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Continuous Models Chapter 4 Bacteria Growth-Revisited • Consider bacteria growing in a nutrient rich medium • Variables QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. – 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 QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this pict ure. Bacteria Growth Revisited • Now suppose that bacteria densities are observed at two closely spaced time points, say t and t + t • If death is negligible, the following statement of balance holds: Bacteria Density @ N(t+t) t +t = Bacteria density @ time t + = N(t) + New bacteria Produced in the Interval t + t - t kN(t) t Bacteria Growth Revisited • Rearrange these terms • Assumptions N(t t) N(t) kN t – 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 Bacteria Growth Revisited • Upon taking the limit N(t t) N(t) dN lim t dt t 0 • The continuous model becomes • Its solution is dN kN dt N(t) N 0e kt Properties of the Model • Doubling Time/Half life: • Steady state – Ne = 0 dN 0 dt • Stability – Ne = 0 is stable if k < 0 – Ne = 0 is unstable if k > 0 ln 2 k Modified Model • Now assume that growth and reproduction depends on the available nutrient concentration • New Variable – C(t) = concentration of available nutrient at time t • Dimensions of C are mass/vol Modified Model • New assumptions – Population growth rate increases linearly with nutrient concentration k(C) C units of nutrient are consumed in producing one new unit of bacteria dC dN dt dt Modified Model • We now have two equations dN CN dt dC dN dt dt • Upon integration we see C(t) N(t) C0 C C0 • So any initial nutrient concentration can only support a fixed amount of bacteria C0 N The Logistic Growth Model • Substitute to find N dN rN1 K dt dN C0 N dt • where r C0 Intrinsic growth rate K C0 Environmental Carrying capacity The Logistic Growth Model • Model • Solution N dN rN1 K dt N 0K N(t) N 0 (K N 0 )ert • Note: as N K, N/K 1 and 1-N/K 0 • Asthe population size approaches K, the population growth rate approaches zero Breakdown • In general, single species population growth models can all be written in the following form dN f (N) Ng(N) dt where g(N) is the actual growth rate. Actual growth rate = Actual birth rate Actual death rate g(N) b(N) d(N) Breakdown • The logistic equations makes d(N) certain assumptions about the relationship between population size and the actual birth and death rates. of the • The actual death rate d 0 population is assumed to increase linearly with population size d(N) d0 N Intrinsic death rate N Breakdown b(N) • The actual birth rate of the population is assumed to decrease linearly with population size b0 b(N) b0 N Intrinsic birth rate b0 N Breakdown dN g(N)N b(N) d(N)N dt dN b0 N d0 N N dt • Rearrange to get: dN b0 d0 N1 N dt b0 d0 Breakdown • Now let Intrinsic growth rate r b0 d0 C0 = Intrinsic birth rate Intrinsic death rate b0 d0 C0 K Carrying capacity Sensitivity of birth and death rate to population size Plot of Actual Birth and Death Rates QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. K Assuming Linearity • Linearity is the simplest way to model the relationship between population size and actual birth and death rates • This may not be the most realistic assumption for many population • A curve of some sort is more likely to be realistic, as the effect of adding individuals may not be felt until some critical threshold in resource per individual has been crossed Solution Profiles General Single Species Models dN f (N) Ng(N) dt • Steady States – Solutions of f(N) = 0 • N = 0 is always a steady state • So must determine when g(N) = 0 for nontrivial steady states – Example N dN rN1 K dt • Steady states are N = 0 and N = K, both always exist. General Single Species Models • Stability – How do small perturbations away from steady state behave? 1. Let N = Ne + n where |n| << 1 2. Substitute into model equation 3. Expand RHS in a Taylor series and simplify 4. Drop all nonlinear terms General Single Species Models • Stability – Once steps 1 - 4 are preformed, you’ll arrive at an equation for the behavior of the small perturbations dn f (N e )n dt – – n(t) grows if f (Ne ) 0 • Therefore N = Ne is unstable • Therefore N = Ne is stable (Ne ) 0 N(t) decays if f n(t) e f (N e )t General Single Species Models • Stability – Analysis shows that stability is completely determined by the slope of the growth function, f(N), evaluated at the steady dN state. • Example dt N dN rN1 K dt 0 K N General Single Species Models • Stability N dN rN1 K dt N f (N) rN 1 K 2N f (N) r1 K f (0) r 0 Ne = 0 is always unstable f (K) r 0 Ne = K is always stable Compare Continuous and Discrete Logistic Model Discrete Continuous dN rN dt Nt 1 rN t Solutions grow or decay -- possible oscillations N t N t 1 rN t 1 K Solutions approach N = 0 or N = K or undergo period doubling bifurcations to chaos Solutions grow or decay --no oscillations N dN rN1 K dt All solutions approach N = K Nondimensionalization • Definition: Nondimensionalization is an informed rescaling of the model equations that replaces dimensional model variables and parameters with nondimensional counterparts Why Nondimensionalize? • To reduce the number of parameters • To allow for direct comparison of the magnitude of parameters • To identify and exploit the presence of small/large parameters • Note: Nondimensionalization is not unique!! How to Nondimensionalize • Perform a dimensional analysis dN rN dt Variables/Dimension N density t time N(0) N0 r0 Parameters/Dimension r 1/time N0 density How to Nondimensionalize • Introduce an arbitrary scaling of all variables N u Bt A • Substitute into the model equation Original Model dN rN dt N(0) N0 Scaled Model du AB rAu d Au(0) N0 How to Nondimensionalize • Choose meaning scales du AB rAu d du r u d B • Let Br A N0 Au(0) N0 N0 u(0) A Time is scaled by the intrinsic growth rate Population size is scaled by the initial size du u d u(0) 1 How to Nondimensionalize du u d u(0) 1 • Note: The parameters of the system are reduced from 2 to 0!! • There are no changes in initial or growth rate that can conditions qualitatively change the behavior of the solutions-- ie no bifurcations!! Nondimensionalize the Logistic Equation