About

Alvin Bayliss is a Professor of Engineering Sciences and Applied Mathematics and also holds a courtesy appointment in Electrical and Computer Engineering at Northwestern University. He serves as the Director of Graduate Studies for Engineering Sciences and Applied Mathematics. His research focuses on the numerical solution of partial differential equations, particularly those modeling ecological systems. He emphasizes the development of accurate and efficient numerical methods and their application to describe the dynamics of ecological systems, including adaptive numerical procedures for finite differences and spectral methods. Currently, his research is centered on computational ecology, where he develops, analyzes, and simulates models of multi-species ecological communities, with a particular interest in cyclic competition systems such as rock-paper-scissors models.

Research topics

• Computer Science
• Ecology
• Biology
• Artificial Intelligence
• Mathematics
• Applied mathematics
• Economics
• Physics
• Statistical physics
• Mathematical analysis
• Paleontology
• Biological system
• Microeconomics
• Geography

Selected publications

• Invasions in a four-species fractured cyclic ecological system

  Mathematical Biosciences · 2025-12-08

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• Fractured alliances in a four-species cyclic ecological system

  Physica D Nonlinear Phenomena · 2024-12-07 · 1 citations

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• Order and Disorder in a Cyclically Competitive Ecological Community

  SIAM Journal on Applied Dynamical Systems · 2023-02-14

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  We consider a deterministic model of a rock-paper-scissors cyclic ecological system. The model accounts for species mobility via Fickian diffusion, as well as interspecies interactions describing the cyclic competition scheme. The system admits both single-species equilibria (saddles) and a three-species coexistence equilibrium state. We focus on the regime where coexistence is unstable. When there is no spatial dependence, the solution of the corresponding ordinary differential equation system admits an attracting heteroclinic cycle consisting of orbits connecting the three single-species saddles in the sequence, to which most solutions with physical initial conditions asymptote. When there is spatial dependence (partial differential equation system), we consider mixed initial conditions (all species cohabit the same spatial region), as well as patch initial conditions, where each species is isolated from the other two, and show that, when interspecies competition is not strong, spatiotemporal chaotic behavior generally occurs. We propose a mechanism for the development of chaos—patch splitting—whereby sufficiently large patches are repeatedly split by the diffusion of competitors into the patch. However, when there is strong interspecies competition, ordered spatiotemporal patterns can occur. We consider both 1D patterns, corresponding to a community confined to a thin, circular annulus, and 2D geometries, simulating a petri dish. We show that, in 1 dimension, traveling arrays of single-species patches, as well as modulated traveling waves, consisting of patches which periodically expand and contract (breather modes), exist. In 2 dimensions, spirals, as well as localized patches chasing each other in the sequence, can occur. We consider length and temporal scales appropriate for a bacterial community in a laboratory setting, qualitatively modeling observed Escherichia coli cyclic bacterial systems.

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• Competing alliances in a four-species cyclic ecosystem

  Applied Mathematics and Computation · 2023-10-17 · 1 citations

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• Coflow filtration combustion waves

  Combustion Theory and Modelling · 2023-06-20 · 1 citations

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  Recently, it has been proposed to develop space power systems based on filtration combustion of metal powders with oxygen supplied by a chemical oxygen generator. The experiments with lithium and magnesium powders at natural infiltration of oxygen have shown propagation of both counterflow and coflow combustion waves. However, natural filtration combustion of metal powders at relatively low pressures is not sufficiently understood. In the present paper, we investigate the natural coflow combustion waves propagating through a porous medium. The porous matrix is made of metal particles that react with oxygen flowing from the open end of the sample to the reaction zone where it is consumed forming a condensed product which also has a porous structure. The gas flow is due to the pressure difference between the pressure at the open end and that in the reaction zone (the so-called natural filtration). The open end is where the sample is ignited, so that the gas flows through the reaction products, i.e. in the same direction as the combustion wave propagates (coflow filtration). Our mathematical model involves the conservation of energy equation and gas mass, solid reactant mass, and gas momentum balances, as well as an equation of state, and appropriate boundary and initial conditions. It is studied analytically under the combustion front approximation. When the reaction zone is close to the open end, there is sufficient amount of oxygen in the reaction zone and the reaction is controlled by kinetic factors (the kinetic regime of propagation). As the reaction moves away from the open end, it is gas supply that becomes a limiting factor (filtration regime). Both kinetic and filtration regimes of propagation as well the transition between them are analytically studied in this paper.

