American Institute of Mathematical Sciences

Advanced
February  2020, 25(2): 701-713. doi: 10.3934/dcdsb.2019262

A note on the Lasota discrete model for blood cell production

Eduardo Liz 1, and  Cristina Lois-Prados 2,

1. 

Departamento de Matemática Aplicada Ⅱ, Universidade de Vigo, 36310 Vigo, Spain
2. 

Instituto de Matemáticas, Universidade de Santiago de Compostela, Campus Vida, 15782 Santiago de Compostela, Spain

Dedicated to Prof. Juan J. Nieto on the occasion of his 60th birthday

Received  January 2019 Revised  March 2019 Published  November 2019

In an attempt to explain experimental evidence of chaotic oscillations in blood cell population, A. Lasota suggested in 1977 a discrete-time one-dimensional model for the production of blood cells, and he showed that this equation allows to model the behavior of blood cell population in many clinical cases. Our main aim in this note is to carry out a detailed study of Lasota's equation, in particular revisiting the results in the original paper and showing new interesting phenomena. The considered equation is also suitable to model the dynamics of populations with discrete reproductive seasons, adult survivorship, overcompensating density dependence, and Allee effects. In this context, our results show the rich dynamics of this type of models and point out the subtle interplay between adult survivorship rates and strength of density dependence (including Allee effects).

Keywords: Discrete-time model, gamma model, Ricker map, stability, bifurcations, boundary collision, essential extinction.
Mathematics Subject Classification: 39A10, 39A30, 92C37, 92D25.
Citation: Eduardo Liz, Cristina Lois-Prados. A note on the Lasota discrete model for blood cell production. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 701-713. doi: 10.3934/dcdsb.2019262
References:
[1]

P. A. Abrams, When does greater mortality increase population size? The long story and diverse mechanisms underlying the hydra effect, Ecol. Lett., 12 (2009), 462-474.  doi: 10.1111/j.1461-0248.2009.01282.x.  Google Scholar

[2]

L. Avilés, Cooperation and non-linear dynamics: An ecological perspective on the evolution of sociality, Evol. Ecol. Res., 1 (1999), 459-477.   Google Scholar

[3]

E. Braverman and E. Liz, Global stabilization of periodic orbits using a proportional feedback control with pulses, Nonlinear Dynam., 67 (2012), 2467-2475.  doi: 10.1007/s11071-011-0160-x.  Google Scholar

[4] F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, New York, 2008.  doi: 10.1093/acprof:oso/9780198570301.001.0001.  Google Scholar
[5]

A. Lasota, Ergodic problems in biology, Asterisque, 50 (1977), 239-250.   Google Scholar

[6]

E. Liz, Complex dynamics of survival and extinction in simple population models with harvesting, Theor. Ecol., 3 (2010), 209-221.  doi: 10.1007/s12080-009-0064-2.  Google Scholar

[7]

E. Liz, A new flexible discrete-time model for stable populations, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2487-2498.  doi: 10.3934/dcdsb.2018066.  Google Scholar

[8]

E. Liz, A global picture of the gamma-Ricker map: A flexible discrete-time model with factors of positive and negative density dependence, Bull. Math. Biol., 80 (2018), 417-434.  doi: 10.1007/s11538-017-0382-2.  Google Scholar

[9]

E. Liz and S. Buedo-Fernández, A new formula to get sharp global stability criteria for one-dimensional discrete-time models, Qual. Theory Dyn. Syst., (2019), 1-12.  doi: 10.1007/s12346-018-00314-4.  Google Scholar

[10]

E. Liz and D. Franco, Global stabilization of fixed points using predictive control, Chaos, 20 (2010), 023124, 9 pp. doi: 10.1063/1.3432558.  Google Scholar

[11]

E. Liz and A. Ruiz-Herrera, The hydra effect, bubbles, and chaos in a simple discrete population model with constant effort harvesting, J. Math. Biol., 65 (2012), 997-1016.  doi: 10.1007/s00285-011-0489-2.  Google Scholar

[12]

P. J. Mitkowski, Chaos in the Ergodic Theory Approach in the Model of Disturbed Erythropoiesis, Ph.D. Thesis, AGH University of Science and Technology, Cracow, 2011. Google Scholar

[13] T. J. Quinn and R. B. Deriso, Quantitative Fish Dynamics,, Oxford University Press, New York, 1999.   Google Scholar
[14]

S. J. Schreiber, Chaos and population disappearances in simple ecological models, J. Math. Biol., 42 (2001), 239-260.  doi: 10.1007/s002850000070.  Google Scholar

[15]

S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209.  doi: 10.1016/S0040-5809(03)00072-8.  Google Scholar

[16]

M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the red-blood cell system, Mat. Stos., 6 (1976), 25-40.   Google Scholar

[17]

A. A. Yakubu, N. Li, J. M. Conrad and M. L. Zeeman, Constant proportion harvest policies: Dynamic implications in the Pacific halibut and Atlantic cod fisheries, Math. Biosci., 232 (2011), 66-77.  doi: 10.1016/j.mbs.2011.04.004.  Google Scholar

show all references

References:
[1]

