# American Institute of Mathematical Sciences

January  2015, 20(1): 129-152. doi: 10.3934/dcdsb.2015.20.129

## Quiescent phases and stability in discrete time dynamical systems

 1 Mathematics, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany

Received  February 2014 Revised  May 2014 Published  November 2014

We study coupled maps where a map representing an active phase' is coupled to the identity which represents a quiescent phase'. The resulting system in double dimension is a natural analogue of differential equations with quiescent phases that have been thoroughly studied. In the continuous time case quiescent phases with equal rates for all components stabilize against the onset of Hopf bifurcations (but not against eigenvalues passing through zero) while unequal rates may induce Hopf bifurcations unless the Jacobian matrix has a strong stability' property. Here we show that similar effects occur in the discrete time case. In the case of equal rates we determine the exact stability boundary as an algebraic curve of fourth order. It is shown that large quiescence rates may completely inhibit period doubling bifurcations. If the rates are unequal, quiescent phases may destabilize a stationary point. In this case we find (for two components) a notion of strong stability' for the Jacobian matrix such that the stationary point cannot be excited. Discrete time predator prey models serve as examples for the damping and excitation phenomena.
Citation: Karl P. Hadeler. Quiescent phases and stability in discrete time dynamical systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 129-152. doi: 10.3934/dcdsb.2015.20.129
##### References:
 [1] T. Alarcón and H. J. Jensen, Quiescence, a mechanism for escaping the effects of drug on cell populations, J. Roy. Soc. Interface, 8 (2011), 99-106. doi: 10.1098/rsif.2010.0130. [2] O. Arino, E. Sanchez and G. F. Webb, Necessary and sufficient conditions for asynchronous exponential growth in age structured cell populations with quiescence, J. Math. Anal. Appl., 215 (1997), 499-513. doi: 10.1006/jmaa.1997.5654. [3] J. R. Beddington, C. A. Free and J. H. Lawton, Dynamic complexity in predator-prey models framed in difference equations, Nature, 255 (1975), 58-60. doi: 10.1038/255058a0. [4] L. Bilinsky and K. P. Hadeler, Quiescence stabilizes predator-prey relations, J. Biological Dynamics, 3 (2009), 196-208. doi: 10.1080/17513750802590707. [5] C. A. Cobbold, J. Roland and M. A. Lewis, The impact of parasitoid emergence time on host-parasitoid population dynamics, Theoretical Population Biology, 75 (2009), 201-215. doi: 10.1016/j.tpb.2009.02.004. [6] J. Dyson, R. Villela-Bressan and G. F. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells, Mathematical Biosciences, 177-178 (2002), 73-83. doi: 10.1016/S0025-5564(01)00097-9. [7] I. Gerstmann and K. P. Hadeler, The discrete Rosenzweig model, Mathematical Biosciences, 98 (1990), 49-72. doi: 10.1016/0025-5564(90)90011-M. [8] M. Gyllenberg and G. F. Webb, Age-size structure in populations with quiescence, Mathematical Biosciences 86 (1987), 67-95. doi: 10.1016/0025-5564(87)90064-2. [9] M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J. of Mathematical Biology, 28 (1990), 671-694. doi: 10.1007/BF00160231. [10] K. P. Hadeler, Homogeneous equations with a quiescent phase, Math. Models Natural Phenomena, 3 (2008), 115-125. doi: 