How target waves emerge in population dynamics

arXiv:0807.4390v1 [physics.bio-ph] 28 Jul 2008 How target waves emerge in population dynamics with cyclical interactions Luo-Luo Jiang1, Tao Zhou1,2, Xin Huang3, Bing-Hong Wang1 1 Department of Modern Physics, University of Science and Technology of China, Hefei 230026, PR China 2 Department of Physics, University of Fribourg, Chemin du Muse 3, CH-1700 Fribourg, Switzerland 3 Department of Physics, University of Science and Technology of China, Hefei 230026, PR China E-mail: bhwang@ustc.edu.cn Abstract. Based on a multi-agent model, we investigate how target waves emerge from a population dynamics with cyclical interactions among three species. We show that the periodically injecting source in a small central area can generate target waves in a two-dimensional lattice system. By detecting the temporal period of species’ concentration at the central area, three modes of target waves can be distinguished. Those different modes result from the competition between local and global oscillations induced by cyclical interactions: Mode A corresponds to a synchronization of local and global oscillations, Mode B results from an intermittent synchronization, and Mode C corresponds to the case when the frequency of the local oscillation is much higher than that of the global oscillation. This work provides insights into pattern formation in biologic and ecologic systems that are totally different from the extensively studied diffusion systems driven by chemical reactions. PACS numbers: 87.23.Cc, 05.10.Ln, 87.18.Hf How target waves emerge in population dynamics 2 1. Introduction Spatially distributed excitable systems are widely investigated owning to their biological significance of long-rang signal transmission through self-sustained waves [1, 2, 3, 4, 5, 6]. Pattern formation of wave propagation in excitable systems have been well studied [7]. Especially, target waves induced by noise have been found in the recurrent single species’ population dynamics [6]. However, the formation mechanism of target waves in populations dynamics driven by multi-species’ competitive interactions are not very clear. Competitive interactions in natural and social systems consisted of many elements (e.g., various biological species, political parties, businesses, coupled reaction chemical components, bacterial production bacteria, etc.) play an important role in evolutionary processes and pattern formations [8, 9, 10, 11, 12, 13], and lead to the emergence of spatial patterns. In particular, target waves are commonly observed in those systems [14, 15, 16, 17, 18]. Previous works reveal that patterns could emerge in reaction- diffusion systems if one of the substances diffuses much faster than others [19, 20, 21]. For example, the bromous acid diffuses much faster than ferrion in the Belousev-Zhabotinsky (BZ) reaction and the cyclic AMP diffuses much faster than membrane receptor in the dictyostelium discoideum. However, many pattern formations of mobile population in ecosystems, such as migrating animals and running bacteria, can not be explained by the reaction-diffusion mechanism, since the diffusion speed induced by individual mobility is the same for all substances [22, 23]. Furthermore, partial differential equations (PDEs) have been proposed to describe the evolution of pattern formation in reaction-diffusion systems, such as the Oregonator model for BZ reaction [24]. Based on those PDEs, one can analyze the stability of patterns by using the mean field theory, but the reliable detailed information is quite limited. Therefore, it is necessary to use an agent-based models to describe the pattern formation of mobile population [11]. The cyclic predator-prey model provides a terse description of competition among population of different species. Experimental study [22] has revealed that the mechanism of rack-paper-scissors game can promote the biodiversity of three strains of Escherichia coli. Very recently, Reichenbach et al proposed a rack-paper-scissors game of mobile populations [8], where the individual mobility displays a critical effect on species diversity. When mobility is below a certain value, all species coexist and form spiral waves. In contrast, above this threshold biodiversity is jeopardized. Indeed, previous investigations show that the cyclical competition mechanism and low mobility of individuals maintain the biodivesity in ecosystems [8, 25, 22, 26, 27]. In this paper, we investigate the pattern formation based on a cyclic predator-prey model with mobile individuals. Target waves are observed from the recurrent dynamics driven by three species’ competitive interactions. As a result of global oscillation, a transition from disordered state to an ordered spacial structure occurs with the increasing of injection period in the vortex of target waves. By detecting the temporal period of species’ concentration at the central area, three modes of target waves can be distinguished. How target waves emerge in population dynamics 3 Those

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