1
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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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2
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Lu Y, Wang X, Wu M, Shi L, Park J. Effects of species vigilance on coexistence in evolutionary dynamics of spatial rock-paper-scissors game. CHAOS (WOODBURY, N.Y.) 2022; 32:093116. [PMID: 36182385 DOI: 10.1063/5.0103247] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/15/2022] [Accepted: 08/15/2022] [Indexed: 06/16/2023]
Abstract
Recognizing surrounding situations, such as enemy attacks, which can be realized by predator-prey relationships, is one of the common behaviors of the population in ecosystems. In this paper, we explore the relationship between such species' behavior and biodiversity in the spatial rock-paper-scissors game by employing the ecological concept "vigilance." In order to describe the vigilance process, we adopt a multiplex structure where two distinct layers describe virtual and physical interactions. By investigating the process of evolution in species, we also found that species with different vigilance go together. In addition, by utilizing the dynamic time warping method, we found that species with the same vigilance have consistent behavior, but species with different vigilance have diverse behavior. Our findings may lead to broader interpretations of mechanisms promoting biodiversity via vigilance in species ecosystems.
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Affiliation(s)
- Yikang Lu
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
| | - Xiaoyue Wang
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
| | - Mengjie Wu
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
| | - Lei Shi
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
| | - Junpyo Park
- Department of Applied Mathematics, Kyung Hee University, Yongin 17104, Republic of Korea
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3
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Lu Y, Shen C, Wu M, Du C, Shi L, Park J. Enhancing coexistence of mobile species in the cyclic competition system by wildlife refuge. CHAOS (WOODBURY, N.Y.) 2022; 32:081104. [PMID: 36049906 DOI: 10.1063/5.0093342] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/28/2022] [Accepted: 07/22/2022] [Indexed: 06/15/2023]
Abstract
We investigate evolving dynamics of cyclically competing species on spatially extended systems with considering a specific region, which is called the "wildlife refuge," one of the institutional ways to preserve species biodiversity. Through Monte-Carlo simulations, we found that the refuge can play not groundbreaking but an important role in species survival. Species coexistence is maintained at a moderate mobility regime, which traditionally leads to the collapse of coexistence, and eventually, the extinction is postponed depending on the competition rate rather than the portion of the refuge. Incorporating the extinction probability and Fourier transform supported our results in both stochastic and analogous ways. Our findings may provide valuable evidence to assist fields of ecological/biological sciences in understanding the presence and construction of refuges for wildlife with associated effects on species biodiversity.
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Affiliation(s)
- Yikang Lu
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
| | - Chen Shen
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
| | - Mengjie Wu
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
| | - Chunpeng Du
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
| | - Lei Shi
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
| | - Junpyo Park
- Department of Applied Mathematics, Kyung Hee University, Yongin 17104, Republic of Korea
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4
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Wang X, Lu Y, Shi L, Park J. The effect of territorial awareness in a three-species cyclic predator-prey model. Sci Rep 2022; 12:1821. [PMID: 35110669 PMCID: PMC8810777 DOI: 10.1038/s41598-022-05845-0] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2021] [Accepted: 01/19/2022] [Indexed: 11/24/2022] Open
Abstract
Recognizing territories is essential to decide behavior of population either human or animals, and interaction between groups or individuals according to the territorial awareness is universal. Understanding various mechanisms which affect on such species behaviors can be possible by evolutionary games, and in particular, the rock–paper–scissors (RPS) game has been played a key role as a paradigmatic model to explore biodiversity from microbiota to societies. Among paramount mechanisms in systems of RPS, the role of intraspecific interaction has been recently noted in terms of promoting biodiversity. Since intraspecific interaction is defined by an invasive reaction between individuals in the same group, the interaction may be also sensitive to the territorial awareness. To explore how territorial awareness-based intraspecific interaction can affect species biodiversity, we endow species with the mechanism in the classic RPS game. By means of the Monte-Carlo method, we find the phenomenon that the presence of species’ territorial awareness has an impact on intraspecific interaction which ultimately affects species biodiversity. At the same time, we also find that territorial awareness can play a significant role to the average waiting time for extinction which is numerically elucidated by exploiting the quantity: interface width statistic. Unlike prior research that concentrated solely on the relationship between interaction frequency and species diversity, our results shed lights on the important role of territorial awareness in models of RPS, and they reveal fascinating evolutionary outcomes in structured populations that are a unique consequence of such awareness behavior.