P. A. Abrams, When does greater mortality increase population size? The long story and diverse mechanisms underlying the hydra effect, Ecol. Lett., 12 (2009), 462-474.  doi: 10.1111/j.1461-0248.2009.01282.x.  Google Scholar

[2]

L. Avilés, Cooperation and non-linear dynamics: An ecological perspective on the evolution of sociality, Evol. Ecol. Res., 1 (1999), 459-477.   Google Scholar

[3]

E. Braverman and E. Liz, Global stabilization of periodic orbits using a proportional feedback control with pulses, Nonlinear Dynam., 67 (2012), 2467-2475.  doi: 10.1007/s11071-011-0160-x.  Google Scholar

[4] F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, New York, 2008.  doi: 10.1093/acprof:oso/9780198570301.001.0001.  Google Scholar
[5]

A. Lasota, Ergodic problems in biology, Asterisque, 50 (1977), 239-250.   Google Scholar

[6]

E. Liz, Complex dynamics of survival and extinction in simple population models with harvesting, Theor. Ecol., 3 (2010), 209-221.  doi: 10.1007/s12080-009-0064-2.  Google Scholar

[7]

E. Liz, A new flexible discrete-time model for stable populations, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2487-2498.  doi: 10.3934/dcdsb.2018066.  Google Scholar

[8]

E. Liz, A global picture of the gamma-Ricker map: A flexible discrete-time model with factors of positive and negative density dependence, Bull. Math. Biol., 80 (2018), 417-434.  doi: 10.1007/s11538-017-0382-2.  Google Scholar

[9]

E. Liz and S. Buedo-Fernández, A new formula to get sharp global stability criteria for one-dimensional discrete-time models, Qual. Theory Dyn. Syst., (2019), 1-12.  doi: 10.1007/s12346-018-00314-4.  Google Scholar

[10]

E. Liz and D. Franco, Global stabilization of fixed points using predictive control, Chaos, 20 (2010), 023124, 9 pp. doi: 10.1063/1.3432558.  Google Scholar

[11]

E. Liz and A. Ruiz-Herrera, The hydra effect, bubbles, and chaos in a simple discrete population model with constant effort harvesting, J. Math. Biol., 65 (2012), 997-1016.  doi: 10.1007/s00285-011-0489-2.  Google Scholar

[12]

P. J. Mitkowski, Chaos in the Ergodic Theory Approach in the Model of Disturbed Erythropoiesis, Ph.D. Thesis, AGH University of Science and Technology, Cracow, 2011. Google Scholar

[13] T. J. Quinn and R. B. Deriso, Quantitative Fish Dynamics,, Oxford University Press, New York, 1999.   Google Scholar
[14]

S. J. Schreiber, Chaos and population disappearances in simple ecological models, J. Math. Biol., 42 (2001), 239-260.  doi: 10.1007/s002850000070.  Google Scholar

[15]

S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209.  doi: 10.1016/S0040-5809(03)00072-8.  Google Scholar

[16]

M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the red-blood cell system, Mat. Stos., 6 (1976), 25-40.   Google Scholar

[17]

A. A. Yakubu, N. Li, J. M. Conrad and M. L. Zeeman, Constant proportion harvest policies: Dynamic implications in the Pacific halibut and Atlantic cod fisheries, Math. Biosci., 232 (2011), 66-77.  doi: 10.1016/j.mbs.2011.04.004.  Google Scholar

