Interactions between local dynamics and dispersal: Insights from single species models.

*(English)*Zbl 0894.92029Summary: Population persistence in patchy environments is expected to result from an interaction between local density dependence, dispersal, and spatial heterogeneity. Using two-patch models of single species, I explore two aspects of this interaction that have hitherto received little attention. First, how is the interaction affected when local density dependence changes from negative (logistic) to positive (Allee)? Second, how does dispersal mortality influence persistence? When local dynamics are logistic, dispersal changes the strength of negative density dependence within patches without much between-patch effect. For example, dispersal mortality reduces the population growth rate and counteracts the tendency towards complex dynamics. Density-dependent disperal amplifies the nonlinearity in the growth rate, thus opposing the stabilizing effects of dispersal mortality.

Spatial heterogeneity has little or no effect on stability. In contrast to the logistic, dispersal under Allee dynamics creates between-patch effects that bring about a qualitative change. Patches that fall below the Allee threshold are rescued from extinction by immigrants from patches that are above the threshold. Density-dependent dispersal enhances this rescue effect. Persistence below the extinction threshold is contingent on spatial heterogeneity. Dispersal mortality is of little or no consequence.

Spatial heterogeneity has little or no effect on stability. In contrast to the logistic, dispersal under Allee dynamics creates between-patch effects that bring about a qualitative change. Patches that fall below the Allee threshold are rescued from extinction by immigrants from patches that are above the threshold. Density-dependent dispersal enhances this rescue effect. Persistence below the extinction threshold is contingent on spatial heterogeneity. Dispersal mortality is of little or no consequence.

##### Keywords:

logistic growth; Allee effects; density-dependent dispersal; population persistence; spatial heterogeneity; dispersal mortality
PDF
BibTeX
XML
Cite

\textit{P. Amarasekare}, Theor. Popul. Biol. 53, No. 1, 44--59 (1998; Zbl 0894.92029)

Full Text:
DOI

##### References:

[1] | Allee, W.C., Animal aggregations: a study in general sociology, (1931), Univ. of Chicago Press Chicago |

[2] | Allen, J.C.; Schaffer, W.M.; Rosko, D., Chaos reduces species extinctions by amplifying local population noise, Nature, 364, 229-232, (1993) |

[3] | Bascompte, J.; Sole, R.V., Spatially induced bifurcations in single-species population dynamics, J. anim. ecol., 63, 256-264, (1994) |

[4] | Bulmer, M.G., Theory of perdator – prey oscillations, Theor. popul. biol., 47, 137-150, (1976) · Zbl 0352.92015 |

[5] | Chesson, P.L., Models for spatially distributed populations: the effect of within-patch variability, Theor. popul. biol., 19, 288-325, (1981) · Zbl 0472.92015 |

[6] | Doebli, M., Dispersal and dynamics, Theor. popul. biol., 47, 82-106, (1995) |

[7] | Freedman, H.I., Deterministic mathematical models in population ecology, (1987), HIFR Consulting Edmonton · Zbl 0448.92023 |

[8] | Gyllenberg, M.; Soderbacka, G.; Ericsson, S., Does migration stabilize local population dynamics? analysis of a discrete metapopulation model, Math. biosci., 118, 25-49, (1993) · Zbl 0784.92019 |

[9] | Hanski, I.; Kuussaari, M.; Nieminen, M., Metapopulation structure and migration in the butterfly, Melitaea cinixa, ecology, 75, 747-762, (1994) |

[10] | Hassell, M.P.; Miramontes, O.; Rohani, P.; May, R.M., Appropriate formulations for dispersal in spatially structured models: comments on bascompte and sole, J. anim. ecol., 64, 662-664, (1995) |

[11] | Hastings, A., Complex interactions between dispersal and dynamics: lessons from coupled logistic equations, Ecology, 74, 1362-1372, (1993) |

[12] | Holt, R.D., Population dynamics in two-patch environments: some anomalous consequences of an optimal habitat distribution, Theor. popul. biol., 28, 181-208, (1985) · Zbl 0584.92022 |

[13] | Kareiva, P.M., Population dynamics in spatially complex environments: theory and data, Philos. trans. R. soc. London B, 330, 175-190, (1990) |

[14] | Karlin, S.; MacGregor, J., Polymorphisms for genetic and ecological systems with weak coupling, Theor. popul. biol., 3, 210-238, (1972) · Zbl 0262.92007 |

[15] | Kruess, A.; Tscharntke, T, Habitat fragmentation, species loss, and biological control, Science, 264, 1581-1584, (1994) |

[16] | Levin, S.A., Dispersion and population interactions, Am. nat., 108, 207-228, (1974) |

[17] | Levins, R., Some demographic and genetic consequences of environmental heterogeneity for biological control, Bull. entomol. soc. am., 15, 237-240, (1969) |

[18] | Levins, R., Extinction, Some mathematical problems in biology, (1979), Am. Math. Soc Providence, p. 77-107 |

[19] | Lewis, M.A.; Kareiva, P., Allee dynamics and the spread of invading organisms, Theor. popul. biol., 43, 141-158, (1993) · Zbl 0769.92025 |

[20] | May, R.M., Stability and complexity in model ecosystems, (1974), Princeton Univ. Press Princeton |

[21] | McCallum, H.I., Effects of immigration on chaotic population dynamics, J. theor. biol., 154, 227-284, (1992) |

[22] | Murdoch, W.W.; Briggs, C.J.; Nisbet, R.M.; Gurney, W.S.C.; Stewart-Oaten, A., Aggregation and stability in metapopulation models, Am. nat., 140, 41-58, (1992) |

[23] | Murray, J.D., Mathematical biology, (1989), Springer-Verlag New York · Zbl 0682.92001 |

[24] | Nisbet, R.M.; Gurney, W.S.C., Modeling fluctuating populations, (1982), Wiley New York · Zbl 0593.92013 |

[25] | Odell, G.M., Qualitative theory of systems of ordinary differential equations, including phase plane analysis and the use of Hopf bifurcation theorem, (), 649-727 |

[26] | Pulliam, H.R., Sources, sinks and population regulation, Am. nat., 132, 652-661, (1988) |

[27] | Ruxton, G., Linked populations can still be chaotic, Oikos, 68, 347-348, (1993) |

[28] | Ruxton, G., Low levels of immigration between chaotic populations can reduce system extinctions by inducing asynchronous regular cycles, Proc. R. soc. London B, 256, 189-193, (1994) |

[29] | Stacey, P.B.; Taper, M., Environmental variation and the persistence of small populations, Ecol. appl., 2, 18-29, (1992) |

[30] | Vandermeer, J.H., Generalized models of two species interactions: A graphical analysis, Ecology, 54, 809-818, (1972) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.