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Complicated dynamics of tumor-immune system interaction model with distributed time delay

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  • In this paper, we propose a distributed delay model to investigate the dynamics of the interactions between tumor and immune system. And we choose a special form of delay kernel which combines two delay kernels: a monotonic delay kernel representing a fading memory and a nonmonotonic delay kernel describing a peaking memory. Then, we discuss the effect of such delay kernel on system dynamics. The results show that the introduction of nonmonotonic delay kernel does not change the stability of tumor-free equilibrium, but it can induce stability switches of tumor-presence equilibrium and cause a rich pattern of dynamical behaviors including stabilization. Moreover, our numerical simulation results reveal that the nonmonotonic delay kernel has more complicated effects on the stability compared with the monotonic delay kernel.

    Mathematics Subject Classification: Primary: 92B05, 92D25; Secondary: 37N25, 34C23.


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  • Figure 1.  The black curves represent relatively small $ b $ and imply strong delay. The red curves represent relatively large $ b $ and imply weak delay. (a) The dashed curves correspond to the monotonic function $ F_1(t) $ with respect to $ t $. (b) The solid curves correspond to the nonmonotonic function $ F_2(t) $ with respect to $ t $

    Figure 2.  (a) The monotonic decreasing function $ D_4 $ with respect to $ c_2 $. (b) The quadratic function $ H(D_4) $ with respect to $ D_4 $

    Figure 3.  Log plots for the stability regions of tumor-presence equilibrium $ E^+ $ and stability boundary curves representing by black curves in $ c_2-b $ parameter plane. (a) $ E^+ $ is stable in the area outside these two stability boundary curves, while $ E^+ $ is unstable in the area between these two stability boundary curves. (b) $ E^+ $ is stable in the area below the stability boundary curve, while $ E^+ $ is unstable in the area above the stability boundary curve

    Figure 4.  Stability boundary curves of $ E^+ $ are shown in $ \rho-\omega $ parameter plane for different $ b $. $ E^+ $ is stable below each stability boundary curve, while $ E^+ $ is unstable above each stability boundary curve. (a) We fix $ c_2 = 0.2(<0.5) $. (b) We fix $ c_2 = 0.8(>0.5) $

    Figure 5.  Stability boundary curves of $ E^+ $ are shown in $ \rho-\omega $ parameter plane for different $ c_2 $. $ E^+ $ is stable below each stability boundary curve, while $ E^+ $ is unstable above each stability boundary curve. (a) We fix $ b = 0.001 $. (b) We fix $ b = 0.1 $

    Figure 7.  Schematic diagram of $ F(t) $ with respect to $ t $ for different $ c_2 $ and fixed $ b $

    Figure 6.  The function $ F(t) $ with respect to $ t $. The black curves are given for relatively small $ b $ and imply strong delay. The red curves are given for relatively large $ b $ and imply weak delay. (a) We fix $ c_2 = 0.2(<0.5) $. The dashed curves correspond to relatively small $ c_2 $. (b) We fix $ c_2 = 0.8(>0.5) $. The solid curves correspond to relatively large $ c_2 $

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