生态系统中时空动力学行为的研究
摘要
生态系统是典型的远离热力学平衡态的复杂适应系统,由于自组织,生态系统可以形成复杂的随时间动态演化的时空斑图,这类时空斑图的形成可以为我们提供许多生态问题的重要认识,比如：单一物种的空间动态演化,物种之间的竞争、捕食、寄生、互利共生作用对物种在空间分布的影响等等。因此,研究这类斑图对于理解生态系统的演化以及生物多样性的维持具有重要的意义。
本文主要以循环竞争的种群动力学模型为基础,通过蒙特卡洛(Monte Carlo)方法并结合描述种群相互作用的平均场模型来模拟和分析生态系统中的自组织现象,研究物种间的长程相互作用对斑图形成及生物多样性的影响。在此基础上,通过引入植物和动物间的互利关系,进一步研究了植物和动物间互利网络的不同结构对斑图形成和生物多样性的影响。主要研究内容及结果如下：
首先,以Barkley模型为对象,研究了可激介质的非均匀性对螺旋波斑图形成的影响。该模型中各参数与可激介质的属性密切相关,通过参数涨落的正态分布来刻画非均匀性,数值研究了单参数以及多参数涨落的正态分布情形下螺旋波斑图的形成。研究表明,可激介质的非均匀性对于螺旋波波纹的粗细及疏密程度有较大影响。参数涨落分布的方差越大,形成的螺旋波波纹越粗糙。对于两参数均匀分布的极端情形,当参数分布大于某一范围时,无法形成螺旋波。这些都与螺旋波旋转的角频率密切相关。螺旋波旋转的角频率越大,螺旋波波纹越粗,同时波纹越密集；反之,螺旋波旋转角频越小,螺旋波波纹越细,同时波纹越稀疏。另外,研究了双层耦合的非均匀介质中螺旋波的同步现象。研究发现,在耦合强度较弱的情况下,可以观察到相同步,进一步增加耦合强度,同步消失,但继续增加耦合强度达到一定值的时候,双层系统可以达到完全同步。研究还发现,非均匀介质相对于均匀介质的耦合,更容易达到同步。
其次,在二维网格上利用受限的Newman-Watts网络研究了三物种循环竞争的种群动力学模型。通过限制Newman-Watts网络的最大连接距离和长程连接概率来描述物种间的长程相互作用。研究发现,随着长程连接的最大距离和长程连接概率的增加,系统自组织形成的螺旋波的波长不断增大,当波长增大到一定程度后,系统失稳,生物多样性丧失。我们进一步比较了在有长程相互作用和没有长程相互作用的情况下,系统的稳定情况,并计算了空间关联函数和关联长度,进一步说明了长程相互作用能够提高物种的流动性,从而使系统变得不稳定而更容易导致物种的灭绝。
最后,我们利用双层网格研究了植物和动物间的互利网络结构对生态系统的自组织斑图和生物多样性的影响。让动物之间为循环竞争关系,植物之间为平等的竞争关系,同时让动物具有流动性,通过比较动物和植物间不同的互利共生网络结构对系统自组织斑图和生物多样性的影响,结果发现,嵌入式的互利共生网络结构相对于全连接的和随机连接的共生网络结构来说,不利于系统的稳定性。这主要是因为嵌入式的互利共生网络结构是一种极不对称结构,从而导致某些植物物种受到较多的互利作用具有较强的竞争力,而另外一些植物物种由于受到的互利作用较小从而在竞争中处于劣势而灭绝。
在本文的最后,对我们的工作作了总结,并对将来的研究作了展望。
Ecosystem is a typical complex adaptive system far from thermody-namic equilibrium. Owing to its self-organization, ecosystem can produce complex spatio-temporal patterns. The spatio-temporal patterns possess special physical properties. The study on those patterns is of importance for understanding of biodiversity.
In this thesis, we study the phenomenon of self-organization in ecosys-tems with Monte Carlo simulations and mean field models based on cyclic dynamics. One of our focus lies on the understanding of the influence of long-range interactions between species on the pattern formation and bio-diversity. The others are the understanding of the plant-animal mutualistic networks on the pattern formation and biodiversity.
Firstly, the pattern formation of spiral waves in inhomogeneous ex-citable medium is investigated. The normal distribution of parameters is introduced to depict the inhomogeneous medium. It is found that the parameter fluctuations play an important role in the formation of spiral pattern. For a larger variance of the parameter fluctuations, the spiral waves are rough. In the case of the uniform distributions for two param-eters, spiral wave cannot be observed for the larger range of fluctuations. It is conjectured that these results are induced by the rotating frequency of spiral wave for different parameters. For the larger rotating frequency, spiral wave is crowded, but is sparse for small frequency. Furthermore, we studied synchronization behaviours of spiral waves in a two-layer coupled inhomogeneous excitable system. It was found that phase synchronization can be observed under weak coupling strength. By increasing the coupling strength, the synchronization is broken down. With the further increase of the coupling strength, complete synchronization and phase synchronization occur again. We also found that the inhomogeneity in excitable systems is helpful to the synchronization.
