# 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 & 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. doi: 10.1098/rsif.2010.0130. Google Scholar [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. doi: 10.1006/jmaa.1997.5654. Google Scholar [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. doi: 10.1038/255058a0. Google Scholar [4] L. Bilinsky and K. P. Hadeler, Quiescence stabilizes predator-prey relations,, J. Biological Dynamics, 3 (2009), 196. doi: 10.1080/17513750802590707. Google Scholar [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. doi: 10.1016/j.tpb.2009.02.004. Google Scholar [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), 177. doi: 10.1016/S0025-5564(01)00097-9. Google Scholar [7] I. Gerstmann and K. P. Hadeler, The discrete Rosenzweig model,, Mathematical Biosciences, 98 (1990), 49. doi: 10.1016/0025-5564(90)90011-M. Google Scholar [8] M. Gyllenberg and G. F. Webb, Age-size structure in populations with quiescence,, Mathematical Biosciences 86 (1987), 86 (1987), 67. doi: 10.1016/0025-5564(87)90064-2. Google Scholar [9] M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence,, J. of Mathematical Biology, 28 (1990), 671. doi: 10.1007/BF00160231. Google Scholar [10] K. P. Hadeler, Homogeneous equations with a quiescent phase,, Math. Models Natural Phenomena, 3 (2008), 115. doi: 10.1051/mmnp:2008044. Google Scholar [11] K. P. Hadeler, Quiescent phases and stability,, Linear Algebra and its Applications, 428 (2008), 1620. doi: 10.1016/j.laa.2007.10.008. Google Scholar [12] K. P. Hadeler, Quiescence, excitability, and heterogeneity in ecological models,, J. of Mathematical Biology, 66 (2013), 649. doi: 10.1007/s00285-012-0590-1. Google Scholar [13] K. P. Hadeler and T. Hillen, Coupled dynamics and quiescent states,, in Math Everywhere (eds. G. Aletti, (2007), 7. doi: 10.1007/978-3-540-44446-6_2. Google Scholar [14] K. P. Hadeler, T. Hillen and M. Lewis, Biological modeling with quiescent phases,, in Spatial Ecology, (2009). Google Scholar [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. Google Scholar [16] K. P. Hadeler and F. Lutscher, Quiescent phases with distributed exit times,, Discrete and Continuous Dynamical Systems B, 17 (2012), 849. doi: 10.3934/dcdsb.2012.17.849. Google Scholar [17] T. Hillen, Transport equations with resting phases,, European J. Appl. Math., 14 (2003), 613. doi: 10.1017/S0956792503005291. Google Scholar [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. doi: 10.1080/17513750903528192. Google Scholar [19] W. Jäger, S. Krömker and B. Tang, Quiescence and transient growth dynamics in chemostat models,, Mathematical Biosciences, 119 (1994), 225. doi: 10.1016/0025-5564(94)90077-9. Google Scholar [20] M. Kot, Elements of Mathematical Ecology,, Cambridge University Press, (2001). doi: 10.1017/CBO9780511608520. Google Scholar [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. Google Scholar [22] X. Liu and X. Dongmei, Complex dynamic behavior of a discrete-time predator-prey system,, Chaos, 32 (2007), 80. doi: 10.1016/j.chaos.2005.10.081. Google Scholar [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. doi: 10.1016/j.nonrwa.2012.07.011. Google Scholar [24] T. Malik and H. Smith, A resource-based model of microbial quiescence,, J. of Mathematical Biology, 53 (2006), 231. doi: 10.1007/s00285-006-0003-4. Google Scholar [25] M. Marden, The Geometry of the Zeroes of a Polynomial in a Complex Variable,, AMS, (1949). Google Scholar [26] J. D. Murray, Mathematical Biology,, Biomathematics, (1989). doi: 10.1007/978-3-662-08539-4. Google Scholar [27] M. Neubert, P. Klepac and P. van den Driessche, Stabilizing dispersal delays in predator-prey metapopulation models,, Theoretical Population Biology, 61 (2002), 339. doi: 10.1006/tpbi.2002.1578. Google Scholar [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. doi: 10.1006/tpbi.1995.1020. Google Scholar [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. doi: 10.1006/tpbi.2002.1584. Google Scholar [30] P. Olofsson, A stochastic model of a cell population with quiescence,, J. of Biological Dynamics, 2 (2008), 386. doi: 10.1080/17513750801956305. Google Scholar