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Affiliation(s)
- Xiaoyue Wang
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, 650221, Yunnan, China
| | - Yikang Lu
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, 650221, Yunnan, China
| | - Lei Shi
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, 650221, Yunnan, China.
| | - Junpyo Park
- Department of Applied Mathematics, Kyung Hee University, Yongin, 17104, Republic of Korea.
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5
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Avelino PP, de Oliveira BF, Trintin RS. Lotka-Volterra versus May-Leonard formulations of the spatial stochastic rock-paper-scissors model: The missing link. Phys Rev E 2022; 105:024309. [PMID: 35291086 DOI: 10.1103/physreve.105.024309] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/06/2021] [Accepted: 02/04/2022] [Indexed: 06/14/2023]
Abstract
The rock-paper-scissors (RPS) model successfully reproduces some of the main features of simple cyclic predator-prey systems with interspecific competition observed in nature. Still, lattice-based simulations of the spatial stochastic RPS model are known to give rise to significantly different results, depending on whether the three-state Lotka-Volterra or the four-state May-Leonard formulation is employed. This is true independently of the values of the model parameters and of the use of either a von Neumann or a Moore neighborhood. In this paper, we introduce a simple modification to the standard spatial stochastic RPS model in which the range of the search of the nearest neighbor may be extended up to a maximum Euclidean radius R. We show that, with this adjustment, the Lotka-Volterra and May-Leonard formulations can be designed to produce similar results, both in terms of dynamical properties and spatial features, by means of an appropriate parameter choice. In particular, we show that this modified spatial stochastic RPS model naturally leads to the emergence of spiral patterns in both its three- and four-state formulations.
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Affiliation(s)
- P P Avelino
- Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal
- Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, PT4169-007 Porto, Portugal
| | - B F de Oliveira
- Departamento de Física, Universidade Estadual de Maringá, Av. Colombo 5790, 87020-900 Maringá, PR, Brazil
| | - R S Trintin
- Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal
- Departamento de Física, Universidade Estadual de Maringá, Av. Colombo 5790, 87020-900 Maringá, PR, Brazil
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6
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Menezes J, Moura B. Mobility-limiting antipredator response in the rock-paper-scissors model. Phys Rev E 2021; 104:054201. [PMID: 34942823 DOI: 10.1103/physreve.104.054201] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/15/2021] [Accepted: 10/18/2021] [Indexed: 11/07/2022]
Abstract
Antipredator behavior is present in many biological systems where individuals collectively react to an imminent attack. The antipredator response may influence spatial pattern formation and ecosystem stability but requires an organism's cost to contribute to the collective effort. We investigate a nonhierarchical tritrophic system, whose predator-prey interactions are described by the rock-paper-scissors game rules. In our spatial stochastic simulations, the radius of antipredator response defines the maximum prey group size that disturbs the predator's action, determining the individual cost to participate in antipredator strategies. We consider that each organism contributes equally to the collective effort, having its mobility limited by the proportion of energy devoted to the antipredator reaction. Our outcomes show that the antipredator response leads to spiral patterns, with the segregation of organisms of the same species occupying departed spatial domains. We found that a less localized antipredator response increases the average size of the single-species patches, improving the protection of individuals against predation. Finally, our findings show that although the increase of the predation risk for a more localized antipredator response, the high mobility constraining benefits species coexistence. Our results may help ecologists understand the mechanisms leading to the stability of biological systems where locality is crucial to behavioral interactions among species.