Secondly, We study the cyclic dominance of three species in two-dimensional constrained Newman-Watts networks with a four-state vari-ant of the rock-paper-scissors game. By limiting the maximal connection distance Rmax in Newman-Watts networks with the long-range connection probability p, we depict more realistically the stochastic interactions among species within ecosystems. When we fix mobility and vary the value of p or Rmax, the Monte Carlo simulations show that the spiral waves grow in size, and the system becomes unstable and biodiversity is lost with increasing p or Rmax.We compared extinctions with or without long-range connec-tions and computed spatial correlation functions and correlation length. We conclude that long-range connections could improve the mobility of species, drastically changing their crossover to extinction and making the system more unstable.
Thirdly, we study the effects of mutualistic network structure of plants and their animal pollinators on the self-organized patterns and biodiversity with two-layer coupled lattices. In this model, plants and animals compete for resources within their respective groups. The animals exhibit cyclic dominance and the plants compete all to all. At the same time, the animals may move randomly. Through comparing the effects of three kinds of plant-animal mutualistic networks on pattern formation and biodiversity, we found that nested mutualism would lead to instability.
At last, we make a conclusion and outlook of this thesis.
本文主要以循环竞争的种群动力学模型为基础,通过蒙特卡洛(Monte Carlo)方法并结合描述种群相互作用的平均场模型来模拟和分析生态系统中的自组织现象,研究物种间的长程相互作用对斑图形成及生物多样性的影响。在此基础上,通过引入植物和动物间的互利关系,进一步研究了植物和动物间互利网络的不同结构对斑图形成和生物多样性的影响。主要研究内容及结果如下：
首先,以Barkley模型为对象,研究了可激介质的非均匀性对螺旋波斑图形成的影响。该模型中各参数与可激介质的属性密切相关,通过参数涨落的正态分布来刻画非均匀性,数值研究了单参数以及多参数涨落的正态分布情形下螺旋波斑图的形成。研究表明,可激介质的非均匀性对于螺旋波波纹的粗细及疏密程度有较大影响。参数涨落分布的方差越大,形成的螺旋波波纹越粗糙。对于两参数均匀分布的极端情形,当参数分布大于某一范围时,无法形成螺旋波。这些都与螺旋波旋转的角频率密切相关。螺旋波旋转的角频率越大,螺旋波波纹越粗,同时波纹越密集；反之,螺旋波旋转角频越小,螺旋波波纹越细,同时波纹越稀疏。另外,研究了双层耦合的非均匀介质中螺旋波的同步现象。研究发现,在耦合强度较弱的情况下,可以观察到相同步,进一步增加耦合强度,同步消失,但继续增加耦合强度达到一定值的时候,双层系统可以达到完全同步。研究还发现,非均匀介质相对于均匀介质的耦合,更容易达到同步。
其次,在二维网格上利用受限的Newman-Watts网络研究了三物种循环竞争的种群动力学模型。通过限制Newman-Watts网络的最大连接距离和长程连接概率来描述物种间的长程相互作用。研究发现,随着长程连接的最大距离和长程连接概率的增加,系统自组织形成的螺旋波的波长不断增大,当波长增大到一定程度后,系统失稳,生物多样性丧失。我们进一步比较了在有长程相互作用和没有长程相互作用的情况下,系统的稳定情况,并计算了空间关联函数和关联长度,进一步说明了长程相互作用能够提高物种的流动性,从而使系统变得不稳定而更容易导致物种的灭绝。
最后,我们利用双层网格研究了植物和动物间的互利网络结构对生态系统的自组织斑图和生物多样性的影响。让动物之间为循环竞争关系,植物之间为平等的竞争关系,同时让动物具有流动性,通过比较动物和植物间不同的互利共生网络结构对系统自组织斑图和生物多样性的影响,结果发现,嵌入式的互利共生网络结构相对于全连接的和随机连接的共生网络结构来说,不利于系统的稳定性。这主要是因为嵌入式的互利共生网络结构是一种极不对称结构,从而导致某些植物物种受到较多的互利作用具有较强的竞争力,而另外一些植物物种由于受到的互利作用较小从而在竞争中处于劣势而灭绝。
在本文的最后,对我们的工作作了总结,并对将来的研究作了展望。
Ecosystem is a typical complex adaptive system far from thermody-namic equilibrium. Owing to its self-organization, ecosystem can produce complex spatio-temporal patterns. The spatio-temporal patterns possess special physical properties. The study on those patterns is of importance for understanding of biodiversity.
