Computer simulations and numerical methods in two-species models of the spatial community

Egor G. Galkin, Viktor K. Zelenkov, Alexey A. Nikitin

Abstract


This publication begins a series of works devoted to the comparison of the results of numerical methods for solving integral equations and the results of computer simulations using Poisson processes. The pros and cons of these two approaches are highlighted. The main subject of study is the model of two-species communities proposed in the works of W. Dickman and R. Lowe. The mathematical formulation of this model for an infinite domain is described. A modification of this model for work in a limited area is given. In the future, this is required for stochastic modeling. Further, the most important moments of the stochastic model are briefly described, and the features of the computer simulations built for it are noted – the algorithm used is described and the data structures used are noted. At the end of the work is a number of pictures, with the results of the above comparisons. The comparison is made within the framework of two classical scenarios of two-species models -- Competition-colonization trade-off (Bay-Run scenario) and heteromyopia (People-and-Mosquitoes scenario). These drawings show very similar results, as well as some discrepancies. All this allows us to hope that the work in this direction will be very fruitful in the future.


Full Text:

PDF (Russian)

References


Law R., Dieckmann U. Moment approximations of individual-based models // The Geometry of Ecological Interactions: Simplifying Spatial Complexity / Ed. by U. Dieckmann, R. Law, J. Metz. Cambridge University Press, 2000. P. 252–270.

Law R., Murrel D., Dieckmann U. Population Growth in Space and Time: Spatial Logistic Equations// Ecology. 2003, V.84, N.1, P.252-262. URL: http://www.jstor.org/stable/3108013.

Murrell D. J., Law R. Heteromyopia and the spatial coexistence of similar competitors // Ecology Letters. 2003. V.6, N.1. P.48–59. URL: http://dx.doi.org/10.1046/j.1461-0248.2003.00397.x.

Velazquez Jorge, Garrahan Juan P., Eichhorn Markus P. Spatial Complementarity and the Coexistence of Species // PLOS ONE. 2014. 12. V.9, N.12. P.1–20. URL:https://doi.org/10.1371/journal.pone.0114979

Nikitin A. A., Savostianov A. S. Nontrivial stationary points of two-species self-structuring communities // Moscow University Computational Mathematics and Cybernetics. 2017. Vol. 41, no. 3. P. 122–129.

Nikitin A. A. O zamykanii prostranstvennyh momentov v biologicheskoj modeli, i integral'nyh uravnenijah, k kotorym ono privodit // International Journal of Open Information Technologies. 2018. T. 6, No 10. S. 1–8.

Bodrov A.G., Nikitin A.A. Issledovanie integral'nogo uravnenija plotnosti biologicheskogo vida v prostranstvah razlichnyh razmernostej // Vestn. Mosk. un-ta. Ser. 15. Vychisl. matem. i kibern. 2015. No 4. S. 7–13. (Bodrov A. G., Nikitin A. A. Examining the biological species steady-state density equation in spaces with different dimensions // Moscow Univ. Comput. Math. and Cybern. 2015.39. N 4. P. 157–162.)

Ohta, T. (2013). Molecular Evolution: Nearly Neutral Theory. In eLS, (Ed.). doi:10.1002/9780470015902.a0001801.pub4

Ohta T. Near-neutrality in evolution of genes and gene regulation. Proc Natl Acad Sci U S A. 2002;99(25):16134–16137. doi:10.1073/pnas.252626899

Raghib, M., Hill, N.A. & Dieckmann, U. J. Math. Biol. (2011) 62: 605. https://doi.org/10.1007/s00285-010-0345-9

Rand, David. (2009). Correlation Equations and Pair Approximations for Spatial Ecologies. 10.1002/9781444311501.ch4.

B. Błaszczyszyn, Factorial moment expansion for stochastic systems, Stochastic Processes and their Applications, Volume 56, Issue 2, 1995, Pages 321-335, ISSN 0304-4149, https://doi.org/10.1016/0304-4149(94)00071-Z.

D.J. Daley; David Vere-Jones (12 November 2007). An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure. Springer Science & Business Media. pp. 13–14. ISBN 978-0-387-21337-8.


Refbacks

  • There are currently no refbacks.


Abava  Absolutech Fruct 2020

ISSN: 2307-8162