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Affiliation(s)
- J Menezes
- Escola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte Caixa Postal 1524, 59072-970 Natal, RN, Brazil.,Institute for Biodiversity and Ecosystem Dynamics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
| | - B Moura
- Departamento de Engenharia Biomédica, Universidade Federal do Rio Grande do Norte Av. Senador Salgado Filho, 300, 59078-970 Natal, RN, Brazil.,Edmond and Lily Safra International Neuroscience Institute, Santos Dumont Institute Av Santos Dumont, 1560, 59280-000 Macaiba, RN, Brazil
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7
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Menezes J. Antipredator behavior in the rock-paper-scissors model. Phys Rev E 2021; 103:052216. [PMID: 34134300 DOI: 10.1103/physreve.103.052216] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/28/2021] [Accepted: 05/08/2021] [Indexed: 11/07/2022]
Abstract
When faced with an imminent risk of predation, many animals react to escape consumption. Antipredator strategies are performed by individuals acting as a group to intimidate predators and minimize the damage when attacked. We study the antipredator prey response in spatial tritrophic systems with cyclic species dominance using the rock-paper-scissors game. The impact of the antipredator behavior is local, with the predation probability reducing exponentially with the number of prey in the predator's neighborhood. In contrast to the standard Lotka-Volterra implementation of the rock-paper-scissors model, where no spiral waves appear, our outcomes show that the antipredator behavior leads to spiral patterns from random initial conditions. The results show that the predation risk decreases exponentially with the level of antipredator strength. Finally, we investigate the coexistence probability and verify that antipredator behavior may jeopardize biodiversity for high mobility. Our findings may help biologists to understand ecosystems formed by species whose individuals behave strategically to resist predation.
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Affiliation(s)
- J Menezes
- Escola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte Caixa Postal 1524, 59072-970 Natal, RN, Brazil and Institute for Biodiversity and Ecosystem Dynamics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
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8
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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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9
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Park J. Emergence of oscillatory coexistence with exponentially decayed waiting times in a coupled cyclic competition system. CHAOS (WOODBURY, N.Y.) 2019; 29:071107. [PMID: 31370425 DOI: 10.1063/1.5118833] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2019] [Accepted: 07/15/2019] [Indexed: 06/10/2023]
Abstract
Interpatch migration between two environments is generally considered as a spatial concept and can affect species biodiversity in each patch by inducing flux of population such as inflow and outflow quantities of species. In this paper, we explore the effect of interpatch migration, which can be generally considered as a spatial concept and may affect species biodiversity between two different patches in the perspective of the macroscopic level by exploiting the coupling of two systems, where each patch is occupied by cyclically competing three species who can stably coexist by exhibiting periodic orbits. For two simple scenarios of interpatch migration either single or all species migration, we found that two systems with independently stable coexisting species in each patch are eventually synchronized, and oscillatory behaviors of species densities in two patches become identical, i.e., the synchronized coexistence emerges. In addition, we find that, whether single or all species interpatch migration occurs, the waiting time for the synchronization is exponentially decreasing as the coupling strength is intensified. Our findings suggest that the synchronized behavior of species as a result of migration between different patches can be easily predicted by the coupling of systems and additional information such as waiting times and sensitivity of initial densities.
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Affiliation(s)
- Junpyo Park
- Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 44919, South Korea
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10
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Park J. Nonlinear dynamics with Hopf bifurcations by targeted mutation in the system of rock-paper-scissors metaphor. CHAOS (WOODBURY, N.Y.) 2019; 29:033102. [PMID: 30927841 DOI: 10.1063/1.5081966] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2018] [Accepted: 02/05/2019] [Indexed: 06/09/2023]
Abstract
The role of mutation, which is an error process in gene evolution, in systems of cyclically competing species has been studied from various perspectives, and it is regarded as one of the key factors for promoting coexistence of all species. In addition to naturally occurring mutations, many experiments in genetic engineering have involved targeted mutation techniques such as recombination between DNA and somatic cell sequences and have studied genetic modifications through loss or augmentation of cell functions. In this paper, we investigate nonlinear dynamics with targeted mutation in cyclically competing species. In different ways to classic approaches of mutation in cyclic games, we assume that mutation may occur in targeted individuals who have been removed from intraspecific competition. By investigating each scenario depending on the number of objects for targeted mutation analytically and numerically, we found that targeted mutation can lead to persistent coexistence of all species. In addition, under the specific condition of targeted mutation, we found that targeted mutation can lead to emergences of bistable states for species survival. Through the linear stability analysis of rate equations, we found that those phenomena are accompanied by Hopf bifurcation which is supercritical. Our findings may provide more global perspectives on understanding underlying mechanisms to control biodiversity in ecological/biological sciences, and evidences with mathematical foundations to resolve social dilemmas such as a turnover of group members by resigning with intragroup conflicts in social sciences.