In this thesis, we study the phenomenon of self-organization in ecosys-tems with Monte Carlo simulations and mean field models based on cyclic dynamics. One of our focus lies on the understanding of the influence of long-range interactions between species on the pattern formation and bio-diversity. The others are the understanding of the plant-animal mutualistic networks on the pattern formation and biodiversity.
Firstly, the pattern formation of spiral waves in inhomogeneous ex-citable medium is investigated. The normal distribution of parameters is introduced to depict the inhomogeneous medium. It is found that the parameter fluctuations play an important role in the formation of spiral pattern. For a larger variance of the parameter fluctuations, the spiral waves are rough. In the case of the uniform distributions for two param-eters, spiral wave cannot be observed for the larger range of fluctuations. It is conjectured that these results are induced by the rotating frequency of spiral wave for different parameters. For the larger rotating frequency, spiral wave is crowded, but is sparse for small frequency. Furthermore, we studied synchronization behaviours of spiral waves in a two-layer coupled inhomogeneous excitable system. It was found that phase synchronization can be observed under weak coupling strength. By increasing the coupling strength, the synchronization is broken down. With the further increase of the coupling strength, complete synchronization and phase synchronization occur again. We also found that the inhomogeneity in excitable systems is helpful to the synchronization.
Secondly, We study the cyclic dominance of three species in two-dimensional constrained Newman-Watts networks with a four-state vari-ant of the rock-paper-scissors game. By limiting the maximal connection distance Rmax in Newman-Watts networks with the long-range connection probability p, we depict more realistically the stochastic interactions among species within ecosystems. When we fix mobility and vary the value of p or Rmax, the Monte Carlo simulations show that the spiral waves grow in size, and the system becomes unstable and biodiversity is lost with increasing p or Rmax.We compared extinctions with or without long-range connec-tions and computed spatial correlation functions and correlation length. We conclude that long-range connections could improve the mobility of species, drastically changing their crossover to extinction and making the system more unstable.
Thirdly, we study the effects of mutualistic network structure of plants and their animal pollinators on the self-organized patterns and biodiversity with two-layer coupled lattices. In this model, plants and animals compete for resources within their respective groups. The animals exhibit cyclic dominance and the plants compete all to all. At the same time, the animals may move randomly. Through comparing the effects of three kinds of plant-animal mutualistic networks on pattern formation and biodiversity, we found that nested mutualism would lead to instability.
At last, we make a conclusion and outlook of this thesis.
引文
[1]邬建国,耗散结构、等级理论和生态系统,应用生态学报,2,181,1991.
[2]R. V. Sole, J. Bascompte, Self-Organization in Complex Ecosystems. Princeton University Press,2006.
[3]张知彬,王祖望和李典谟,生态复杂性研究综述与展望。生态学报,18:433-441,1998.
[4]G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems:From Dissipative Structures to Order through fluctuations. Wiley& Sons, New York,1977.
[5]P. Cilliers. Complexity and Postmodernism:Understanding Complex Systems. Routledge, London,1998.
[6]H. Haken, Synergetics:An introduction:Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry, and Biology. Berlin Heidelberg:Springer-Verlag,1977.
[7]颜泽贤,范冬萍,张华夏,系统科学导论-复杂性探索。北京：人民出版社,2006.
[8]欧阳颀,反应扩散系统中的斑图动力学,上海科技教育出版社,2000.
[9]G. A. Cowan, D. Pines, and D. Meltzer, editors, Complexity:Metaphors, Models, and Re-ality. Perseus Books, Reading, Massachusetts,1994.
[10]M. M. Waldrop, Complexity:The Emerging Science at the Edge of Order and Chaos, Simon & Schuster, New York,1992.
[11]M. Gell-Mann, The Quark and the Jaguar:Adventures in the Simple and the Complex. W.H. Freeman and Company, New York,1994.
[12]A. J. Lotka, Elements of Physical Biology, Williams and Wilkins:Baltimore,1925.
[13]V. Volterra, Variations and Fluctuations of the Number of Individuals in Animal Species Living Together, Animal Ecology,409,1926.
[14]A. J. Nicholson and V. A. Bailey, The Balance of Animal Populations Ⅰ, Proceedings of the Zoological Society of London 3,551,1935.
[15]张大勇,理论生态学研究,高等教育出版社,2000.
[16]O. E. Rhodes, R. K. Chester, and M. H. Smith, (eds.), Population Dynamics in Ecological Space and Time, Chicage University Press, Chicago,1996.
[17]J. Bascompte and R. V. Sole, Modeling Spatiotemporal Dynamics in Ecology. Springer-Verlag, New York,1998.
[18]L. A. Segel, and J. Jackson, Dissipative Structure:An Explanantion and an Ecological Example. Journal of Theoretical Biology,37,545,1972.