show all references

##### 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. doi: 10.1098/rsif.2010.0130. Google Scholar [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. doi: 10.1006/jmaa.1997.5654. Google Scholar [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. doi: 10.1038/255058a0. Google Scholar [4] L. Bilinsky and K. P. Hadeler, Quiescence stabilizes predator-prey relations,, J. Biological Dynamics, 3 (2009), 196. doi: 10.1080/17513750802590707. Google Scholar [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. doi: 10.1016/j.tpb.2009.02.004. Google Scholar [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), 177. doi: 10.1016/S0025-5564(01)00097-9. Google Scholar [7] I. Gerstmann and K. P. Hadeler, The discrete Rosenzweig model,, Mathematical Biosciences, 98 (1990), 49. doi: 10.1016/0025-5564(90)90011-M. Google Scholar [8] M. Gyllenberg and G. F. Webb, Age-size structure in populations with quiescence,, Mathematical Biosciences 86 (1987), 86 (1987), 67. doi: 10.1016/0025-5564(87)90064-2. Google Scholar [9] M. Gyllenberg and G. F. Webb, A nonlinear structured population model of tumor growth with quiescence,, J. of Mathematical Biology, 28 (1990), 671. doi: 10.1007/BF00160231. Google Scholar [10] K. P. Hadeler, Homogeneous equations with a quiescent phase,, Math. Models Natural Phenomena, 3 (2008), 115. doi: 10.1051/mmnp:2008044. Google Scholar [11] K. P. Hadeler, Quiescent phases and stability,, Linear Algebra and its Applications, 428 (2008), 1620. doi: 10.1016/j.laa.2007.10.008. Google Scholar [12] K. P. Hadeler, Quiescence, excitability, and heterogeneity in ecological models,, J. of Mathematical Biology, 66 (2013), 649. doi: 10.1007/s00285-012-0590-1. Google Scholar [13] K. P. Hadeler and T. Hillen, Coupled dynamics and quiescent states,, in Math Everywhere (eds. G. Aletti, (2007), 7. doi: 10.1007/978-3-540-44446-6_2. Google Scholar [14] K. P. Hadeler, T. Hillen and M. Lewis, Biological modeling with quiescent phases,, in Spatial Ecology, (2009). Google Scholar [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. Google Scholar [16] K. P. Hadeler and F. Lutscher, Quiescent phases with distributed exit times,, Discrete and Continuous Dynamical Systems B, 17 (2012), 849. doi: 10.3934/dcdsb.2012.17.849. Google Scholar [17] T. Hillen, Transport equations with resting phases,, European J. Appl. Math., 14 (2003), 613. doi: 10.1017/S0956792503005291. Google Scholar [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. doi: 10.1080/17513750903528192. Google Scholar [19] W. Jäger, S. Krömker and B. Tang, Quiescence and transient growth dynamics in chemostat models,, Mathematical Biosciences, 119 (1994), 225. doi: 10.1016/0025-5564(94)90077-9. Google Scholar [20] M. Kot, Elements of Mathematical Ecology,, Cambridge University Press, (2001). doi: 10.1017/CBO9780511608520. Google Scholar [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. Google Scholar [22] X. Liu and X. Dongmei, Complex dynamic behavior of a discrete-time predator-prey system,, Chaos, 32 (2007), 80. doi: 10.1016/j.chaos.2005.10.081. Google Scholar [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. doi: 10.1016/j.nonrwa.2012.07.011. Google Scholar [24] T. Malik and H. Smith, A resource-based model of microbial quiescence,, J. of Mathematical Biology, 53 (2006), 231. doi: 10.1007/s00285-006-0003-4. Google Scholar [25] M. Marden, The Geometry of the Zeroes of a Polynomial in a Complex Variable,, AMS, (1949). Google Scholar [26] J. D. Murray, Mathematical Biology,, Biomathematics, (1989). doi: 10.1007/978-3-662-08539-4. Google Scholar [27] M. Neubert, P. Klepac and P. van den Driessche, Stabilizing dispersal delays in predator-prey metapopulation models,, Theoretical Population Biology, 61 (2002), 339. doi: 10.1006/tpbi.2002.1578. Google Scholar [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. doi: 10.1006/tpbi.1995.1020. Google Scholar [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. doi: 10.1006/tpbi.2002.1584. Google Scholar [30] P. Olofsson, A stochastic model of a cell population with quiescence,, J. of Biological Dynamics, 2 (2008), 386. doi: 10.1080/17513750801956305. Google Scholar
 [1] Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041 [2] Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979 [3] Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045 [4] Yunshyong Chow, Sophia Jang. Neimark-Sacker bifurcations in a host-parasitoid system with a host refuge. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1713-1728. doi: 10.3934/dcdsb.2016019 [5] Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507 [6] Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002 [7] Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031 [8] Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325 [9] Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121 [10] Jicai Huang, Yijun Gong, Shigui Ruan. Bifurcation analysis in a predator-prey model with constant-yield predator harvesting. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2101-2121. doi: 10.3934/dcdsb.2013.18.2101 [11] Qing Zhu, Huaqin Peng, Xiaoxiao Zheng, Huafeng Xiao. Bifurcation analysis of a stage-structured predator-prey model with prey refuge. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2195-2209. doi: 10.3934/dcdss.2019141 [12] Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035 [13] Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065 [14] Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489 [15] Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997 [16] Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 [17] John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 [18] Yuncherl Choi, Jongmin Han, Chun-Hsiung Hsia. Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1933-1957. doi: 10.3934/dcdsb.2015.20.1933 [19] Karl-Peter Hadeler, Frithjof Lutscher. Quiescent phases with distributed exit times. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 849-869. doi: 10.3934/dcdsb.2012.17.849 [20] Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026

2018 Impact Factor: 1.008

## Metrics

• PDF downloads (9)
• HTML views (0)
• Cited by (1)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]