Figure 2.1.  Graphs of the map $ F $ for $ \gamma = 0.3 $, $ c = 0.6 $, and different values of $ \sigma $: $ \sigma = 0.6<\sigma^*\approx0.774 $ in red ($ F $ is increasing), and $ \sigma = 0.9>\sigma^* $ in blue ($ F $ has two critical points). The dashed line is the graph of $ y = x $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.2.  Graphs of the map $ F' $ (solid, blue) and the line $ y = 0 $ (dashed, black), with c = 1. (a): $ \gamma = 0.25<1 $, $ \sigma = 0.8>\sigma^*\approx0.707 $; (b): $ \gamma = 2>1 $, $ \sigma = 0.9>\sigma^*\approx0.841 $. In both cases, there are two intersection points, that determine the critical points of $ F $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.1.  Graphs of the map $ F $ for $ \gamma = 8 $, $ \sigma = 0.8 $, and different values of $ c $: $ c = c^*\approx0.425 $ in blue (one positive fixed point), $ c = 0.4<c^* $ in red (no positive fixed points), and $ c = 0.5>c^* $ in black (two positive fixed points). The dashed line is the graph of $ y = x $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.1.  Main bifurcation boundaries and regions with different dynamical behavior for equation (1.2) with $ c = 0.47 $, in the parameter plane $ (\gamma,\sigma) $. The two solid lines represent the extinction boundary (red color) and the stability boundary of the largest positive equilibrium (blue color). The vertical dashed line $ \gamma = 1 $ (from $ \sigma = 0 $ to $ \sigma = c = 0.47 $) is the border between global stability of the unique positive equilibrium and a bistability region, in which both the largest positive equilibrium and the extinction equilibrium are asymptotically stable
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.2.  Bifurcation diagram showing a bubble for equation (1.2) with $ c = 0.47 $, $ \gamma = 7.65 $, using $ \sigma $ as the bifurcation parameter. Black dashed lines correspond to unstable equilibria
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.3.  Bifurcation diagrams for equation (1.2) with $ c = 0.47 $, using $ \sigma $ as the bifurcation parameter. Black dashed lines correspond to unstable equilibria. For more details, see the text. (a): $ \gamma = 7 $; (b): $ \gamma = 8 $; (c): $ \gamma = 8.5 $ and $ \sigma\in (0,1) $; (d): magnification for $ \gamma = 8.5 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.4.  Bifurcation diagrams for equation (1.2) with $ c = 0.47 $ and different values of $ \sigma $, using $ \gamma $ as the bifurcation parameter. Black dashed lines correspond to unstable equilibria, and red dashed lines to unstable 2-periodic orbits. (a): There are neither oscillations nor extinction windows for $ \sigma = 0.05 $. (b): An extinction window for $ \sigma = 0.1 $. (c): Multiple extinction windows for $ \sigma = 0.9 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Ferenc A. Bartha, Ábel Garab. Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model. Journal of Computational Dynamics, 2014, 1 (2) : 213-232. doi: 10.3934/jcd.2014.1.213
[2]

H. L. Smith, X. Q. Zhao. Competitive exclusion in a discrete-time, size-structured chemostat model. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 183-191. doi: 10.3934/dcdsb.2001.1.183
[3]

Eduardo Liz. A new flexible discrete-time model for stable populations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2487-2498. doi: 10.3934/dcdsb.2018066
[4]

E. Cabral Balreira, Saber Elaydi, Rafael Luís. Local stability implies global stability for the planar Ricker competition model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 323-351. doi: 10.3934/dcdsb.2014.19.323
[5]

Nika Lazaryan, Hassan Sedaghat. Extinction and the Allee effect in an age structured Ricker population model with inter-stage interaction. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 731-747. doi: 10.3934/dcdsb.2018040
[6]

Sie Long Kek, Mohd Ismail Abd Aziz, Kok Lay Teo, Rohanin Ahmad. An iterative algorithm based on model-reality differences for discrete-time nonlinear stochastic optimal control problems. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 109-125. doi: 10.3934/naco.2013.3.109
[7]

Yun Kang. Permanence of a general discrete-time two-species-interaction model with nonlinear per-capita growth rates. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2123-2142. doi: 10.3934/dcdsb.2013.18.2123
[8]

Ka Chun Cheung, Hailiang Yang. Optimal investment-consumption strategy in a discrete-time model with regime switching. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 315-332. doi: 10.3934/dcdsb.2007.8.315
[9]

Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023
[10]

Deepak Kumar, Ahmad Jazlan, Victor Sreeram, Roberto Togneri. Partial fraction expansion based frequency weighted model reduction for discrete-time systems. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 329-337. doi: 10.3934/naco.2016015
[11]

Sie Long Kek, Kok Lay Teo, Mohd Ismail Abd Aziz. Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 207-222. doi: 10.3934/naco.2012.2.207
[12]

John E. Franke, Abdul-Aziz Yakubu. Periodically forced discrete-time SIS epidemic model with disease induced mortality. Mathematical Biosciences & Engineering, 2011, 8 (2) : 385-408. doi: 10.3934/mbe.2011.8.385
[13]

S. R.-J. Jang. Allee effects in a discrete-time host-parasitoid model with stage structure in the host. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 145-159. doi: 10.3934/dcdsb.2007.8.145
[14]

Sie Long Kek, Mohd Ismail Abd Aziz. Output regulation for discrete-time nonlinear stochastic optimal control problems with model-reality differences. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 275-288. doi: 10.3934/naco.2015.5.275
[15]

Dan Zhang, Xiaochun Cai, Lin Wang. Complex dynamics in a discrete-time size-structured chemostat model with inhibitory kinetics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3439-3451. doi: 10.3934/dcdsb.2018327
[16]

Vladimir Răsvan. On the central stability zone for linear discrete-time Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 734-741. doi: 10.3934/proc.2003.2003.734
[17]

Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 199-208. doi: 10.3934/dcdsb.2017010
[18]

Hun Ki Baek, Younghae Do. Dangerous Border-Collision bifurcations of a piecewise-smooth map. Communications on Pure & Applied Analysis, 2006, 5 (3) : 493-503. doi: 10.3934/cpaa.2006.5.493
[19]

Zhiying Qin, Jichen Yang, Soumitro Banerjee, Guirong Jiang. Border-collision bifurcations in a generalized piecewise linear-power map. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 547-567. doi: 10.3934/dcdsb.2011.16.547
[20]

Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (51)
  • HTML views (44)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences