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**Population diffusion in a two-patch environment.**
*(English)*
Zbl 0672.92021

Summary: A model of a single-species population diffusing in a two-patch environment is proposed. It is shown that there exists a positive, monotonic, continuous steady-state solution with continuous flux, in the cases of both reservoir and no-flux boundary conditions, that is asymptotically stable. In the case of patches with equal carrying capacities, it is shown that the uniform steady state is globally asymptotically stable.

### MSC:

92D40 | Ecology |

35Q99 | Partial differential equations of mathematical physics and other areas of application |

35B35 | Stability in context of PDEs |

35B40 | Asymptotic behavior of solutions to PDEs |

### Keywords:

single-species population; two-patch environment; positive, monotonic, continuous steady-state solution; reservoir and no-flux boundary conditions; uniform steady state; globally asymptotically stable
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\textit{H. I. Freedman} et al., Math. Biosci. 95, No. 1, 111--123 (1989; Zbl 0672.92021)

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### References:

[1] | Allen, L.J., Persistence, extinction and critical patch number for island populations, J. math. biol., 24, 617-625, (1987) · Zbl 0603.92019 |

[2] | Bailey, P.; Shampine, L.F.; Waltman, P.E., Nonlinear two point boundary value problems, (1968), Academic New York · Zbl 0169.10502 |

[3] | Bernfeld, S.R.; Lakshmikantham, V., An introduction to nonlinear boundary value problems, (1974), Academic New York · Zbl 0286.34018 |

[4] | Cantrell, R.S.; Cosner, C., Diffusive logistic equations with indefinite weights: population models in disrupted environments, (1989), (preprint) · Zbl 0711.92020 |

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

[6] | Freedman, H.I.; Shukla, J.B., The effect of a predator resource on a diffusive predator prey system, Natural resources modeling, (1989), (in press) · Zbl 0850.92061 |

[7] | Hallam, T.G., A temporal study of diffusion effects on a population modelled by quadratic growth, Nonlin. anal. theory methods appl., 3, 123-133, (1979) · Zbl 0419.35048 |

[8] | Kierstead, H.; Slobodkin, L.B., The size of water masses containing plankton blooms, J. mar. res., 12, 141-147, (1953) |

[9] | Levin, S.A., Population models and community structure in heterogeneous environments, (), 295-321 |

[10] | McMurtrie, R., Persistence and stability of single-species and prey-predator systems in spatially heterogeneous environments, Math. biosci., 39, 11-51, (1978) · Zbl 0384.92011 |

[11] | Pacala, S.W.; Roughgarden, J., Spatial heterogeneity and interspecific competition, Theor. pop. biol., 21, 92-113, (1982) · Zbl 0492.92017 |

[12] | Pao, C.V., Coexistence and stability of a competition-diffusion system in population dynamics, J. math. anal. appl., 83, 54-76, (1981) · Zbl 0479.92013 |

[13] | Shigesada, N.; Roughgarden, J., The role of rapid dispersal in the population dynamics of competition, Theor. pop. biol., 21, 353-372, (1982) · Zbl 0494.92021 |

[14] | Skellam, J.G., Random dispersal in theoretical populations, Biometrika, 36, 199-218, (1951) · Zbl 0043.14401 |

[15] | Smith, O.L., The influence of environmental gradients on ecosystem stability, Am. nat., 116, 1-24, (1980) |

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