Abstract
A spatially explicit metapopulation model with density-dependent dispersal is proposed in order to study the stability of synchronous dynamics. A stability criterion is obtained based on the computation of the transversal Liapunov number of attractors on the synchronous invariant manifold. We examine in detail a special case of density-dependent dispersal rule where migration does not occur if the patch density is below a certain critical density, while the fraction of individuals that migrate to other patches is kept constant if the patch density is above the threshold level. Comparisons with density-independent migration models indicate that this simple density-dependent dispersal mechanism reduces the stability of synchronous dynamics. We were able to quantify exactly this loss of stability through the frequency that synchronous trajectories are above the critical density.
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Allen, J.C., Schaffer, W.M., Rosko, D., 1993. Chaos reduces species extinction by aplifying local population noise. Nature 364, 229–232.
Amarasekare, P., 2004. The role of density-dependent dispersal in source-sink dynamics. J. Theor. Biol. 226, 159–168.
Blasius, B., Huppert, A., Stone, L., 1999. Complex dynamics and phase synchronization in spatially extended ecological systems. Nature 399, 353–359.
Caswell, H., 1989. Matrix Population Models, Sinauer, Sunderland, MA.
Cazelles, B., 2001. Dynamics with riddled basins of attraction in models of interacting populations. Chaos, Solitons Fractals 12, 301–311.
Cazelles, B., Bottani, S., Stone, L., 2001. Unexpected coherence and conservation. Proc. R. Soc. Lond. B268, 2595–2602.
De Castro, M.L., Silva, J.A.L., Justo, D.R., 2005. Stability in a age-structured metapopulaton model. J. Math. Biol. (Online).
Gonzalez, A., Lawton, J.H., Gilbert, F.S., Blackburn, T.M., Evans-Freke, I., 1998. Metapopulation dynamics, abundance, and distribution in a microecosystem. Science 281, 2045–2047.
Denno, R.F., Peterson, M.A., 1995. Density-dependent dispersal and its consequences for population dynamics. In: Capuccino, N., Price, P.W. (Eds.), Population Dynamics New Approaches and Synthesis. Academic Press, London, pp. 113–130.
Doebeli, M., Ruxton, G.D., 1998. Stabilization through spatial pattern formation in metapopulations with long-range dispersal. Proc. R. Soc. Lond. B265, 1325–1332.
Earn, D.J.D., Levin, S.A., Rohani, P., 2000. Coherence and conservation. Science 290, 1360–1364.
Eckmann, J.P., Ruelle, D., 1985. Ergodic theory of chaos and strange attractors. Am. Phys. Soc. 57, 617–656.
Hanski, I., Gilpin, M.E., 1997. Metapopulation Biology: Ecology Genetics and Evolution, Academic Press, London.
Harrison, M.A., Lai, Y.C., Holt, R.D., 2001. Dynamical mechanism for coexistence of dispersing species. J. Theor. Biol. 213, 53–72.
Heagy, J.F., Platt, N., Hammel, S.M., 1994. Characterization of on-off intermittency. Phys. Rev. E 49, 1140–1150.
Heino, M., Kaitala, V., Ranta, E., Lindström, J., 1997. Synchronous dynamics and rates of extiction in spatially structured populations. Proc. R. Soc. Lond. B264, 481–486.
Huang, Y., Diekmann, O., 2003. Interspecific influence on mobility and Turing instability. Bull. Math. Biol. 65, 143–156.
Huang, Y., Diekmann, O., 2004. Double-jump migration and diffusive instability. Bull. Math. Biol. 66, 487–504.
Jang, S.R.J., Mitra, A.K., 2000. Equilibrium stability of single—species metapopulations. Bull. Math. Biol. 62, 155–161.
Jansen, V.A.A., Lloyd, A., 2000. Local stability analysis of spatially homogeneous solutions of multi-patch systems. J. Math. Biol. 41, 232–252.
Lancaster, P., Tismenetsky, M., 1985. The Theory of Matrices, Academic Press, London.
Lloyd, A.L., Jansen, V.A.A., 1994. Spatiotemporal dynamics of epidemics: Synchrony in metapopulation models. Math. Biosci. 188, 1–16.
Matthysen, E., 2005. Density-dependent dispersal in birds and mammals. Ecography 28, 403–416.
May, R.M., 1976. Simple mathematical models with very complicated dynamics. Nature 261, 459–467.
Nagai, Y., Lai, Y.-C., 1997. Periodic-orbit theory of the blowout bifurcation. Phys. Rev. E56(4), 4031–4041.
Ott, E., 1994. Chaos in Dynamical Systems. Cambridge University Press, New York.
Ott, E., Sommerer, J.C., 1994. Blowout bifurcations: The occurence of riddled basins and on-off intermittency. Physics Letters A 188, 39–47.
Platt, N., Spiegel, E.A., Tresser, C., 1993. On-off intermittency: A mechanism for bursting. Phys. Rev. Lett. 70, 279–282.
Rohani, P.R., May, R.M., Hassel, M.P., 1996. Metapopulations and equilibrium stability: The effects of spatial structure. J. Theor. Biol. 181, 97–109.
Rohani, P., Ruxton, G.D., 1999. Dispersal-induced instabilities in host-parasitoid metapopulations. Theor. Popu. Biol. 55, 23–36.
Rosenblum, M.G., Plkovsky, A.S., Kurths, J., 1996. Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 76, 1804–1807.
Ruxton, G.D., 1996. Density-dependent migration and stability in a system of linked populations. Bull. Math. Biol. 58, 643–660.
Ruxton, G.D., Rohani, P., 1998. Fitness-dependent dispersal in metapopulations and its consequence for persistence and synchrony. J. Anim. Ecol. 67, 530–539.
Scheuring, I., Jánosi, I.M., 1996. When two and two make four: A structured population without chaos. J. Theor. Biol. 178, 89–97.
Silva, J.A.L., De Castro, M.L., Justo, D.A.R., 2000. Synchronism in a metapopulation model. Bull. Math. Biol. 62, 337–349.
Silva, J.A.L., De Castro, M.L., Justo, D.A.R., 2001. Stability in a metapopulation model with density-dependent dispersal. Bull. Math. Biol. 63, 485–506.
Simberloff, D., Farr, J.A., Cox, J., Mehlman, D.W., 1992. Movement Corridors: conservaton bargains or poor investments? Conserv. Biol. 6, 493–504.
Solé, R.V., Gamarra, J.P.G., 1998. Chaos, dispersal and extinction in coupled ecosystems. J. Theor. Biol. 193, 539–541.
Tilman, D., Kareiva, P., 1997. Spatial Ecology: The Role of Space in Population Dynamics and Iterspecific Interactions. Princeton University Press, Princeton.
Ylikarjula, J., Alaja, S., Laakso, J., Tesar, D., 2000. Effects of patch number and dispersal patterns on population dynamics and synchrony. J. Theor. Biol. 207, 377–387.
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Silva, J.A.L., Giordani, F.T. Density-dependent migration and synchronism in metapopulations. Bull. Math. Biol. 68, 451–466 (2006). https://doi.org/10.1007/s11538-005-9054-8
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DOI: https://doi.org/10.1007/s11538-005-9054-8