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Swailem M, Täuber UC. Computing macroscopic reaction rates in reaction-diffusion systems using Monte Carlo simulations. Phys Rev E 2024; 110:014124. [PMID: 39160995 DOI: 10.1103/physreve.110.014124] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2024] [Accepted: 06/21/2024] [Indexed: 08/21/2024]
Abstract
Stochastic reaction-diffusion models are employed to represent many complex physical, biological, societal, and ecological systems. The macroscopic reaction rates describing the large-scale, long-time kinetics in such systems are effective, scale-dependent renormalized parameters that need to be either measured experimentally or computed by means of a microscopic model. In a Monte Carlo simulation of stochastic reaction-diffusion systems, microscopic probabilities for specific events to happen serve as the input control parameters. To match the results of any computer simulation to observations or experiments carried out on the macroscale, a mapping is required between the microscopic probabilities that define the Monte Carlo algorithm and the macroscopic reaction rates that are experimentally measured. Finding the functional dependence of emergent macroscopic rates on the microscopic probabilities (subject to specific rules of interaction) is a very difficult problem, and there is currently no systematic, accurate analytical way to achieve this goal. Therefore, we introduce a straightforward numerical method of using lattice Monte Carlo simulations to evaluate the macroscopic reaction rates by directly obtaining the count statistics of how many events occur per simulation time step. Our technique is first tested on well-understood fundamental examples, namely, restricted birth processes, diffusion-limited two-particle coagulation, and two-species pair annihilation kinetics. Next we utilize the thus gained experience to investigate how the microscopic algorithmic probabilities become coarse-grained into effective macroscopic rates in more complex model systems such as the Lotka-Volterra model for predator-prey competition and coexistence, as well as the rock-paper-scissors or cyclic Lotka-Volterra model and its May-Leonard variant that capture population dynamics with cyclic dominance motifs. Thereby we achieve a more thorough and deeper understanding of coarse graining in spatially extended stochastic reaction-diffusion systems and the nontrivial relationships between the associated microscopic and macroscopic model parameters, with a focus on ecological systems. The proposed technique should generally provide a useful means to better fit Monte Carlo simulation results to experimental or observational data.
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Affiliation(s)
- Mohamed Swailem
- Department of Physics & Center for Soft Matter and Biological Physics, MC 0435, Robeson Hall, 850 West Campus Drive, Virginia Tech, Blacksburg, Virginia 24061, USA
| | - Uwe C Täuber
- Department of Physics & Center for Soft Matter and Biological Physics, MC 0435, Robeson Hall, 850 West Campus Drive, Virginia Tech, Blacksburg, Virginia 24061, USA
- Faculty of Health Sciences, Virginia Tech, Blacksburg, Virginia 24061, USA
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Lu Y, Wang X, Du C, Wang Y, Geng Y, Shi L, Park J. Understanding the role of neutral species by means of high-order interaction in the rock-paper-scissors dynamics. Phys Rev E 2024; 109:014313. [PMID: 38366519 DOI: 10.1103/physreve.109.014313] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2023] [Accepted: 01/05/2024] [Indexed: 02/18/2024]
Abstract
The existence of neutral species carries profound ecological implications that warrant further investigation. In this paper, we study the impact of neutral species on biodiversity in a spatial tritrophic system of cyclic competition, in which the neutral species are identified as the fourth species that may affect the competition process of the other three species under the rock-paper-scissors (RPS) rule. Extensive simulations showed that neutral species can promote coexistence in a high mobility regime within the system. When coexistence occurs, we found that the state can be maintained by two mechanisms: Species can either (i) adhere to traditional RPS rule or (ii) form patches to resist invasion. Our findings might aid in understanding the impact of neutral species on biodiversity in ecosystems.
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Affiliation(s)
- Yikang Lu
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
- Institute for Biocomputation and Physics of Complex Systems, University of Zaragoza, 50018 Zaragoza, Spain
| | - Xiaoyue Wang
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
| | - Chunpeng Du
- School of Mathematics, Kunming University, Kunming, 650214, China
| | - Yanan Wang
- Institute for Biocomputation and Physics of Complex Systems, University of Zaragoza, 50018 Zaragoza, Spain
- School of Economics and Management, Beihang University, Beijing 100191, China
| | - Yini Geng
- School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China
| | - Lei Shi
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
- Interdisciplinary Research Institute of Data Science, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China
| | - Junpyo Park
- Department of Applied Mathematics, College of Applied Sciences, Kyung Hee University, Yongin 17104, Republic of Korea
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Szolnoki A, Chen X. Emerging solutions from the battle of defensive alliances. Sci Rep 2023; 13:8472. [PMID: 37231065 DOI: 10.1038/s41598-023-35746-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/21/2023] [Accepted: 05/23/2023] [Indexed: 05/27/2023] Open
Abstract
Competing strategies in an evolutionary game model, or species in a biosystem, can easily form a larger unit which protects them from the invasion of an external actor. Such a defensive alliance may have two, three, four or even more members. But how effective can be such formation against an alternative group composed by other competitors? To address this question we study a minimal model where a two-member and a four-member alliances fight in a symmetric and balanced way. By presenting representative phase diagrams, we systematically explore the whole parameter range which characterizes the inner dynamics of the alliances and the intensity of their interactions. The group formed by a pair, who can exchange their neighboring positions, prevail in the majority of the parameter region. The rival quartet can only win if their inner cyclic invasion rate is significant while the mixing rate of the pair is extremely low. At specific parameter values, when neither of the alliances is strong enough, new four-member solutions emerge where a rock-paper-scissors-like trio is extended by the other member of the pair. These new solutions coexist hence all six competitors can survive. The evolutionary process is accompanied by serious finite-size effects which can be mitigated by appropriately chosen prepared initial states.
