Persistence and stability for a class of forced positive nonlinear delay-differential systems. (English) Zbl 1480.34096
Summary: Persistence and stability properties are considered for a class of forced positive nonlinear delay-differential systems which arise in mathematical ecology and other applied contexts. The inclusion of forcing incorporates the effects of control actions (such as harvesting or breeding programmes in an ecological setting), disturbances induced by seasonal or environmental variation, or migration. We provide necessary and sufficient conditions under which the states of these models are semi-globally persistent, uniformly with respect to the initial conditions and forcing terms. Under mild assumptions, the model under consideration naturally admits two steady states (equilibria) when unforced: the origin and a unique non-zero steady state. We present sufficient conditions for the non-zero steady state to be stable in a sense which is reminiscent of input-to-state stability, a stability notion for forced systems developed in control theory. In the absence of forcing, our input-to-sate stability concept is identical to semi-global exponential stability.
MSC:
| 34K20 | Stability theory of functional-differential equations |
| 34K12 | Growth, boundedness, comparison of solutions to functional-differential equations |
| 34K25 | Asymptotic theory of functional-differential equations |
| 34K35 | Control problems for functional-differential equations |
| 92D25 | Population dynamics (general) |
| 92E20 | Classical flows, reactions, etc. in chemistry |
| 93D09 | Robust stability |
| 93D10 | Popov-type stability of feedback systems |
Keywords:
delay-differential systems; density-dependent population models; environmental forcing; forced systems; input-to-state stability; persistence; positive systems; semi-global exponential stabilityReferences:
| [1] | Allwright, D. J., A global stability criterion for simple control loops, J. Math. Biol., 4, 363-373 (1977) · Zbl 0372.93042 · doi:10.1007/BF00275084 |
| [2] | Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences (1994), Philadelphia: SIAM, Philadelphia · Zbl 0815.15016 · doi:10.1137/1.9781611971262 |
| [3] | Bill, A.; Guiver, C.; Logemann, H.; Townley, S., Stability of non-negative Lur’e systems, SIAM J. Control Optim., 54, 1176-1211 (2016) · Zbl 1381.93081 · doi:10.1137/140994599 |
| [4] | Blythe, S. P.; Nisbet, R. M.; Gurney, W. S.C., Instability and complex dynamic behaviour in population models with long time delays, Theor. Popul. Biol., 22, 147-176 (1982) · Zbl 0493.92019 · doi:10.1016/0040-5809(82)90040-5 |
| [5] | Brauer, F.; Castillo-Chavez, C., Mathematical Models in Population Biology and Epidemiology (2012), New York: Springer, New York · Zbl 1302.92001 · doi:10.1007/978-1-4614-1686-9 |
| [6] | Cull, P.; Allen, L. J.S.; Aulbach, B.; Elaydi, S.; Sacker, R., Enveloping implies global stability, Difference Equations and Discrete Dynamical Systems, 71-85 (2005), Hackensack: World Sci. Publ., Hackensack · Zbl 1094.39005 · doi:10.1142/9789812701572_0006 |
| [7] | Curtain, R. F.; Zwart, H. J., An Introduction to Infinite-Dimensional Linear Systems Theory (1995), New York: Springer, New York · Zbl 0839.93001 · doi:10.1007/978-1-4612-4224-6 |
| [8] | Dashkovskiy, S. N.; Efimov, D. V.; Sontag, E. D., Input-to-state stability and allied system properties, Autom. Remote Control, 72, 1579-1614 (2011) · Zbl 1235.93221 · doi:10.1134/S0005117911080017 |
| [9] | Eager, E. A.; Rebarber, R., Sensitivity and elasticity analysis of a Lur’e system used to model a population subject to density-dependent reproduction, Math. Biosci., 282, 34-45 (2016) · Zbl 1352.92122 · doi:10.1016/j.mbs.2016.09.016 |
