biosim lec6

biosim lec6 - Continuous Models Read Chapter 4.1 4.5 and...

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Continuous Models Read: Chapter 4.1, 4.5 and 4.7 Mathematical Models in Biology (2005), Leah E delstein-Keshet Supplementary material for Dimensional Analysis TAs: Review of linear ODE (e.g. Chapter 4.8)
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Bacteria Growth-Revisited • Consider bacteria growing in a nutrient rich medium • Variables – Time, t – N(t) = bacteria density at time t • Dimension of N(t) is # cells/vol. • Parameters – k = growth/reproduction rate per unit time • Dimension of k is 1/time
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Bacteria Growth Revisited • Now suppose that bacteria densities are observed at two closely spaced time points, say t and t + Δ t • If death is negligible, the following statement of balance holds: Bacteria Density @ t + Δ t Bacteria density @ time t New bacteria Produced in the Interval t + Δ t - t = + N(t+ Δ t) = N(t) + kN(t) Δ t
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Bacteria Growth Revisited • Rearrange these terms • Assumptions – N(t) is large--addition of one or several new cells is of little consequence – There is no new mass generated at distinct intervals of time, ie cell growth and reproduction is not correlated. • Under these assumptions we can say that N(t) changes continuously N ( t + Δ t ) N ( t ) Δ t = kN
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Bacteria Growth Revisited • Upon taking the limit • The continuous model becomes • Its solution is Δ t 0 lim N ( t + Δ t ) N ( t ) Δ t = dN dt dN dt = kN N ( t ) = N 0 e kt
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Properties of the Model • Doubling Time/Half life: • Steady state N e = 0 • Stability – N e = 0 is stable if k < 0 – N e = 0 is unstable if k > 0 ln2 k dN dt = 0
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Modified Model
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biosim lec6 - Continuous Models Read Chapter 4.1 4.5 and...

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