Research article

Bifurcations in discontinuous mathematical models with control strategy for a species


  • Received: 29 September 2021 Accepted: 05 December 2021 Published: 10 December 2021
  • In this paper a preliminary mathematical model is proposed, by means of a system of ordinary differential equations, for the growth of a species. In this case, the species does not interact with another species and is divided into two stages, those that have or have not reached reproductive maturity, with natural and control mortality for both stages. When performing a qualitative analysis to determine conditions in the parameters that allow the extinction or preservation of the species, a modification is made to the model when only control is assumed for each of the stages if the number of species in that stage is above a critical value. These studies are carried out by bifurcation analysis with respect to two parameters: control for each stage and their critical values. It is concluded that for certain conditions in their parameters, the dynamics in each of the controlled stages converge to their critical values.

    Citation: Christian Cortés García. Bifurcations in discontinuous mathematical models with control strategy for a species[J]. Mathematical Biosciences and Engineering, 2022, 19(2): 1536-1558. doi: 10.3934/mbe.2022071

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  • In this paper a preliminary mathematical model is proposed, by means of a system of ordinary differential equations, for the growth of a species. In this case, the species does not interact with another species and is divided into two stages, those that have or have not reached reproductive maturity, with natural and control mortality for both stages. When performing a qualitative analysis to determine conditions in the parameters that allow the extinction or preservation of the species, a modification is made to the model when only control is assumed for each of the stages if the number of species in that stage is above a critical value. These studies are carried out by bifurcation analysis with respect to two parameters: control for each stage and their critical values. It is concluded that for certain conditions in their parameters, the dynamics in each of the controlled stages converge to their critical values.



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    [1] Y. A. Kuznetsov, Elements of applied bifurcation theory, Springer Science and Business Media, (2013).
    [2] J. D. Murray, Mathematical biology: I. an introduction, Springer, (2002).
    [3] L. Edelstein-Keshet, Mathematical models in biology, Society for Industrial and Applied Mathematics, (2005).
    [4] S. Y. Tang, Y. N. Xiao, Biological dynamics of single species, Science Press, (2008).
    [5] C. V. Marín, A. M. Loaiza, H. Zapata, L. Alape, Modelado de estrategias para el control químico y biológico del Aedes aegypti (Diptera: Culicidae), Math. Educ. Univer., 19 (2011), 63–78. doi: 10.1590/S1413-73722008000300013. doi: 10.1590/S1413-73722008000300013
    [6] J. García, A. M. Loaiza, Un modelo de crecimiento poblacional De ædes ægypti con capacidad de carga Logística, J. Math. Theory Appl., 25 (2018), 79–113. doi: 10.15517/RMTA.V1I25.32233. doi: 10.15517/RMTA.V1I25.32233
    [7] A. F. Filippov, Diferential equations with discontinuous righthand sides, Mathematics and its Applications, Kluwer Academic Publishers Group, (1988).
    [8] F. Dercole, A. Gragnani, S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models, Theor. Popul. Biol., 72 (2007), 197–213. doi: 10.1016/j.tpb.2007.06.003. doi: 10.1016/j.tpb.2007.06.003
    [9] T. Zhao, Y. Xiao, Non-smooth plant disease models with economic thresholds, Math. Biosci., 241 (2013), 34–48. doi: 10.1016/j.mbs.2012.09.005. doi: 10.1016/j.mbs.2012.09.005
    [10] A. L. Wang, Y. N. Xiao, R. Smith, Using non-smooth models to determine thresholds for microbial pest management, J. Math. Biol., 78 (2019), 1389–1424. doi: 10.1007/s00285-018-1313-z. doi: 10.1007/s00285-018-1313-z
    [11] H. Zhou, X. Wang, S. Tang, Global dynamics of non-smooth Filippov pest-natural enemy system with constant releasing rate, Math. Biosci. Eng., 16 (2019), 7327–7361. doi: 0.3934/mbe.2019366.
    [12] C. Cortés, Bifurcaciones en modelo gause depredador-presa con discontinuidad, J. Math. Theory Appl., 28 (2021), 183–208. doi: 10.15517/rmta.v28i2.36084. doi: 10.15517/rmta.v28i2.36084
    [13] Y. Kuznetsov, S. Rinaldi, A. Gragnani, One-parameter bifurcations in planar filippov systems, Int. J. Bifurcation Chaos, 13 (2003), 2157–2188. doi: 10.1142/S0218127403007874. doi: 10.1142/S0218127403007874
    [14] F. Dercole, A. Gragnani, Y. Kuznetsov, S. Rinaldi, Numerical sliding bifurcation analysis: an application to a relay control system, IEEE Trans. Circuits Systems I Fund. Theory Appl., 50 (2003), 1058–1063. doi: 110.1109/TCSI.2003.815214.
    [15] F. Dercole, Y. Kuznetsov, SlideCont: An auto97 driver for bifurcation analysis of filippov systems, ACM Trans. Math. Soft, 31 (2005), 95–119. doi: 10.1145/1055531.1055536. doi: 10.1145/1055531.1055536
    [16] M. Guardia, T. M. Seara, M. A. Teixeira, Generic bifurcations of low codimension of planar Filippov systems, J. Differ. Equ., 250 (2010), 1967–2023. doi: 10.1016/j.jde.2010.11.016. doi: 10.1016/j.jde.2010.11.016
    [17] M. Antali, G. Stepan, Sliding and crossing dynamics in extended Filippov systems, SIAM J. Appl. Dyn. Syst., 17 (2018), 823–858. doi: 10.1137/17M1110328. doi: 10.1137/17M1110328
    [18] J. Sotomayor, Lições de equações diferenciais ordinárias, Instituto de Matemática Pura e Aplicada, (1980).
    [19] W. Li, J. Ji, L. Huang, J. Wang, Bifurcations and dynamics of a plant disease system under non-smooth control strategy, Nonlinear Dyn., 2020 (2020), 1–21. doi: 10.1007/s11071-020-05464-2. doi: 10.1007/s11071-020-05464-2
    [20] W. Li, J. Ji, L. Huang, Dynamics of a controlled discontinuous computer worm system, Proc. Am. Math. Soc., 148 (2020), 4389–4403. doi: 10.1090/proc/15095. doi: 10.1090/proc/15095
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