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Affiliation(s)
- Junpyo Park
- Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 44919, South Korea
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11
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Park J, Do Y, Jang B. Multistability in the cyclic competition system. CHAOS (WOODBURY, N.Y.) 2018; 28:113110. [PMID: 30501221 DOI: 10.1063/1.5045366] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2018] [Accepted: 10/11/2018] [Indexed: 06/09/2023]
Abstract
Cyclically competition models have been successful to gain an insight of biodiversity mechanism in ecosystems. There are, however, still limitations to elucidate complex phenomena arising in real competition. In this paper, we report that a multistability occurs in a simple rock-paper-scissor cyclically competition model by assuming that intraspecific competition depends on the logistic growth of each species density. This complex stability is absent in any cyclically competition model, and we investigate how the proposed intraspecific competition affects biodiversity in the existing society of three species through macroscopic and microscopic approaches. When the system is multistable, we show basins of the asymptotically stable heteroclinic cycle and stable attractors to demonstrate how the survival state is determined by initial densities of three species. Also, we find that the multistability is associated with a subcritical Hopf bifurcation. This surprising finding will give an opportunity to interpret rich dynamical phenomena in ecosystems which may occur in cyclic competition systems with different types of interactions.
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Affiliation(s)
- Junpyo Park
- Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of Korea
| | - Younghae Do
- Department of Mathematics, KNU-Center for Nonlinear Dynamics, Kyungpook National University, Daegu 41566, Republic of Korea
| | - Bongsoo Jang
- Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of Korea
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12
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Park J. Asymmetric interplay leads to robust coexistence by means of a global attractor in the spatial dynamics of cyclic competition. CHAOS (WOODBURY, N.Y.) 2018; 28:081103. [PMID: 30180598 DOI: 10.1063/1.5048468] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/14/2018] [Accepted: 07/27/2018] [Indexed: 06/08/2023]
Abstract
In the past decade, there have been many efforts to understand the species interplay with biodiversity in cyclic games within the macro and microscopic levels. In this direction, mobility and intraspecific competition have been found to be the main factors promoting coexistence in spatially extended systems. In this paper, we explore the relevant effect of asymmetric competitions coupled with mobility on the coexistence of cyclically competing species. By examining the coexistence probability, we have found that mobility can facilitate coexistence in the limited cases of asymmetric competition and can be well predicted by the basin structure of the deterministic system. In addition, it is found that mobility can have beneficial and harmful effects on coexistence when all competitions occur asymmetrically. We also found that the coexistence in the spatial dynamics ultimately becomes a global attractor. We hope to provide insights into the associated effects of asymmetric interplays on species coexistence in a spatially extended system and understand the biodiversity of asymmetrically competitive species under more complex competition structures.
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Affiliation(s)
- Junpyo Park
- Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 44919, South Korea
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13
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Park J. Biodiversity in the cyclic competition system of three species according to the emergence of mutant species. CHAOS (WOODBURY, N.Y.) 2018; 28:053111. [PMID: 29857686 DOI: 10.1063/1.5021145] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Understanding mechanisms which promote or hinder existing ecosystems are important issues in ecological sciences. In addition to fundamental interactions such as competition and migration among native species, existing ecosystems can be easily disturbed by external factors, and the emergence of new species may be an example in such cases. The new species which does not exist in a current ecosystem can be regarded as either alien species entered from outside or mutant species born by mutation in existing normal species. Recently, as existing ecosystems are getting influenced by various physical/chemical external factors, mutation due to anthropogenic and environmental factors can occur more frequently and is thus attracting much attention for the maintenance of ecosystems. In this paper, we consider emergences of mutant species among self-competing three species in the cyclic dominance. By defining mutation as the birth of mutant species, we investigate how mutant species can affect biodiversity in the existing ecosystem. Through microscopic and macroscopic approaches, we have found that the society of existing normal species can be disturbed by mutant species either the society is maintained accompanying with the coexistence of all species or jeopardized by occupying of mutant species. Due to the birth of mutant species, the existing society may be more complex by constituting two different groups of normal and mutant species, and our results can be contributed to analyze complex ecosystems of many species. We hope our findings may propose a new insight on mutation in cyclic competition systems of many species.
