The dynamics of two diffusively coupled predator-prey populations.

I analyze the dynamics of predator and prey populations living in two patches. Within a patch the prey grow logistically and the predators have a Holling type II functional response. The two patches are coupled through predator migration. The system can be interpreted as a simple predator-prey metapopulation or as a spatially explicit predator-prey system. Asynchronous local dynamics are presumed by metapopulation theory. The main question I address is when synchronous and when asynchronous dynamics arise. Contrary to biological intuition, for very small migration rates the oscillations always synchronize. For intermediate migration rates the synchronous oscillations are unstable and I found periodic, quasi-periodic, and intermittently chaotic attractors with asynchronous dynamics. For large predator migration rates, attractors in the form of equilibria or limit cycles exist in which one of the patches contains no prey. The dynamical behavior of the system is described using bifurcation diagrams. The model shows that spatial predator-prey populations can be regulated through the interplay of local dynamics and migration.

[1]  Alan Hastings,et al.  Complex interactions between dispersal and dynamics: Lessons from coupled logistic equations , 1993 .

[2]  William G. Wilson,et al.  Mobility versus density-limited predator-prey dynamics on different spatial scales , 1991, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[3]  C. Huffaker.  Experimental studies on predation : dispersion factors and predator-prey oscillations , 1958 .

[4]  M. Holyoak.  Effects of nutrient enrichment on predator–prey metapopulation dynamics , 2000 .

[5]  Alan Hastings,et al.  Chaos in three species food chains , 1994 .

[6]  R. Macarthur,et al.  Graphical Representation and Stability Conditions of Predator-Prey Interactions , 1963, The American Naturalist.

[7]  J. L. Jackson,et al.  Dissipative structure: an explanation and an ecological example. , 1972, Journal of theoretical biology.

[8]  Lundberg,et al.  Synchronicity in population systems: cause and consequence mixed. , 1999, Trends in ecology & evolution.

[9]  J. Ripa,et al.  Analysing the Moran effect and dispersal: their significance and interaction in synchronous population dynamics , 2000 .

[10]  Lloyd,et al.  Synchronicity, chaos and population cycles: spatial coherence in an uncertain world. , 1999, Trends in ecology & evolution.

[11]  T. Clutton‐Brock,et al.  Noise and determinism in synchronized sheep dynamics , 1998, Nature.

[12]  A. Winfree.  The geometry of biological time , 1991 .

[13]  A. Hastings,et al.  Metapopulation Dynamics and Genetics , 1994 .

[14]  Andrew D. Taylor.  Metapopulations, Dispersal, and Predator‐Prey Dynamics: An Overview , 1990 .

[15]  Y. Kuznetsov,et al.  Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps: physics , 1993 .

[16]  V. Jansen,et al.  Local stability analysis of spatially homogeneous solutions of multi-patch systems , 2000, Journal of mathematical biology.

[17]  Michael P. Hassell,et al.  Spatial structure and chaos in insect population dynamics , 1991, Nature.

[18]  C. Stringer,et al.  Mass-spectrometric U-series dates for Israeli Neanderthal/early modern hominid sites , 1993, Nature.

[19]  G. Nachman.  An acarine predator‐prey metapopulation system inhabiting greenhouse cucumbers , 1991 .

[20]  W. Schaffer,et al.  Chaos reduces species extinction by amplifying local population noise , 1993, Nature.

[21]  W. Wilson,et al.  Pattern Formation and the Spatial Scale of Interaction between Predators and Their Prey. , 1998, Theoretical population biology.

[22]  I. Cattadori,et al.  The Moran effect: a cause of population synchrony. , 1999, Trends in ecology & evolution.

[23]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[24]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[25]  G D Ruxton,et al.  Spatial self-organisation in ecology: pretty patterns or robust reality? , 1997, Trends in ecology & evolution.

[26]  Bai-Lian Li,et al.  The Geometry of Ecological Interactions: Simplifying Spatial Complexity , 2002 .