| [10] | Eager, E. A.; Rebarber, R.; Tenhumberg, B., Global asymptotic stability of plant-seed bank models, J. Math. Biol., 69, 1-37 (2014) · Zbl 1321.92067 · doi:10.1007/s00285-013-0689-z |
| [11] | Faria, T.; Röst, G., Persistence, permanence and global stability for a \(n\)-dimensional Nicholson system, J. Dyn. Differ. Equ., 26, 723-744 (2014) · Zbl 1316.34086 · doi:10.1007/s10884-014-9381-2 |
| [12] | Franco, D.; Logemann, H.; Perán, J., Global stability of an age-structured population model, Syst. Control Lett., 65, 30-36 (2014) · Zbl 1285.93077 · doi:10.1016/j.sysconle.2013.11.012 |
| [13] | Franco, D.; Guiver, C.; Logemann, H.; Perán, J., Semi-global persistence and stability for a class of forced discrete-time population models, Physica D, 360, 46-61 (2017) · Zbl 1378.92056 · doi:10.1016/j.physd.2017.08.001 |
| [14] | Franco, D.; Guiver, C.; Logemann, H.; Perán, J., Boundedness, persistence and stability for classes of forced difference equations arising in population ecology, J. Math. Biol., 79, 1029-1076 (2019) · Zbl 1422.37067 · doi:10.1007/s00285-019-01388-7 |
| [15] | Freedman, H. I.; So, J. W.-H., Persistence in discrete semidynamical systems, SIAM J. Math. Anal., 20, 930-938 (1989) · Zbl 0676.92011 · doi:10.1137/0520062 |
| [16] | Goodwin, B. C., Oscillatory behavior of enzymatic control processes, Adv. Enzyme Regul., 3, 425-439 (1965) · doi:10.1016/0065-2571(65)90067-1 |
| [17] | Guiver, C.; Logemann, H.; Opmeer, M. R., Infinite-dimensional Lur’e systems: input-to-state stability and convergence properties, SIAM J. Control Optim., 57, 334-365 (2019) · Zbl 1405.93204 · doi:10.1137/17M1150426 |
| [18] | Gurney, W. S.C.; Blythe, S. P.; Nisbet, R. M., Nicholson’s blowflies revisited, Nature, 287, 17-21 (1980) · doi:10.1038/287017a0 |
| [19] | Haddad, W. M.; Chellaboina, V.; Hui, Q., Nonnegative and Compartmental Dynamical Systems (2010), Princeton: Princeton University Press, Princeton · Zbl 1184.93001 · doi:10.1515/9781400832248 |
| [20] | Hadeler, K. P.; Bocharov, G., Where to put delays in population models, in particular in the neutral case, Can. Appl. Math. Q., 11, 159-173 (2003) · Zbl 1089.34065 |
| [21] | Hale, J. K.; Verduyn Lunel, S. M., Introduction to Functional Differential Equations (1993), New York: Springer, New York · Zbl 0787.34002 · doi:10.1007/978-1-4612-4342-7 |
| [22] | Jayawardhana, B.; Logemann, H.; Ryan, E. P., The circle criterion and input-to-state stability: new perspectives on a classical result, IEEE Control Syst. Mag., 31, 32-67 (2011) · Zbl 1395.93473 · doi:10.1109/MCS.2011.941143 |
| [23] | Khalil, H. K., Nonlinear Systems (2014), Harlow: Pearson, Harlow |
| [24] | Kiss, G.; Röst, G., Controlling Mackey-Glass chaos, Chaos, 27 (2017) · Zbl 1390.34195 · doi:10.1063/1.5006922 |
| [25] | Krasnosel’skij, M. A.; Lifshits, J. A.; Sobolev, A. V., Positive Linear Systems: The Method of Positive Operators (1989), Berlin: Heldermann Verlag, Berlin · Zbl 0674.47036 |
| [26] | Krause, U., Positive Dynamical Systems in Discrete Time (2015), Berlin: de Gruyter, Berlin · Zbl 1326.37003 · doi:10.1515/9783110365696 |
| [27] | Landahl, H. D., Some conditions for sustained oscillations in biochemical chains with feedback inhibition, Bull. Math. Biophys., 31, 775-787 (1969) · Zbl 0181.47801 · doi:10.1007/BF02477786 |
| [28] | Logemann, H.; Ryan, E. P., Volterra functional differential equations: existence, uniqueness, and coninuation of solutions, Am. Math. Mon., 117, 490-511 (2010) · Zbl 1210.34085 · doi:10.4169/000298910x492790 |