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Affiliation(s)
- Junpyo Park
- Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 44919, South Korea
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14
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Kim B, Park J. Basins of distinct asymptotic states in the cyclically competing mobile five species game. CHAOS (WOODBURY, N.Y.) 2017; 27:103117. [PMID: 29092432 DOI: 10.1063/1.4998984] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We study the dynamics of cyclic competing mobile five species on spatially extended systems originated from asymmetric initial populations and investigate the basins for the three possible asymptotic states, coexistence of all species, existences of only two independent species, and the extinction. Through extensive numerical simulations, we find a prosperous dependence on initial conditions for species biodiversity. In particular, for fixed given equal densities of two relevant species, we find that only five basins for the existence of two independent species exist and they are spirally entangled for high mobility. A basin of coexistence is outbreaking when the mobility parameter is decreased through a critical value and surrounded by the other five basins. For fixed given equal densities of two independent species, however, we find that basin structures are not spirally entangled. Further, final states of two independent species are totally different. For all possible considerations, the extinction state is not witnessed which is verified by the survival probability. To provide the validity of basin structures from lattice simulations, we analyze the system in mean-field manners. Consequently, results on macroscopic levels are matched to direct lattice simulations for high mobility regimes. These findings provide a good insight into the fundamental issue of the biodiversity among many species than previous cases.
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Affiliation(s)
- Beomseok Kim
- Department of Mathematics, KNU-Center for Nonlinear Dynamics, Kyungpook National University, Daegu 41566, South Korea
| | - Junpyo Park
- Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 44919, South Korea
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15
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Emergence of unusual coexistence states in cyclic game systems. Sci Rep 2017; 7:7465. [PMID: 28785001 PMCID: PMC5547111 DOI: 10.1038/s41598-017-07911-4] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/04/2017] [Accepted: 07/04/2017] [Indexed: 11/08/2022] Open
Abstract
Evolutionary games of cyclic competitions have been extensively studied to gain insights into one of the most fundamental phenomena in nature: biodiversity that seems to be excluded by the principle of natural selection. The Rock-Paper-Scissors (RPS) game of three species and its extensions [e.g., the Rock-Paper-Scissors-Lizard-Spock (RPSLS) game] are paradigmatic models in this field. In all previous studies, the intrinsic symmetry associated with cyclic competitions imposes a limitation on the resulting coexistence states, leading to only selective types of such states. We investigate the effect of nonuniform intraspecific competitions on coexistence and find that a wider spectrum of coexistence states can emerge and persist. This surprising finding is substantiated using three classes of cyclic game models through stability analysis, Monte Carlo simulations and continuous spatiotemporal dynamical evolution from partial differential equations. Our finding indicates that intraspecific competitions or alternative symmetry-breaking mechanisms can promote biodiversity to a broader extent than previously thought.
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16
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Szolnoki A, Perc M. Biodiversity in models of cyclic dominance is preserved by heterogeneity in site-specific invasion rates. Sci Rep 2016; 6:38608. [PMID: 27917952 PMCID: PMC5137108 DOI: 10.1038/srep38608] [Citation(s) in RCA: 37] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/05/2016] [Accepted: 11/09/2016] [Indexed: 12/20/2022] Open
Abstract
Global, population-wide oscillations in models of cyclic dominance may result in the collapse of biodiversity due to the accidental extinction of one species in the loop. Previous research has shown that such oscillations can emerge if the interaction network has small-world properties, and more generally, because of long-range interactions among individuals or because of mobility. But although these features are all common in nature, global oscillations are rarely observed in actual biological systems. This begets the question what is the missing ingredient that would prevent local oscillations to synchronize across the population to form global oscillations. Here we show that, although heterogeneous species-specific invasion rates fail to have a noticeable impact on species coexistence, randomness in site-specific invasion rates successfully hinders the emergence of global oscillations and thus preserves biodiversity. Our model takes into account that the environment is often not uniform but rather spatially heterogeneous, which may influence the success of microscopic dynamics locally. This prevents the synchronization of locally emerging oscillations, and ultimately results in a phenomenon where one type of randomness is used to mitigate the adverse effects of other types of randomness in the system.