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• Nonlinear evolution of cellular flame instabilities

  2022-09-15

  book-chapter1st authorCorresponding

  We describe cellular dames in a cylindrical geometry and focus on rotating waves traveling along the dame front. We consider two distinctly different parameter ranges, adiabatic dames with Le < 1 and nonadiabatic dames with Le > 1, where the Lewis number Le is the ratio of thermal diffusivity to mass diffusivity of a dedcient component of the mixture. Both regimes exhibit transitions from uniformly rotating traveling waves (TWs) to modulated traveling waves (MTWs) and further transitions to more complex behavior, ultimately leading to chaos. We describe several different scenarios

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• Cyclic Ecological Systems with an Exceptional Species

  Applied Mathematics and Computation · 2022 · 2 citations

  • Computer Science
  • Ecology
  • Biology
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• Counterflow combustion waves in short samples of metal powders at natural filtration of oxygen

  Combustion Theory and Modelling · 2022-04-26 · 1 citations

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  Combustion of a porous solid fuel is considered. An exothermic reaction takes place between the fuel and a gaseous oxidiser which is delivered to the reaction zone by filtration through the pores in the sample from an open end toward which the combustion wave propagates (counterflow filtration). The gas reacts with the solid fuel to form a solid product. The gas filtration is due to the pressure difference between the ambient pressure at the open end and the pressure in the reaction zone where the gas is being consumed (referred to as natural filtration). A 1D mathematical model based on equations describing conservation of energy, gas mass, solid reactant mass, and gas momentum, as well as an equation of state, and appropriate boundary and initial conditions is formulated and analytically studied taking advantage of the separation of length scales in the process. When the reaction zone is sufficiently far from the open end, the combustion wave propagates at a constant speed and has a time-independent structure, while when the reaction is close to the open end (closer than the filtration length), the structure of the combustion wave and its speed become time dependent. Both cases are discussed in the paper though the main emphasis is on short samples, in which the combustion wave is affected by the gas flow from the open end during the entire propagation process. A specific example of interest involves magnesium as the solid fuel and oxygen as the gaseous oxidiser.

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• Asymptotic analysis of the bistable Lotka-Volterra competition-diffusion system

  Applied Mathematics and Computation · 2022 · 2 citations

  • Statistical physics
  • Mathematics
  • Applied mathematics
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• Beyond rock–paper–scissors systems — Deterministic models of cyclic ecological systems with more than three species

  Physica D Nonlinear Phenomena · 2020 · 19 citations

  1st authorCorresponding
  • Computer Science
  • Statistical physics
  • Artificial Intelligence
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Recent grants

• EMSW21-RTG: Applied Mathematics Training Program for Interdisciplinary Research in Science and Engineering

  NSF · $2.3M · 2007–2014

• A Model for Combustion in Strongly Stratified Environments

  NSF · $274k · 2002–2007

Frequent coauthors

• L. Maestrello

  59 shared

• B. J. Matkowsky

  56 shared

• Eli Turkel

  Tel Aviv University

  41 shared

• V. A. Volpert

  13 shared

• A. P. Aldushin

  11 shared

• Paresh Parikh

  Gujarat Kidney Foundation

  11 shared

• M. Minkoff

  9 shared

• Abdelkader Frendi

  University of Alabama in Huntsville

  9 shared

Labs

• Bayliss LaboratoryPI

Awards & honors

• Clarence ver Steeg Graduate Faculty Award
• McCormick Teaching Award
• ISI Highly Cited Researcher