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Modifications of the  model

How n influences the model:

  • n=0: exponential growth — no suppression at all

  • n=1: standard logistic growth

  • n>1: sharper transition to saturation — growth slows down rapidly as tumor nears capacity

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Acknowledging the Effects of the Tumor's Environment 

Population growth over time

  • c(t) represents the size of a tumor or population (scaled by carrying capacity K)

  • The curve shows how quickly the tumor grows, how it slows down, and whether it stabilizes (plateaus)

  • Different values of n affect how early or late the slowing happens:

    • Small n → slower tapering off, smoother curve

    • Large n → growth appears fast at first, then suddenly levels off

Per Capita Growth Rate f(c) Over Time

  • This shows how fast each tumor cell (or individual) contributes to overall growth

  • This rate decreases over time as the tumor gets larger (due to space/nutrient limitations)

  • The shape of the drop depends on n:

    • Higher n causes the growth rate to drop more abruptly (stronger inhibition as tumor grows)

    • Lower n means the decline is more gradual (weaker suppression)

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Figure 1: This represents the effect of different values of n on tumor growth when lambda(tumor growth rate)=0.25

Introduction of the Immune Killing Term

In a tumor growth model immune attack is represented by adding a killing term.It gives a more realistic and accurate depiction of tumor cells in the body.This is because as the tumor cells grow,the immune cells especially the cytotoxic T cells recognize them and actively  kill them slowing down the growth of the tumor

  • γ= killing rate by immune cells

  • I = concentration of immune cells

  • c=Tumor cell population

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Figure 2: This represents the effect of the immune killing term on different values of lambda (tumor growth rate)

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