[19]D. Alonso, F. Bartumeus, and J. Catalan, Mutual Interference between Predators Can Give Rise to Turing Spatial Structures, Ecology83,28,2002.
[20]M. P. Hassell, H. N. Comins, and R. M. May, Spatial Structure and Chaos in Insect Popu-lation Dynamics, Nature,353 255,1991.
[21]A. White, M. Begon, and R. G. Bowers, Host-Pathogen System in a Spatially Patchy Environment, Proceedings of the Royal Society of London B,263,325,1996.
[22]J. Bascompte, R. V. Sole, and N. Martinez., Population Cycles and Spatial Patterns in Snowshoe Hares:An Individual-Oriented Simulation, Journal of Theoretical Biology 187, 213,1997.
[23]P. Rohani, T. J. Lewis, D. Grunbaum, and G. D. Ruxton, Spatial Self-Organization in Ecology:Pretty Patterns or Robust Reality?, Trends in Ecology and Evolution,12,70, 1997.
[24]T. Reichenbach, M. Mobilia, and E. Frey, Noise and correlations in a spatial population model with cyclic competition. Phys. Rev. Lett.,99,238105,2007.
[25]T. Reichenbach, M. Mobilia, and E. Frey, Self-organization of mobile populations in cyclic competition, J. Theor. Biol.254,368,2008.
[26]A. T. Winfree, When Time Breaks Down (Princeton University Press, Princeton, NJ),1987.
[27]M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys.65851,1993.
[28]A. S. Mikhailov, K. Showalter, Control of waves, patterns and turbulence in chemical systems, Phys. Rep.425 79,2006.
[29]M. Hildebrand, M. Bar and M. Eiswirth, Statistics of Topological Defects and Spatiotem-poral Chaos in a Reaction-Diffusion System, Phys. Rev. Lett 75,1503,1995.
[30]N. D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys.51, 591,1979.
[31]K. Tomita, A. Ito and T. Ohta, Simplified model for Belousov-Zhabotinsky reaction, J. Theor. Biol.68,4,1977.
[32]M. Bar, N. Gottschalk, M. Eiswirth and G.Ertl, Spiral waves in a surface reaction:Model calculations, J. Chem. Phys.100,1202,1994.
[33]Zhang H., Cao Z. J. and Wu N. J. et al, Suppress Winfree Turbulence by Local Forcing Excitable Systems, Phys. Rev. Lett.94,188301,2005.
[34]P. C. Dauby, Th. Desaive and H. Croisier et al, Standing waves in the FitzHugh-Nagumo model of cardiac electrical activity, Phys. Rev. E 73,021908,2006.
[35]Wang P Y, Xie P, Yin H W, Control of spiral waves and turbulent states in a cardiac model by travelling-wave perturbations, Chin. Phys.12,674,2003.
(36]马军,靳伍银,易鸣,李延龙,时变反应扩散系统中螺旋波和湍流的控制,物理学报57,5,2008.
[37]Liu F C, Wang X F, Li X C et al, Controlling spiral wave with target wave in oscillatory systems, Chin. Phys.16,2640,2007.
[38]Yunfan Gong, David J. Christini, Antispiral Waves in Reaction-Diffusion Systems, Phys. Rev. Lett.90,088302,2003.
[39]L. Brusch, E. M. Nicola and M. Bar, Comment on "Antispiral Waves in Reaction-Diffusion Systems", Phys. Rev. Lett.92,089801,2004.
[40]A. T. Winfree, Electrical turbulence in three-dimensional heart muscle, Science 266,1003, 1994.
[41]M. P. Nash, A. V. Panfilov, Prog. Biophy.& Mole. Biol. Electromechanical model of ex-citable tissue to study reentrant cardiac arrhythmias,85,501,2004.
[42]Zhang Guo-Yong, Ma Jun, Gan Zheng Ning, and Chen Yong, Spiral wave in an inhomoge-neous excitable medium. Acta Physica Sinica 57:6815-09,2008(张国勇,马军,甘正宁,陈勇,非均匀可激介质中的螺旋波,物理学报57:6815,2008).
[43]Z. G. Zhang, Frustration effect on synchronization and chaos in coupled oscillators, Chin. Phys.10,08,2001.
[44]M. Hildebrand, J. X. Cui, E. Mihaliuk, J. C. Wang and K, Showalter, Synchronization of spatiotemporal patterns in locally coupled excitable media, Phys. Rev. E 68,026205,2003.
[45]L. C. Yu, Y. Chen and P. Zhang, Frequency and phase synchronization of two coupled neurons with channel noise, Eur. Phys. J. B 59,249,2007.
[46]P. Parmananda, Generalized synchronization of spatiotemporal chemical chaos, Phys. Rev. E 56,1595,1997.