10.1051/mmnp:2008044. [11] K. P. Hadeler, Quiescent phases and stability, Linear Algebra and its Applications, 428 (2008), 1620-1627. doi: 10.1016/j.laa.2007.10.008. [12] K. P. Hadeler, Quiescence, excitability, and heterogeneity in ecological models, J. of Mathematical Biology, 66 (2013), 649-684. doi: 10.1007/s00285-012-0590-1. [13] K. P. Hadeler and T. Hillen, Coupled dynamics and quiescent states, in Math Everywhere (eds. G. Aletti, M. Burger, A. Micheletti and D. Morale), Springer, Berlin, 2007, 7-23. doi: 10.1007/978-3-540-44446-6_2. [14] K. P. Hadeler, T. Hillen and M. Lewis, Biological modeling with quiescent phases, in Spatial Ecology, (eds. C. Cosner and S. Cantrell and S. Ruan), Taylor and Francis, 2009, chapter 5. [15] K. P. Hadeler and M. A. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment, Canadian Applied Mathematics Quarterly, 10 (2002), 473-499. [16] K. P. Hadeler and F. Lutscher, Quiescent phases with distributed exit times, Discrete and Continuous Dynamical Systems B, 17 (2012), 849-869. doi: 10.3934/dcdsb.2012.17.849. [17] T. Hillen, Transport equations with resting phases, European J. Appl. Math., 14 (2003), 613-636. doi: 10.1017/S0956792503005291. [18] A. N. W. Hone, M. V. Irle and G. W. Thurura, On the Naimark-Sacker bifurcation in a discrete predator-prey system, J. Biological Dynamics, 4 (2010), 594-606. doi: 10.1080/17513750903528192. [19] W. Jäger, S. Krömker and B. Tang, Quiescence and transient growth dynamics in chemostat models, Mathematical Biosciences, 119 (1994), 225-239. doi: 10.1016/0025-5564(94)90077-9. [20] M. Kot, Elements of Mathematical Ecology, Cambridge University Press, 2001 doi: 10.1017/CBO9780511608520. [21] M. A. Lewis and G. Schmitz, Biological invasion of an organism with separate mobile and stationary states: Modelling and analysis, Forma, 11 (1996), 1-25. [22] X. Liu and X. Dongmei, Complex dynamic behavior of a discrete-time predator-prey system, Chaos, Solitons & Fractals, 32 (2007), 80-94. doi: 10.1016/j.chaos.2005.10.081. [23] F. Lutscher and V. M. Nguyen, Traveling waves in discrete models of biological populations with sessile stages, Nonlinear Analysis: Real World Applications, 14 (2013), 495-506. doi: 10.1016/j.nonrwa.2012.07.011. [24] T. Malik and H. Smith, A resource-based model of microbial quiescence, J. of Mathematical Biology, 53 (2006), 231-252. doi: 10.1007/s00285-006-0003-4. [25] M. Marden, The Geometry of the Zeroes of a Polynomial in a Complex Variable, AMS, New York, 1949. [26] J. D. Murray, Mathematical Biology, Biomathematics, 19, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4. [27] M. Neubert, P. Klepac and P. van den Driessche, Stabilizing dispersal delays in predator-prey metapopulation models, Theoretical Population Biology, 61 (2002), 339-347. doi: 10.1006/tpbi.2002.1578. [28] M. G. Neubert, M. Kot and M. A. Lewis, Dispersal and pattern formation in a discrete-time predator-prey model, Theoretical Population Biology, 48 (1995), 7-43. doi: 10.1006/tpbi.1995.1020. [29] T. J. Newman, J. Antonovics and H. M. Wilbur, Population dynamics with a refuge: Fractal basins and the suppression of chaos, Theoretical Population Biology, 62 (2002), 121-128. doi: 10.1006/tpbi.2002.1584. [30] P. Olofsson, A stochastic model of a cell population with quiescence, J. of Biological Dynamics, 2 (2008), 386-391. doi: 10.1080/17513750801956305.