| [29] | Logemann, H.; Ryan, E. P., Ordinary Differential Equations: Analysis, Qualitative Theory and Control (2014), London: Springer, London · Zbl 1300.34001 · doi:10.1007/978-1-4471-6398-5 |
| [30] | Mackey, M. C.; Glass, L., Oscillation and chaos in physiological control systems, Science, 197, 287-289 (1977) · Zbl 1383.92036 · doi:10.1126/science.267326 |
| [31] | Monk, N. A.M., Oscillatory expression of Hes1, p53, and NF-\( \kappa B\) driven by transcriptional time delays, Curr. Biol., 13, 1409-1413 (2003) · doi:10.1016/S0960-9822(03)00494-9 |
| [32] | Nicholson, A. J., An outline of the dynamics of animal populations, Aust. J. Zool., 2, 9-65 (1954) · doi:10.1071/ZO9540009 |
| [33] | Rebarber, R.; Tenhumberg, B.; Townley, S., Global asymptotic stability density dependent integral population projection models, Theor. Popul. Biol., 81, 81-87 (2012) · Zbl 1322.92062 · doi:10.1016/j.tpb.2011.11.002 |
| [34] | Rubió-Massegú, J.; Mañosa, V., On the enveloping method and the existence of global Lyapunov functions, J. Differ. Equ. Appl., 13, 1029-1035 (2007) · Zbl 1130.39009 · doi:10.1080/10236190701403895 |
| [35] | Sarkans, E.; Logemann, H., Input-to-state stability for Lur’e systems, Math. Control Signals Syst., 27, 439-465 (2015) · Zbl 1327.93343 · doi:10.1007/s00498-015-0147-0 |
| [36] | Sarkans, E.; Logemann, H., Input-to-state stability for discrete-time Lur’e systems, SIAM J. Control Optim., 54, 1739-1768 (2016) · Zbl 1349.93351 · doi:10.1137/130939067 |
| [37] | Smith, H. L., An Introduction to Delay Differential Equations with Applications to the Life Sciences (2011), New York: Springer, New York · Zbl 1227.34001 · doi:10.1007/978-1-4419-7646-8 |
| [38] | Smith, H. L.; Thieme, H. R., Dynamical Systems and Population Persistence (2011), Providence: Am. Math. Soc., Providence · Zbl 1214.37002 |
| [39] | Smith, H. L.; Thieme, H. R., Persistence and global stability for a class of discrete-time structured population models, Discrete Contin. Dyn. Syst., Ser. A, 33, 4627-4646 (2013) · Zbl 1306.37099 · doi:10.3934/dcds.2013.33.4627 |
| [40] | Sontag, E. D., Smooth stabilization implies coprime factorization, IEEE Trans. Autom. Control, 34, 435-443 (1989) · Zbl 0682.93045 · doi:10.1109/9.28018 |
| [41] | Sontag, E. D., Mathematical Control Theory: Deterministic Finite Dimensional Systems (1998), New York: Springer, New York · Zbl 0945.93001 · doi:10.1007/978-1-4612-0577-7 |
| [42] | Sontag, E. D.; Nistri, P.; Stefani, G., Input-to-state stability: basic concepts and results, Nonlinear and Optimal Control Theory, 163-220 (2006), Berlin: Springer, Berlin · Zbl 1175.93001 |
| [43] | Terrell, W. J., Stability and Stabilization (2009), Princeton: Princeton University Press, Princeton · Zbl 1170.34001 · doi:10.1515/9781400833351 |
| [44] | Townley, S.; Rebarber, R.; Tenhumberg, B., Feedback control systems analysis of density dependent population dynamics, Syst. Control Lett., 61, 309-315 (2012) · Zbl 1238.93036 · doi:10.1016/j.sysconle.2011.11.014 |
| [45] | Tyson, J. J.; Othmer, H. G., The dynamics of feedback control circuits in biochemical pathways, Prog. Theor. Biol., 5, 1-62 (1978) · Zbl 0448.92010 |
| [46] | Varga, R. S., Matrix Iterative Analysis (2000), Berlin: Springer, Berlin · Zbl 0998.65505 · doi:10.1007/978-3-642-05156-2 |
| [47] | Vidyasagar, M., Nonlinear Systems Analysis (1993), Englewood Cliffs: Prentice-Hall, Englewood Cliffs · Zbl 0900.93132 |
| [48] | Walter, W., Ordinary Differential Equations (1998), New York: Springer, New York · Zbl 0991.34001 · doi:10.1007/978-1-4612-0601-9 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
