Solution of a Nonlinear Integral Equation Arising in the Moment Approximation of Spatial Logistic Dynamics
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 spatia...
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MDPI AG
2024-12-01
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| author | Mikhail Nikolaev Alexey Nikitin Ulf Dieckmann |
| author_facet | Mikhail Nikolaev Alexey Nikitin Ulf Dieckmann |
| author_sort | Mikhail Nikolaev |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-bc63f2af81514c1895d886d35bcbb888 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-bc63f2af81514c1895d886d35bcbb8882024-12-27T14:38:21ZengMDPI AGMathematics2227-73902024-12-011224403310.3390/math12244033Solution of a Nonlinear Integral Equation Arising in the Moment Approximation of Spatial Logistic DynamicsMikhail Nikolaev0Alexey Nikitin1Ulf Dieckmann2Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, RussiaFaculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, RussiaComplexity Science and Evolution Unit, Okinawa Institute of Science and Technology Graduate University, Onna 904-0412, JapanWe 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.https://www.mdpi.com/2227-7390/12/24/4033nonlinear integral equationsspatial logistic dynamicsindividual-based modelsfixed-point |
| spellingShingle | Mikhail Nikolaev Alexey Nikitin Ulf Dieckmann Solution of a Nonlinear Integral Equation Arising in the Moment Approximation of Spatial Logistic Dynamics Mathematics nonlinear integral equations spatial logistic dynamics individual-based models fixed-point |
| title | Solution of a Nonlinear Integral Equation Arising in the Moment Approximation of Spatial Logistic Dynamics |
| title_full | Solution of a Nonlinear Integral Equation Arising in the Moment Approximation of Spatial Logistic Dynamics |
| title_fullStr | Solution of a Nonlinear Integral Equation Arising in the Moment Approximation of Spatial Logistic Dynamics |
| title_full_unstemmed | Solution of a Nonlinear Integral Equation Arising in the Moment Approximation of Spatial Logistic Dynamics |
| title_short | Solution of a Nonlinear Integral Equation Arising in the Moment Approximation of Spatial Logistic Dynamics |
| title_sort | solution of a nonlinear integral equation arising in the moment approximation of spatial logistic dynamics |
| topic | nonlinear integral equations spatial logistic dynamics individual-based models fixed-point |
| url | https://www.mdpi.com/2227-7390/12/24/4033 |
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