Permanence conditions for models of population dynamics with switches and delay

A. Yu. Aleksandrov

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 16 (2020) no. 2, pp. 88-99

Résumé

Some classes of discrete and continuous generalized Volterra models of population dynamics with parameter switching and constant delay are studied. It is assumed that there are relationships of the type “symbiosis”, “compensationism” or “neutralism” between any two species in a biological community. The goal of the work is to obtain sufficient conditions for the permanence of such models. Original constructions of common Lyapunov—Krasovsky functionals are proposed for families of subsystems corresponding to the switched systems under consideration. Using the constructed functionals, conditions are derived that guarantee permanence for any admissible switching laws and any constant nonnegative delay. These conditions are constructive and are formulated in terms of the existence of a positive solution for an auxiliary system of linear algebraic inequalities. It should be noted that, in the proved theorems, the persistence of the systems is ensured by the positive coefficients of natural growth and the beneficial effect of populations on each other, whereas the ultimate boundedness of species numbers is provided by the intraspecific competition. An example is presented demonstrating the effectiveness of the developed approaches.

Détail

Keywords: population dynamics, ultimate boundedness, switches, delay, Lyapunov—Krasovskii functional.
Mots-clés : permanence

Bibliographie

[1] Yu. A. Pykh, Equilibrium and stability in models of population dynamics, Nauka, M., 1983, 182 pp. (In Russian)

[2] Yu. M. Svirezhev, D. O. Logofet, Stability of biolojical communities, Nauka, M., 1978, 352 pp. (In Russian) | MR

[3] J. Hofbauer, K. Sigmund, Evolutionary games, population dynamics, Cambridge University Press, Cambridge, 1998, 323 pp. | MR | Zbl

[4] N. F. Britton, Essential mathematical biology, Springer, London–Berlin–Heidelberg, 2003, 335 pp. | MR | Zbl

[5] T. Yoshizawa, Stability theory by Liapunov's second method, The Math. Soc. of Japan, Tokyo, 1966, 223 pp. | MR

[6] S. L. Podval'nyi, V. V. Provotorov, “Start control of a parabolic system with distributed parameters on a graph”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2015, no. 3, 126–142 (In Russian)

[7] V. V. Provotorov, V. I. Ryazhskikh, Yu. A. Gnilitskaya, “Unique weak solvability of a nonlinear initial boundary value problem with distributed parameters in a netlike domain”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 13:3 (2017), 264–277 | MR

[8] A. Aleksandrov, O. Mason, “Diagonal stability of a class of discrete-time positive switched systems with delay”, IET Control Theory Applications, 12:6 (2018), 812–818 | DOI | MR

[9] F. Chen, L. Wu, Z. Li, “Permanence and global attractivity of the discrete Gilpin-Ayala type population model”, Computers and Mathematics with Applications, 53 (2007), 1214–1227 | DOI | MR | Zbl

[10] Z. Lu, W. Wang, “Permanence and global attractivity for Lotka-Volterra difference systems”, J. Math. Biol., 39 (1999), 269–282 | DOI | MR | Zbl

[11] E. Kaszkurewicz, A. Bhaya, Matrix diagonal stability in systems and computation, Birkhauser, Boston–Basel–Berlin, 1999, 267 pp. | MR

[12] F. Chen, “Some new results on the permanence and extinction of nonautonomous Gilpin-Ayala type competition model with delays”, Nonlinear Analysis: Real World Applications, 7 (2006), 1205–1222 | DOI | MR | Zbl

[13] V. I. Zubov, “Independence of evolutionary development of species from aftereffects”, Proceedings of Russian Academy of Sciences, 323:4 (1992), 632–635 (In Russian) | Zbl

[14] G. Lu, Zh. Lu, Y. Enatsu, “Permanence for Lotka-Volterra systems with multiple delays”, Nonlinear Analysis: Real World Applications, 12:5 (2011), 2552–2560 | DOI | MR | Zbl

[15] J. Bao, X. Mao, G. Yin, C. Yuan, “Competitive Lotka-Volterra population dynamics with jumps”, Nonlinear Analysis, 74 (2011), 6601–6616 | DOI | MR | Zbl

[16] C. Zhu, G. Yin, “On hybrid competitive Lotka-Volterra ecosystems”, Nonlinear Analysis, 71 (2009), 1370–1379 | DOI | MR

[17] A. Yu. Aleksandrov, E. B. Aleksandrova, A. V. Platonov, “Ultimate boundedness conditions for a hybrid model of population dynamics”, Proc. 21st Mediterranean conference on Control and Automation (June 25-28, 2013, Platanias-Chania, Crite, Greece), 2013, 622–627 | DOI

[18] A. Yu. Aleksandrov, Y. Chen, A. V. Platonov, “Permanence and ultimate boundedness for discretetime switched models of population dynamics”, Nonlinear Dynamics and Systems Theory, 14:1 (2014), 1–10 | MR | Zbl

[19] A. Yu. Aleksandrov, A. V. Platonov, “On the ultimate boundedness and permanence of solutions for a class of discree-time switched models of population dynamics”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2014, no. 1, 5–16 (In Russian)

[20] Y. Nakata, Y. Muroya, “Permanence for nonautonomous Lotka-Volterra cooperative systems with delays”, Nonlinear Analysis, 11 (2010), 528–534 | DOI | MR | Zbl

[21] G. Lu, Z. Lu, X. Lian, “Delay effect on the permanence for Lotka-Volterra cooperative system”, Nonlinear Analysis, 11 (2010), 2810–2816 | DOI | MR | Zbl

[22] R. Xu, Z. Wang, “Periodic solutions of a nonautonomous predator-prey system with stage structure and time delays”, J. Comput. Appl. Math., 196 (2006), 70–86 | DOI | MR | Zbl

[23] Z. Jingru, L. Chen, “Permanence and global stability for a two-species cooperative system with time delays in a two-patch environment”, Comput. Math. Appl., 32 (1996), 101–108 | MR | Zbl

[24] R. Xu, M. A. J. Chaplain, F. A. Davidson, “Permanence and periodicity of a delayed ratiodependent predator-prey model with stage structure”, J. Math. Anal. Appl., 303 (2005), 602–621 | DOI | MR | Zbl

[25] A. Yu. Aleksandrov, A. A. Vorob'eva, E. P. Kolpak, “On the diagonal stability of some classes of complex systems”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 14:2 (2018), 72–88 (In Russian) | MR

[26] D. Liberzon, A. S. Morse, “Basic problems in stability and design of switched systems”, IEEE Control Systems Magazine, 19:15 (1999), 59–70 | MR | Zbl

[27] R. Shorten, F. Wirth, O. Mason, K. Wulf, C. King, “Stability criteria for switched and hybrid systems”, SIAM Rev., 49:4 (2007), 545–592 | DOI | MR | Zbl

[28] A. Aleksandrov, O. Mason, “Absolute stability and Lyapunov-Krasovskii functionals for switched nonlinear systems with time-delay”, Journal of the Franklin Institute, 351 (2014), 4381–4394 | DOI | MR | Zbl