental fluctuations where predators engage with susceptible and infected prey, adopting Holling type II and ratio-dependent functional responses, respectively. During the study of the considered model, we identify each potential equilibrium point and its local stability criterion. The basic reproduction number has been computed, and the backward bifurcation about the disease-free equilibrium point was analyzed. The article illustrates Hopf bifurcation, global stability at the endemic equilibrium point, and their graphical depiction. We look over the variations in the dynamics of non-delay, delayed, and stochastic systems, revealing that a fixed level of temporal delay results in chaotic motion for the increasing strength of the saturation constant yet is potentially controlled by the predator growth rate. To study the dynamic behavior of the solution of the considered system and verify all theoretical results, we use numerical simulation and minutely analyze the influence of model parameters on the solution of the considered system. The stochastic transition is studied by varying the strength of stochastic fluctuation and the effect of delay.

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Local and global analysis of a discrete model describing the second-order digital filter with nonlinear signal processors

Yang,

Li,

Yang

et al. 2024

Int. J. Mod. Phys. C

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The paper devotes to the synthesis of local and global analysis of a discrete model describing the second-order digital filter with nonlinear signal processors. The discrete model gives rise to a two-dimensional non-invertible map, whose basins of attraction have complicated topological structures due to the intrinsic multi-stability. The influences of joint parameters on the local dynamics are presented in great details. Both theoretical and numerical results are plotted on the two-dimensional parametric planes. To show more detailed bifurcation structure, the isoclines are extended to higher periodic orbits for detecting the cusps of resonant entrainments. Invariant manifolds and critical curves are employed to illustrate the global dynamics of the model vividly. The tangency and intersections of invariant manifolds expound the process of erosions of basins of attraction. The global bifurcations of basins of attraction are deduced dynamically by critical curves.

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A Case Study of Multiple Wave Solutions in a Reaction-Diffusion System Using Invariant Manifolds and Global Bifurcations

Cited by 3 publications

References 44 publications

The effect of nonlocal interaction on chaotic dynamics, Turing patterns, and population invasion in a prey–predator model

The effect of nonlocal interaction on chaotic dynamics, Turing patterns, and population invasion in a prey–predator model

Bifurcation and onset of chaos in an eco-epidemiological system with the influence of time delay

Local and global analysis of a discrete model describing the second-order digital filter with nonlinear signal processors

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