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Solution of a Nonlinear Integral Equation Arising in the Moment Approximation of Spatial Logistic Dynamics

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  • Mikhail Nikolaev

    (Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
    Moscow Center for Fundamental and Applied Mathematics, 119071 Moscow, Russia)

  • Alexey Nikitin

    (Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
    Trapeznikov Institute of Control Sciences of the Russian Academy of Sciences, 117997 Moscow, Russia
    Faculty of Computational Mathematics and Cybernetics, Shenzhen MSU-BIT University, Shenzhen 518172, China)

  • Ulf Dieckmann

    (Complexity Science and Evolution Unit, Okinawa Institute of Science and Technology Graduate University, Onna 904-0412, Japan
    Advancing Systems Analysis Program & Evolution and Ecology Program, International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria
    Research Center for Integrative Evolutionary Science, The Graduate University for Advanced Studies, Hayama 240-0193, Japan)

Abstract

We investigate a nonlinear integral equation derived through moment approximation from the individual-based representation of spatial logistic dynamics. The equation describes how the densities of pairs of individuals represented by points in continuous space are expected to equilibrate under spatially explicit birth–death processes characterized by constant fecundity with local natal dispersal and variable mortality determined by local competition. The equation is derived from a moment hierarchy truncated by a moment closure expressing the densities of triplets as a function of the densities of pairs. Focusing on results for individuals inhabiting two-dimensional habitats, we explore the solvability of the equation by introducing a dedicated space of functions that are integrable up to a constant. Using this function space, we establish sufficient conditions for the existence of solutions of the equation within a zero-centered ball. For illustration and further insights, we complement our analytical findings with numerical results.

Suggested Citation

  • Mikhail Nikolaev & Alexey Nikitin & Ulf Dieckmann, 2024. "Solution of a Nonlinear Integral Equation Arising in the Moment Approximation of Spatial Logistic Dynamics," Mathematics, MDPI, vol. 12(24), pages 1-20, December.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:24:p:4033-:d:1550233
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    References listed on IDEAS

    as
    1. W. R. Young & A. J. Roberts & G. Stuhne, 2001. "Reproductive pair correlations and the clustering of organisms," Nature, Nature, vol. 412(6844), pages 328-331, July.
    2. Samraat Pawar & Anthony I. Dell & Van M. Savage, 2012. "Dimensionality of consumer search space drives trophic interaction strengths," Nature, Nature, vol. 486(7404), pages 485-489, June.
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