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##### References:
 [1] T. Alarcón and H. J. Jensen, Quiescence, a mechanism for escaping the effects of drug on cell populations, J. Roy. Soc. Interface, 8 (2011), 99-106. doi: 10.1098/rsif.2010.0130. [2] O. Arino, E. Sanchez and G. F. Webb, Necessary and sufficient conditions for asynchronous exponential growth in age structured cell populations with quiescence, J. Math. Anal. Appl., 215 (1997), 499-513. doi: 10.1006/jmaa.1997.5654. [3] J. R. Beddington, C. A. Free and J. H. Lawton, Dynamic complexity in predator-prey models framed in difference equations, Nature, 255 (1975), 58-60. doi: 10.1038/255058a0. [4] L. Bilinsky and K. P. Hadeler, Quiescence stabilizes predator-prey relations, J. Biological Dynamics, 3 (2009), 196-208. doi: 10.1080/17513750802590707. [5] C. A. Cobbold, J. Roland and M. A. Lewis, The impact of parasitoid emergence time on host-parasitoid population dynamics, Theoretical Population Biology, 75 (2009), 201-215. doi: 10.1016/j.tpb.2009.02.004. [6] J. Dyson, R. Villela-Bressan and G. F. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells, Mathematical Biosciences, 177-178 (2002), 73-83. doi: 10.1016/S0025-5564(01)00097-9. [7] I. Gerstmann and K. P. Hadeler, The discrete Rosenzweig model, Mathematical Biosciences, 98 (1990), 49-72. doi: 10.1016/0025-5564(90)90011-M. [8] M. Gyllenberg and G. F. Webb, Age-size structure in populations with quiescence, Mathematical Biosciences 86 (1987), 67-95. doi: 10.1016/0025-5564(87)90064-2. [9] M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence, J. of Mathematical Biology, 28 (1990), 671-694. doi: 10.1007/BF00160231. [10] K. P. Hadeler, Homogeneous equations with a quiescent phase, Math. Models Natural Phenomena, 3 (2008), 115-125. doi: 10.1051/mmnp:2008044. [11] K. P. Hadeler, Quiescent phases and stability, Linear Algebra and its Applications, 428 (2008), 1620-1627. doi: 10.1016/j.laa.2007.10.008. [12] K. P. Hadeler, Quiescence, excitability, and heterogeneity in ecological models, J. of Mathematical Biology, 66 (2013), 649-684. doi: 10.1007/s00285-012-0590-1. [13] K. P. Hadeler and T. Hillen, Coupled dynamics and quiescent states, in Math Everywhere (eds. G. Aletti, M. Burger, A. Micheletti and D. Morale), Springer, Berlin, 2007, 7-23. doi: 10.1007/978-3-540-44446-6_2. [14] K. P. Hadeler, T. Hillen and M. Lewis, Biological modeling with quiescent phases, in Spatial Ecology, (eds. C. Cosner and S. Cantrell and S. Ruan), Taylor and Francis, 2009, chapter 5. [15] K. P. Hadeler and M. A. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment, Canadian Applied Mathematics Quarterly, 10 (2002), 473-499. [16] K. P. Hadeler and F. Lutscher, Quiescent phases with distributed exit times, Discrete and Continuous Dynamical Systems B, 17 (2012), 849-869. doi: 10.3934/dcdsb.2012.17.849. [17] T. Hillen, Transport equations with resting phases, European J. Appl. Math., 14 (2003), 613-636. doi: 10.1017/S0956792503005291. [18] A. N. W. Hone, M. V. Irle and G. W. Thurura, On the Naimark-Sacker bifurcation in a discrete predator-prey system, J. Biological Dynamics, 4 (2010), 594-606. doi: 10.1080/17513750903528192. [19] W. Jäger, S. Krömker and B. Tang, Quiescence and transient growth dynamics in chemostat models, Mathematical Biosciences, 119 (1994), 225-239. doi: 10.1016/0025-5564(94)90077-9. [20] M. Kot, Elements of Mathematical Ecology, Cambridge University Press, 2001 doi: 10.1017/CBO9780511608520. [21] M. A. Lewis and G. Schmitz, Biological invasion of an organism with separate mobile and stationary states: Modelling and analysis, Forma, 11 (1996), 1-25. [22] X. Liu and X. Dongmei, Complex dynamic behavior of a discrete-time predator-prey system, Chaos, Solitons & Fractals, 32 (2007), 80-94. doi: 10.1016/j.chaos.2005.10.081. [23] F. Lutscher and V. M. Nguyen, Traveling waves in discrete models of biological populations with sessile stages, Nonlinear Analysis: Real World Applications, 14 (2013), 495-506. doi: 10.1016/j.nonrwa.2012.07.011. [24] T. Malik and H. Smith, A resource-based model of microbial quiescence, J. of Mathematical Biology, 53 (2006), 231-252. doi: 10.1007/s00285-006-0003-4. [25] M. Marden, The Geometry of the Zeroes of a Polynomial in a Complex Variable, AMS, New York, 1949. [26] J. D. Murray, Mathematical Biology, Biomathematics, 19, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4. [27] M. Neubert, P. Klepac and P. van den Driessche, Stabilizing dispersal delays in predator-prey metapopulation models, Theoretical Population Biology, 61 (2002), 339-347. doi: 10.1006/tpbi.2002.1578. [28] M. G. Neubert, M. Kot and M. A. Lewis, Dispersal and pattern formation in a discrete-time predator-prey model, Theoretical Population Biology, 48 (1995), 7-43. doi: 10.1006/tpbi.1995.1020. [29] T. J. Newman, J. Antonovics and H. M. Wilbur, Population dynamics with a refuge: Fractal basins and the suppression of chaos, Theoretical Population Biology, 62 (2002), 121-128. doi: 10.1006/tpbi.2002.1584. [30] P. Olofsson, A stochastic model of a cell population with quiescence, J. of Biological Dynamics, 2 (2008), 386-391. doi: 10.1080/17513750801956305.
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