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Affiliation(s)
- Attila Szolnoki
- Institute of Technical Physics and Materials Science, Centre for Energy Research, P.O. Box 49, Budapest, 1525, Hungary.
| | - Xiaojie Chen
- School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China
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Mir H, Stidham J, Pleimling M. Emerging spatiotemporal patterns in cyclic predator-prey systems with habitats. Phys Rev E 2022; 105:054401. [PMID: 35706181 DOI: 10.1103/physreve.105.054401] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/20/2022] [Accepted: 04/15/2022] [Indexed: 06/15/2023]
Abstract
Three-species cyclic predator-prey systems are known to establish spiral waves that allow species to coexist. In this study, we analyze a structured heterogeneous system which gives one species an advantage to escape predation in an area that we refer to as a habitat and study the effect on species coexistence and emerging spatiotemporal patterns. Counterintuitively, the predator of the advantaged species emerges as dominant species with the highest average density inside the habitat. The species given the advantage in the form of an escape rate has the lowest average density until some threshold value for the escape rate is exceeded, after which the density of the species with the advantage overtakes that of its prey. Numerical analysis of the spatial density of each species as well as of the spatial two-point correlation function for both inside and outside the habitats allow a detailed quantitative discussion. Our analysis is extended to a six-species game that exhibits spontaneous spiral waves, which displays similar but more complicated results.
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Affiliation(s)
- Hana Mir
- Department of Physics and Center for Soft Matter and Biological Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA
| | - James Stidham
- Department of Physics and Center for Soft Matter and Biological Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA
| | - Michel Pleimling
- Department of Physics and Center for Soft Matter and Biological Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA
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Bazeia D, Ferreira MJB, Oliveira BFD, Szolnoki A. Environment driven oscillation in an off-lattice May-Leonard model. Sci Rep 2021; 11:12512. [PMID: 34131239 PMCID: PMC8206140 DOI: 10.1038/s41598-021-91994-7] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2021] [Accepted: 05/31/2021] [Indexed: 11/27/2022] Open
Abstract
Cyclic dominance of competing species is an intensively used working hypothesis to explain biodiversity in certain living systems, where the evolutionary selection principle would dictate a single victor otherwise. Technically the May–Leonard models offer a mathematical framework to describe the mentioned non-transitive interaction of competing species when individual movement is also considered in a spatial system. Emerging rotating spirals composed by the competing species are frequently observed character of the resulting patterns. But how do these spiraling patterns change when we vary the external environment which affects the general vitality of individuals? Motivated by this question we suggest an off-lattice version of the tradition May–Leonard model which allows us to change the actual state of the environment gradually. This can be done by introducing a local carrying capacity parameter which value can be varied gently in an off-lattice environment. Our results support a previous analysis obtained in a more intricate metapopulation model and we show that the well-known rotating spirals become evident in a benign environment when the general density of the population is high. The accompanying time-dependent oscillation of competing species can also be detected where the amplitude and the frequency show a scaling law of the parameter that characterizes the state of the environment. These observations highlight that the assumed non-transitive interaction alone is insufficient condition to maintain biodiversity safely, but the actual state of the environment, which characterizes the general living conditions, also plays a decisive role on the evolution of related systems.
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Affiliation(s)
- D Bazeia
- Departamento de Física, Universidade Federal da Paraíba, João Pessoa, PB, 58051-970, Brazil
| | - M J B Ferreira
- Departamento de Física, Universidade Estadual de Maringá, Av. Colombo 5790, Maringá, PR, 87020-900, Brazil
| | - B F de Oliveira
- Departamento de Física, Universidade Estadual de Maringá, Av. Colombo 5790, Maringá, PR, 87020-900, Brazil
| | - A Szolnoki
- Institute of Technical Physics and Materials Science, Centre for Energy Research, P.O. Box 49, Budapest, 1525, Hungary.
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Scott R, Gras R. A simulation study shows impacts of genetic diversity on establishment success of digital invaders in heterogeneous environments. Ecol Modell 2020. [DOI: 10.1016/j.ecolmodel.2020.109173] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023]
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