Abstract
Recently, continuous-time dynamical systems of mosquito populations have been studied. In this paper, we consider a discrete-time dynamical system, generated by an evolution quadratic operator of a mosquito population, and show that this system has two fixed points, which become saddle points under some conditions on the parameters of the system. We construct an evolution algebra, taking its matrix of structural constants equal to the Jacobian of the quadratic operator at a fixed point. Idempotent and absolute nilpotent elements, simplicity properties, and some limit points of the evolution operator corresponding to the evolution algebra are studied. We give some biological interpretations of our results.
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Notes
This equation is given in Lutambi et al. (2013, p. 201) too, but there the non-zero term \(- abehpr\theta \) is missed.
Depending on sign of parameters \(1-{\hat{p}}\), \(1-{\hat{l}}_1-2{\hat{l}}^2\epsilon \) the function \(\varphi \) has 4 forms. For example, in case of part 2) of Lemma 2 the function is \(\varphi (x)=f_{1-{\hat{p}}, p}(f_{1-{\hat{l}}_1-2{\hat{l}}_2\epsilon , a}(f_{1-{\hat{e}}, e}(x)))\), and if \(1-{\hat{p}}<0\), \(1-{\hat{l}}_1-2{\hat{l}}^2\epsilon >0\) then \(\varphi (x)=f_{1-{\hat{p}}, p}(-f_{1-{\hat{l}}_1-2{\hat{l}}_2\epsilon , a}(f_{1-{\hat{e}}, e}(x)))\).
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Acknowledgements
The work partially supported by Projects MTM2016-76327-C3-2-P and MTM2016- 79661-P of the Spanish Ministerio of Economía and Competitividad, and Research Group FQM 199 of the Junta de Andalucía (Spain), all of them include European Union FEDER support; grant 853/2017 Plan Propio University of Granada (Spain); Kazakhstan Ministry of Education and Science, grant 0828/GF4. We thank all (three) referees and Mark Lewis for their suggestions which were helpful to improve readability of the paper. We are very grateful to Farkhod Eshmatov for checking the English of this paper.
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Rozikov, U.A., Velasco, M.V. A discrete-time dynamical system and an evolution algebra of mosquito population. J. Math. Biol. 78, 1225–1244 (2019). https://doi.org/10.1007/s00285-018-1307-x
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DOI: https://doi.org/10.1007/s00285-018-1307-x