[47]L. Junge and U. Parlitz, Phase synchronization of coupled Ginzburg-Landau equations, Phys. Rev. E 62,438,2000.
[48]V. B. Kazantsev, V. I. Nekorkin and D. V. Artyuhin, Synchronization, re-entry, and failure of spiral waves in a two-layer discrete excitable system, Phys. Rev. E 63,016212,2000.
[49]Zhang Guo-Yong, Ma Jun, Yu Lian-Chun, and Chen Yong. Synchronization of spiral waves in a two-layer coupled inhomogeneous excitable system. Chinese Physics B 17:4107-07, 2008.
[50]D. Barkley, Euclidean symmetry and the dynamics of rotating spiral waves, Phys. Rev. Lett.72,164,1994.
[51]D. Margerit and D. Barkley, Selection of Twisted Scroll Waves in Three-Dimensional Ex-citable Media, Phys. Rev. Lett.86,175,2001.
[52]D. Margerit and D. Barkley, Large-excitability asymptotics for scroll waves in three-dimensional excitable media, Phys. Rev. E 66,036214,2002.
[53]R. V. O'Neill, D. L. DeAngelis, J. B. Waide, and T. F. H. Allen, A Hierarchical Concept of Ecosystems. Princeton University Press, Princeton,1986.
[54]S. A. Levin, Ecosystems and the Biosphere as Complex Adaptive Systems. Ecosystems,1, 431,1998.
[55]S. A. Levin, Fragile Dominion:Complexity and the Commons. Perseus Books, Reading, 1999.
[56]J. D. Murray, Mathematical Biology I:An Introduction, Interdisciplinary Applied Mathe-matics,17, Springer-Verlag, New York,2002; Ⅰ. Spatial models and biomedical applications. Interdisciplinary Applied Mathematics,18. Springer-Verlag, New York,2003.
[57]J. Marro and R. Dickman, Nonequilibrium Phase Transitions in Lattice Models. Cambridge University Press:Cambridge,1999.
[58]K. Kaneko, Period-Doubling of Kink-Antikink Patterns, Quasi-Periodicity in Antiferro-like Structures ans Spatial Intermittency in Coupled Map Lattices:Towards a Prelude to a " Field Theory of Chaos", Progress in Theoretical Physics,72:480-86,1984.
[59]R. V. Sole and J. Valls, Order and Chaos in a Two-Dimensional Lotka-Volterra Coupled Map Lattice, Physics Letters A 153:330-36,1991.
[60]R. M. May. Stability and Complexity in Model Ecosystems. Princeton Univ. Press, Prince-ton, second edition,1975.
[61]G. F. Gause, The Struggle for Existence. Baltimore MD:Williams & Wilkins,1934.
[62]G. J. Hardin, The competitive exclusion principle. Science,131 1292,1960.
[63]M. Frean and E. R. Abraham, Rock-scissors-paper and the survival of the weakest, Proc. R. Soc. Lond. B 268,1323,2001.
[64]B. Kerr, M. A. Feldman, M. W. Feldman, and B. J. M. Bohannan, Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors games, Nature 418,171,2002.
[65]T. Reichenbach, M. Mobilia, and E. Frey, Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games, Nature 448,1046,2007.
[66]T. Reichenbach and E. Frey, Instability of spatial patterns and its ambiguous impact on species diversity, Phys. Rev. Lett.101,058102,2008.
[67]J. C. Claussen and A. Traulsen, Cyclic Dominance and Biodiversity in Well-Mixed Popu-lations, Phys. Rev. Lett.100,058104,2008.
[68]M. Peltomaki and M. Alava, Three-and four-state rps games with diffusion, Phys. Rev. E 78,031906,2008.
[69]B. Sinervo and C. M. Lively, The rock-paper-scissors game and the evolution of alternative male strategies, Nature 380,240,1996.
[70]Guo-Yong Zhang, Yong Chen, Wei-Kai Qi, and Shao-Meng Qin, Four-state rock-paper-scissors games in constrained Newman-Watts networks. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics) 79:062901-4,2009.
[71]P. Chesson and J. J. Kuang, The interaction between predation and competition, Nature 456,235,2008.
[72]U. Bastolla, et al., The architecture of mutualistic networks minimizes competition and increases biodiversity, Nature 458,1018,2009.
[73]J. Bascompte, Mutualistic networks, Frontiers in Ecology and the Environment 7:429-436, 2009.
[74]J. Bascompte et al, The nested assembly of plant-animal mutualistic networks, PNAS, 100:9383-9387,2003.
[75]J. Bascompte and P. Jordano, Plant-Animal Mutualistic Networks:The Architecture of Biodiversity, Annu. Rev. Ecol. Evol. Syst,38:567-93,2007.
[76]J. M. Olesen et al, The modularity of pollination networks, PNAS,104:19891-19896,2007.