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Affiliation(s)
- Attila Szolnoki
- Institute of Technical Physics and Materials Science, Centre for Energy Research, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia.,CAMTP - Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia
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Szolnoki A, Perc M. Zealots tame oscillations in the spatial rock-paper-scissors game. Phys Rev E 2016; 93:062307. [PMID: 27415280 DOI: 10.1103/physreve.93.062307] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2016] [Indexed: 06/06/2023]
Abstract
The rock-paper-scissors game is a paradigmatic model for biodiversity, with applications ranging from microbial populations to human societies. Research has shown, however, that mobility jeopardizes biodiversity by promoting the formation of spiral waves, especially if there is no conservation law in place for the total number of competing players. First, we show that even if such a conservation law applies, mobility still jeopardizes biodiversity in the spatial rock-paper-scissors game if only a small fraction of links of the square lattice is randomly rewired. Secondly, we show that zealots are very effective in taming the amplitude of oscillations that emerge due to mobility and/or interaction randomness, and this regardless of whether the later is quenched or annealed. While even a tiny fraction of zealots brings significant benefits, at 5% occupancy zealots practically destroy all oscillations regardless of the intensity of mobility, and regardless of the type and strength of randomness in the interaction structure. Interestingly, by annealed randomness the impact of zealots is qualitatively the same as by mobility, which highlights that fast diffusion does not necessarily destroy the coexistence of species, and that zealotry thus helps to recover the stable mean-field solution. Our results strengthen the important role of zealots in models of cyclic dominance, and they reveal fascinating evolutionary outcomes in structured populations that are a unique consequence of such uncompromising behavior.
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Affiliation(s)
- Attila Szolnoki
- Institute of Technical Physics and Materials Science, Centre for Energy Research, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia
- CAMTP - Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia
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Hummert S, Bohl K, Basanta D, Deutsch A, Werner S, Theissen G, Schroeter A, Schuster S. Evolutionary game theory: cells as players. MOLECULAR BIOSYSTEMS 2015; 10:3044-65. [PMID: 25270362 DOI: 10.1039/c3mb70602h] [Citation(s) in RCA: 65] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Abstract] [Subscribe] [Scholar Register] [Indexed: 01/18/2023]
Abstract
In two papers we review game theory applications in biology below the level of cognitive living beings. It can be seen that evolution and natural selection replace the rationality of the actors appropriately. Even in these micro worlds, competing situations and cooperative relationships can be found and modeled by evolutionary game theory. Also those units of the lowest levels of life show different strategies for different environmental situations or different partners. We give a wide overview of evolutionary game theory applications to microscopic units. In this first review situations on the cellular level are tackled. In particular metabolic problems are discussed, such as ATP-producing pathways, secretion of public goods and cross-feeding. Further topics are cyclic competition among more than two partners, intra- and inter-cellular signalling, the struggle between pathogens and the immune system, and the interactions of cancer cells. Moreover, we introduce the theoretical basics to encourage scientists to investigate problems in cell biology and molecular biology by evolutionary game theory.
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Affiliation(s)
- Sabine Hummert
- Fachhochschule Schmalkalden, Faculty of Electrical Engineering, Blechhammer, 98574 Schmalkalden, Germany
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Mesoscopic interactions and species coexistence in evolutionary game dynamics of cyclic competitions. Sci Rep 2014; 4:7486. [PMID: 25501627 PMCID: PMC4265771 DOI: 10.1038/srep07486] [Citation(s) in RCA: 54] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2014] [Accepted: 11/27/2014] [Indexed: 11/18/2022] Open
Abstract
Evolutionary dynamical models for cyclic competitions of three species (e.g., rock, paper, and scissors, or RPS) provide a paradigm, at the microscopic level of individual interactions, to address many issues in coexistence and biodiversity. Real ecosystems often involve competitions among more than three species. By extending the RPS game model to five (rock-paper-scissors-lizard-Spock, or RPSLS) mobile species, we uncover a fundamental type of mesoscopic interactions among subgroups of species. In particular, competitions at the microscopic level lead to the emergence of various local groups in different regions of the space, each involving three species. It is the interactions among the groups that fundamentally determine how many species can coexist. In fact, as the mobility is increased from zero, two transitions can occur: one from a five- to a three-species coexistence state and another from the latter to a uniform, single-species state. We develop a mean-field theory to show that, in order to understand the first transition, group interactions at the mesoscopic scale must be taken into account. Our findings suggest, more broadly, the importance of mesoscopic interactions in coexistence of great many species.