[27]  Ilkka Hanski,et al.  Single‐species metapopulation dynamics: concepts, models and observations , 1991 .

[28]  G. F. Gause.  The struggle for existence , 1971 .

[29]  C. M. Place,et al.  An Introduction to Dynamical Systems , 1990 .

[30]  V. Jansen,et al.  Phase locking: another cause of synchronicity in predator-prey systems. , 1999, Trends in ecology & evolution.

[31]  Ilkka Hanski,et al.  Metapopulation dynamics : empirical and theoretical investigations , 1991 .

[32]  Mark L. Taper,et al.  Migration within Metapopulations: The Impact upon Local Population Dynamics , 1997 .

[33]  Josef Hofbauer,et al.  Evolutionary Games and Population Dynamics , 1998 .

[34]  Kevin S. McCann,et al.  Bifurcation Structure of a Three-Species Food-Chain Model , 1995 .

[35]  V. Jansen.  Effects of dispersal in a tri-trophic metapopulation model , 1995 .

[36]  Gonzalez,et al.  Metapopulation dynamics, abundance, and distribution in a microecosystem , 1998, Science.

[37]  T. Royama,et al.  Analytical Population Dynamics , 1994, Population and Community Biology Series.

[38]  Robert M. May,et al.  The spatial dynamics of host-parasitoid systems , 1992 .

[39]  Vincent A. A. Jansen,et al.  Regulation of predator-prey systems through spatial interactions:a possible solution to the paradox of enrichment. , 1995 .

[40]  M. Holyoak,et al.  The role of dispersal in predator-prey metapopulation dynamics , 1996 .

[41]  Ulf Dieckmann,et al.  The Geometry of Ecological Interactions: Simplifying Spatial Complexity , 2000 .

[42]  P. Ashwin,et al.  The dynamics ofn weakly coupled identical oscillators , 1992 .

[43]  M. Rosenzweig.  Paradox of Enrichment: Destabilization of Exploitation Ecosystems in Ecological Time , 1971, Science.

[44]  Koenig,et al.  Spatial autocorrelation of ecological phenomena. , 1999, Trends in ecology & evolution.

[45]  A. J,et al.  The Effects of a Pool of Dispersers on Host-parasitoid Systems , 1997 .

[46]  Solé,et al.  Chaos, Dispersal and Extinction in Coupled Ecosystems. , 1998, Journal of theoretical biology.

[47]  M. Rosenzweig,et al.  Stability of enriched aquatic ecosystems. , 1972, Science.

[48]  M. Gyllenberg,et al.  Does migration stabilize local population dynamics? Analysis of a discrete metapopulation model. , 1993, Mathematical biosciences.

[49]  J. D. van der Laan,et al.  Predator—prey coevolution: interactions across different timescales , 1995, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[50]  Y. Pomeau,et al.  Order within chaos , 1986 .

[51]  Tamás Czárán,et al.  Spatiotemporal models of population and community dynamics , 1998 .

[52]  S Rinaldi,et al.  Remarks on food chain dynamics. , 1996, Mathematical biosciences.

[53]  E. J. Maly.  A Laboratory Study of the Interaction Between the Predatory Rotifer Asplanchna and Paramecium , 1969 .

[54]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[55]  Andrew D. Taylor,et al.  Empirical Evidence for Metapopulation Dynamics , 1997 .

[56]  F. Adler.  Migration Alone Can Produce Persistence of Host-Parasitoid Models , 1993, The American Naturalist.

[57]  V. Jansen.  Theoretical aspects of metapopulation dynamics. , 1994 .

[58]  M. Holyoak,et al.  Persistence of an Extinction-Prone Predator-Prey Interaction Through Metapopulation Dynamics , 1996 .

[59]  M. Gilpin,et al.  Metapopulation Biology: Ecology, Genetics, and Evolution , 1997 .