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

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

• 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).

Mathematics Subject Classification: 39A10, 39A30, 92C37, 92D25.

 Citation: • • 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 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 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 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 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 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 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 $\sigma = 0.9$

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