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Szolnoki A, Mobilia M, Jiang LL, Szczesny B, Rucklidge AM, Perc M. Cyclic dominance in evolutionary games: a review. J R Soc Interface 2014; 11:20140735. [PMID: 25232048 PMCID: PMC4191105 DOI: 10.1098/rsif.2014.0735] [Citation(s) in RCA: 143] [Impact Index Per Article: 14.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/07/2014] [Accepted: 08/22/2014] [Indexed: 11/12/2022] Open
Abstract
Rock is wrapped by paper, paper is cut by scissors and scissors are crushed by rock. This simple game is popular among children and adults to decide on trivial disputes that have no obvious winner, but cyclic dominance is also at the heart of predator-prey interactions, the mating strategy of side-blotched lizards, the overgrowth of marine sessile organisms and competition in microbial populations. Cyclical interactions also emerge spontaneously in evolutionary games entailing volunteering, reward, punishment, and in fact are common when the competing strategies are three or more, regardless of the particularities of the game. Here, we review recent advances on the rock-paper-scissors (RPS) and related evolutionary games, focusing, in particular, on pattern formation, the impact of mobility and the spontaneous emergence of cyclic dominance. We also review mean-field and zero-dimensional RPS models and the application of the complex Ginzburg-Landau equation, and we highlight the importance and usefulness of statistical physics for the successful study of large-scale ecological systems. Directions for future research, related, for example, to dynamical effects of coevolutionary rules and invasion reversals owing to multi-point interactions, are also outlined.
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Affiliation(s)
- Attila Szolnoki
- Institute of Technical Physics and Materials Science, Research Centre for Natural Sciences, Hungarian Academy of Sciences, PO Box 49, 1525 Budapest, Hungary
| | - Mauro Mobilia
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
| | - Luo-Luo Jiang
- College of Physics and Electronic Information Engineering, Wenzhou University, 325035 Wenzhou, People's Republic of China
| | - Bartosz Szczesny
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
| | - Alastair M Rucklidge
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
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Szolnoki A, Vukov J, Perc M. From pairwise to group interactions in games of cyclic dominance. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:062125. [PMID: 25019743 DOI: 10.1103/physreve.89.062125] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/10/2014] [Indexed: 06/03/2023]
Abstract
We study the rock-paper-scissors game in structured populations, where the invasion rates determine individual payoffs that govern the process of strategy change. The traditional version of the game is recovered if the payoffs for each potential invasion stem from a single pairwise interaction. However, the transformation of invasion rates to payoffs also allows the usage of larger interaction ranges. In addition to the traditional pairwise interaction, we therefore consider simultaneous interactions with all nearest neighbors, as well as with all nearest and next-nearest neighbors, thus effectively going from single pair to group interactions in games of cyclic dominance. We show that differences in the interaction range affect not only the stationary fractions of strategies but also their relations of dominance. The transition from pairwise to group interactions can thus decelerate and even revert the direction of the invasion between the competing strategies. Like in evolutionary social dilemmas, in games of cyclic dominance, too, the indirect multipoint interactions that are due to group interactions hence play a pivotal role. Our results indicate that, in addition to the invasion rates, the interaction range is at least as important for the maintenance of biodiversity among cyclically competing strategies.
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Affiliation(s)
- Attila Szolnoki
- Institute of Technical Physics and Materials Science, Research Centre for Natural Sciences, Hungarian Academy of Sciences, P. O. Box 49, H-1525 Budapest, Hungary
| | - Jeromos Vukov
- Institute of Technical Physics and Materials Science, Research Centre for Natural Sciences, Hungarian Academy of Sciences, P. O. Box 49, H-1525 Budapest, Hungary
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia
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Brackley CA, Ebenhöh O, Grebogi C, Kurths J, de Moura A, Romano MC, Thiel M. Introduction to focus issue: dynamics in systems biology. CHAOS (WOODBURY, N.Y.) 2010; 20:045101. [PMID: 21198113 DOI: 10.1063/1.3530126] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
The methods of nonlinear systems form an extensive toolbox for the study of biology, and systems biology provides a rich source of motivation for the development of new mathematical techniques and the furthering of understanding of dynamical systems. This Focus Issue collects together a large variety of work which highlights the complementary nature of these two fields, showing what each has to offer the other. While a wide range of subjects is covered, the papers often have common themes such as "rhythms and oscillations," "networks and graph theory," and "switches and decision making." There is a particular emphasis on the links between experimental data and modeling and mathematical analysis.
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Affiliation(s)
- Chris A Brackley
- Institute for Complex Systems and Mathematical Biology, SUPA King's College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom.
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