# American Institute of Mathematical Sciences

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

## A note on the Lasota discrete model for blood cell production

 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  February 2020 Early access  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).

Citation: Eduardo Liz, Cristina Lois-Prados. A note on the Lasota discrete model for blood cell production. Discrete and 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. [2] L. Avilés, Cooperation and non-linear dynamics: An ecological perspective on the evolution of sociality, Evol. Ecol. Res., 1 (1999), 459-477. [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. [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. [5] A. Lasota, Ergodic problems in biology, Asterisque, 50 (1977), 239-250. [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. [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. [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. [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. [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. [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. [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. [13] T. J. Quinn and R. B. Deriso, Quantitative Fish Dynamics,, Oxford University Press, New York, 1999. [14] S. J. Schreiber, Chaos and population disappearances in simple ecological models, J. Math. Biol., 42 (2001), 239-260.  doi: 10.1007/s002850000070. [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. [16] M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the red-blood cell system, Mat. Stos., 6 (1976), 25-40. [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.

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##### 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. [2] L. Avilés, Cooperation and non-linear dynamics: An ecological perspective on the evolution of sociality, Evol. Ecol. Res., 1 (1999), 459-477. [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. [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. [5] A. Lasota, Ergodic problems in biology, Asterisque, 50 (1977), 239-250. [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. [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. [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. [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. [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. [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. [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. [13] T. J. Quinn and R. B. Deriso, Quantitative Fish Dynamics,, Oxford University Press, New York, 1999. [14] S. J. Schreiber, Chaos and population disappearances in simple ecological models, J. Math. Biol., 42 (2001), 239-260.  doi: 10.1007/s002850000070. [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. [16] M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the red-blood cell system, Mat. Stos., 6 (1976), 25-40. [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.
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 $y = x$
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 $F$
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 $y = x$
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
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
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 $\gamma = 8.5$
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 $\sigma = 0.9$
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