[77]E. L. Rezende et al, Non-random coextinctions in phylogenetically structured mutualistic networks, Nature 448:925-928,2007.
[78]戈峰,现代生态学第二版,科学出版社,2008.
[79]P. Ehrlich and P. Raven, Butterflies and plants:a study in coevolution, Evolution,18:586-608,1964.
[80]K. Bawa, Plant-pollinator interactions in tropical rain forests, Annu. Rev. Ecol. Evol. Syst, 21:399-422,1990.
[81]R. Albert and AL. Barabasi, Statistical mechanics of complex networks, Rev. Mod. Phys 74,47,2002.
[82]LA. Amaral, A. Scala, M. Barthelemy, and HE. Stanley, Classes of Small-world networks, Proc. Natl. Acas. Sci. USA 97:11149-52,2000.
[83]MEJ. Newman and DJ. Watts, Renormalization group analysis of the small-world network model, Phys. Lett. A 263,341,1999.
[84]JE. Cohen, F. Briand, CM. Newman, Community Food Webs:Data and Theory, Berlin: Springer-Verlag,1990.
[85]P. Jordano, J. Bascompte, JM. Olesen, Invariant properties in coevolutionary networks of plant-animal interactions. Ecol. Lett.6:69-81,2003.
[2]R. V. Sole, J. Bascompte, Self-Organization in Complex Ecosystems. Princeton University Press,2006.
[3]张知彬,王祖望和李典谟,生态复杂性研究综述与展望。生态学报,18:433-441,1998.
[4]G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems:From Dissipative Structures to Order through fluctuations. Wiley& Sons, New York,1977.
[5]P. Cilliers. Complexity and Postmodernism:Understanding Complex Systems. Routledge, London,1998.
[6]H. Haken, Synergetics:An introduction:Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry, and Biology. Berlin Heidelberg:Springer-Verlag,1977.
[7]颜泽贤,范冬萍,张华夏,系统科学导论-复杂性探索。北京：人民出版社,2006.
[8]欧阳颀,反应扩散系统中的斑图动力学,上海科技教育出版社,2000.
[9]G. A. Cowan, D. Pines, and D. Meltzer, editors, Complexity:Metaphors, Models, and Re-ality. Perseus Books, Reading, Massachusetts,1994.
[10]M. M. Waldrop, Complexity:The Emerging Science at the Edge of Order and Chaos, Simon & Schuster, New York,1992.
[11]M. Gell-Mann, The Quark and the Jaguar:Adventures in the Simple and the Complex. W.H. Freeman and Company, New York,1994.
[12]A. J. Lotka, Elements of Physical Biology, Williams and Wilkins:Baltimore,1925.
[13]V. Volterra, Variations and Fluctuations of the Number of Individuals in Animal Species Living Together, Animal Ecology,409,1926.
[14]A. J. Nicholson and V. A. Bailey, The Balance of Animal Populations Ⅰ, Proceedings of the Zoological Society of London 3,551,1935.
[15]张大勇,理论生态学研究,高等教育出版社,2000.
[16]O. E. Rhodes, R. K. Chester, and M. H. Smith, (eds.), Population Dynamics in Ecological Space and Time, Chicage University Press, Chicago,1996.
[17]J. Bascompte and R. V. Sole, Modeling Spatiotemporal Dynamics in Ecology. Springer-Verlag, New York,1998.
[18]L. A. Segel, and J. Jackson, Dissipative Structure:An Explanantion and an Ecological Example. Journal of Theoretical Biology,37,545,1972.
[19]D. Alonso, F. Bartumeus, and J. Catalan, Mutual Interference between Predators Can Give Rise to Turing Spatial Structures, Ecology83,28,2002.
[20]M. P. Hassell, H. N. Comins, and R. M. May, Spatial Structure and Chaos in Insect Popu-lation Dynamics, Nature,353 255,1991.
[21]A. White, M. Begon, and R. G. Bowers, Host-Pathogen System in a Spatially Patchy Environment, Proceedings of the Royal Society of London B,263,325,1996.
[22]J. Bascompte, R. V. Sole, and N. Martinez., Population Cycles and Spatial Patterns in Snowshoe Hares:An Individual-Oriented Simulation, Journal of Theoretical Biology 187, 213,1997.
[23]P. Rohani, T. J. Lewis, D. Grunbaum, and G. D. Ruxton, Spatial Self-Organization in Ecology:Pretty Patterns or Robust Reality?, Trends in Ecology and Evolution,12,70, 1997.
[24]T. Reichenbach, M. Mobilia, and E. Frey, Noise and correlations in a spatial population model with cyclic competition. Phys. Rev. Lett.,99,238105,2007.
[25]T. Reichenbach, M. Mobilia, and E. Frey, Self-organization of mobile populations in cyclic competition, J. Theor. Biol.254,368,2008.
[26]A. T. Winfree, When Time Breaks Down (Princeton University Press, Princeton, NJ),1987.
[27]M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys.65851,1993.
[28]A. S. Mikhailov, K. Showalter, Control of waves, patterns and turbulence in chemical systems, Phys. Rep.425 79,2006.
[29]M. Hildebrand, M. Bar and M. Eiswirth, Statistics of Topological Defects and Spatiotem-poral Chaos in a Reaction-Diffusion System, Phys. Rev. Lett 75,1503,1995.
[30]N. D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys.51, 591,1979.
[31]K. Tomita, A. Ito and T. Ohta, Simplified model for Belousov-Zhabotinsky reaction, J. Theor. Biol.68,4,1977.
[32]M. Bar, N. Gottschalk, M. Eiswirth and G.Ertl, Spiral waves in a surface reaction:Model calculations, J. Chem. Phys.100,1202,1994.
[33]Zhang H., Cao Z. J. and Wu N. J. et al, Suppress Winfree Turbulence by Local Forcing Excitable Systems, Phys. Rev. Lett.94,188301,2005.
[34]P. C. Dauby, Th. Desaive and H. Croisier et al, Standing waves in the FitzHugh-Nagumo model of cardiac electrical activity, Phys. Rev. E 73,021908,2006.
[35]Wang P Y, Xie P, Yin H W, Control of spiral waves and turbulent states in a cardiac model by travelling-wave perturbations, Chin. Phys.12,674,2003.
(36]马军,靳伍银,易鸣,李延龙,时变反应扩散系统中螺旋波和湍流的控制,物理学报57,5,2008.
[37]Liu F C, Wang X F, Li X C et al, Controlling spiral wave with target wave in oscillatory systems, Chin. Phys.16,2640,2007.
[38]Yunfan Gong, David J. Christini, Antispiral Waves in Reaction-Diffusion Systems, Phys. Rev. Lett.90,088302,2003.
[39]L. Brusch, E. M. Nicola and M. Bar, Comment on "Antispiral Waves in Reaction-Diffusion Systems", Phys. Rev. Lett.92,089801,2004.
[40]A. T. Winfree, Electrical turbulence in three-dimensional heart muscle, Science 266,1003, 1994.
[41]M. P. Nash, A. V. Panfilov, Prog. Biophy.& Mole. Biol. Electromechanical model of ex-citable tissue to study reentrant cardiac arrhythmias,85,501,2004.
[42]Zhang Guo-Yong, Ma Jun, Gan Zheng Ning, and Chen Yong, Spiral wave in an inhomoge-neous excitable medium. Acta Physica Sinica 57:6815-09,2008(张国勇,马军,甘正宁,陈勇,非均匀可激介质中的螺旋波,物理学报57:6815,2008).
[43]Z. G. Zhang, Frustration effect on synchronization and chaos in coupled oscillators, Chin. Phys.10,08,2001.
[44]M. Hildebrand, J. X. Cui, E. Mihaliuk, J. C. Wang and K, Showalter, Synchronization of spatiotemporal patterns in locally coupled excitable media, Phys. Rev. E 68,026205,2003.
[45]L. C. Yu, Y. Chen and P. Zhang, Frequency and phase synchronization of two coupled neurons with channel noise, Eur. Phys. J. B 59,249,2007.
[46]P. Parmananda, Generalized synchronization of spatiotemporal chemical chaos, Phys. Rev. E 56,1595,1997.
[47]L. Junge and U. Parlitz, Phase synchronization of coupled Ginzburg-Landau equations, Phys. Rev. E 62,438,2000.
[48]V. B. Kazantsev, V. I. Nekorkin and D. V. Artyuhin, Synchronization, re-entry, and failure of spiral waves in a two-layer discrete excitable system, Phys. Rev. E 63,016212,2000.
[49]Zhang Guo-Yong, Ma Jun, Yu Lian-Chun, and Chen Yong. Synchronization of spiral waves in a two-layer coupled inhomogeneous excitable system. Chinese Physics B 17:4107-07, 2008.
[50]D. Barkley, Euclidean symmetry and the dynamics of rotating spiral waves, Phys. Rev. Lett.72,164,1994.
[51]D. Margerit and D. Barkley, Selection of Twisted Scroll Waves in Three-Dimensional Ex-citable Media, Phys. Rev. Lett.86,175,2001.
[52]D. Margerit and D. Barkley, Large-excitability asymptotics for scroll waves in three-dimensional excitable media, Phys. Rev. E 66,036214,2002.
[53]R. V. O'Neill, D. L. DeAngelis, J. B. Waide, and T. F. H. Allen, A Hierarchical Concept of Ecosystems. Princeton University Press, Princeton,1986.
[54]S. A. Levin, Ecosystems and the Biosphere as Complex Adaptive Systems. Ecosystems,1, 431,1998.
[55]S. A. Levin, Fragile Dominion:Complexity and the Commons. Perseus Books, Reading, 1999.
[56]J. D. Murray, Mathematical Biology I:An Introduction, Interdisciplinary Applied Mathe-matics,17, Springer-Verlag, New York,2002; Ⅰ. Spatial models and biomedical applications. Interdisciplinary Applied Mathematics,18. Springer-Verlag, New York,2003.
[57]J. Marro and R. Dickman, Nonequilibrium Phase Transitions in Lattice Models. Cambridge University Press:Cambridge,1999.
[58]K. Kaneko, Period-Doubling of Kink-Antikink Patterns, Quasi-Periodicity in Antiferro-like Structures ans Spatial Intermittency in Coupled Map Lattices:Towards a Prelude to a " Field Theory of Chaos", Progress in Theoretical Physics,72:480-86,1984.
[59]R. V. Sole and J. Valls, Order and Chaos in a Two-Dimensional Lotka-Volterra Coupled Map Lattice, Physics Letters A 153:330-36,1991.
[60]R. M. May. Stability and Complexity in Model Ecosystems. Princeton Univ. Press, Prince-ton, second edition,1975.
[61]G. F. Gause, The Struggle for Existence. Baltimore MD:Williams & Wilkins,1934.
[62]G. J. Hardin, The competitive exclusion principle. Science,131 1292,1960.
[63]M. Frean and E. R. Abraham, Rock-scissors-paper and the survival of the weakest, Proc. R. Soc. Lond. B 268,1323,2001.
[64]B. Kerr, M. A. Feldman, M. W. Feldman, and B. J. M. Bohannan, Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors games, Nature 418,171,2002.
[65]T. Reichenbach, M. Mobilia, and E. Frey, Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games, Nature 448,1046,2007.
[66]T. Reichenbach and E. Frey, Instability of spatial patterns and its ambiguous impact on species diversity, Phys. Rev. Lett.101,058102,2008.
[67]J. C. Claussen and A. Traulsen, Cyclic Dominance and Biodiversity in Well-Mixed Popu-lations, Phys. Rev. Lett.100,058104,2008.
[68]M. Peltomaki and M. Alava, Three-and four-state rps games with diffusion, Phys. Rev. E 78,031906,2008.
[69]B. Sinervo and C. M. Lively, The rock-paper-scissors game and the evolution of alternative male strategies, Nature 380,240,1996.
[70]Guo-Yong Zhang, Yong Chen, Wei-Kai Qi, and Shao-Meng Qin, Four-state rock-paper-scissors games in constrained Newman-Watts networks. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics) 79:062901-4,2009.
[71]P. Chesson and J. J. Kuang, The interaction between predation and competition, Nature 456,235,2008.
[72]U. Bastolla, et al., The architecture of mutualistic networks minimizes competition and increases biodiversity, Nature 458,1018,2009.
[73]J. Bascompte, Mutualistic networks, Frontiers in Ecology and the Environment 7:429-436, 2009.
[74]J. Bascompte et al, The nested assembly of plant-animal mutualistic networks, PNAS, 100:9383-9387,2003.
[75]J. Bascompte and P. Jordano, Plant-Animal Mutualistic Networks:The Architecture of Biodiversity, Annu. Rev. Ecol. Evol. Syst,38:567-93,2007.
[76]J. M. Olesen et al, The modularity of pollination networks, PNAS,104:19891-19896,2007.
[77]E. L. Rezende et al, Non-random coextinctions in phylogenetically structured mutualistic networks, Nature 448:925-928,2007.
[78]戈峰,现代生态学第二版,科学出版社,2008.
[79]P. Ehrlich and P. Raven, Butterflies and plants:a study in coevolution, Evolution,18:586-608,1964.
[80]K. Bawa, Plant-pollinator interactions in tropical rain forests, Annu. Rev. Ecol. Evol. Syst, 21:399-422,1990.
[81]R. Albert and AL. Barabasi, Statistical mechanics of complex networks, Rev. Mod. Phys 74,47,2002.
[82]LA. Amaral, A. Scala, M. Barthelemy, and HE. Stanley, Classes of Small-world networks, Proc. Natl. Acas. Sci. USA 97:11149-52,2000.
[83]MEJ. Newman and DJ. Watts, Renormalization group analysis of the small-world network model, Phys. Lett. A 263,341,1999.
[84]JE. Cohen, F. Briand, CM. Newman, Community Food Webs:Data and Theory, Berlin: Springer-Verlag,1990.
[85]P. Jordano, J. Bascompte, JM. Olesen, Invariant properties in coevolutionary networks of plant-animal interactions. Ecol. Lett.6:69-81,2003.