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Angiogenesis model with Erlang distributed delays

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  • We consider the model of angiogenesis process proposed by Bodnar and Foryś (2009) with time delays included into the vessels formation and tumour growth processes. Originally, discrete delays were considered, while in the present paper we focus on distributed delays and discuss specific results for the Erlang distributions. Analytical results concerning stability of positive steady states are illustrated by numerical results in which we also compare these results with those for discrete delays.

    Mathematics Subject Classification: Primary: 34K11, 34K13, 34K18, 37N25; Secondary: 92B05.


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  • Figure 1.  Critical average delay, that is $m/a_\text{cr}$ for various values of $m$ in the dependance on $\delta$ in the case when only the process of tumour growth is delayed; left -graphs for the steady state $D_1$, right -graphs for the steady state $D_3$

    Figure 2.  Solutions of system (1.1) for parameters given by (3.1) and $\tau=10$, with time delay present only in the vessel formation term

    Figure 3.  Solutions of system (1.1) for parameters given by (3.1) and $\tau=10$, with time delay present only in the tumour growth term. Here, for Erlang distribution, the steady state is stable, and solutions for $m=1$ and $m=5$ are almost identical

    Figure 4.  Solutions of system (1.1) for parameters given by (3.1) and $\tau=5$, with time delay present in both terms

    Table 1.  Critical values of $\tau$ at which the positive steady state loses stability

    steady state $D_1$ steady state $D_3$
    $\delta$ 0.332 0.346 0.36 0.368 0.378 0.3 0.332 0.346 0.36 0.368
    discrete 66.7 33.4 29.3 43.6 182 4.49 5.89 7.53 13.0 94.0
    $m=1$ steady state does not lose stability
    $m=2$ 176 54.7 69.1 106.1 460 5.58 9.36 14.4 32.2 284
    $m=5$ 89.9 29.6 37.4 56.6 234 4.03 5.97 8.34 16.6 135
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