Continuous Models
description
Transcript of Continuous Models
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Continuous Models
Chapter 4
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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 + t
• If death is negligible, the following statement of balance holds:
BacteriaDensity @
t +t
Bacteria density @
time t
New bacteriaProduced in theInterval t + t - t
= +
N(t+t) = N(t) + kN(t) 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 + Δt) − N(t)Δt
= kN
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Bacteria Growth Revisited• Upon taking the limit
• The continuous model becomes
• Its solution is
€
t →0lim N(t + Δt) − N(t)
Δt= dN
dt
€
dNdt
= kN
€
N(t) = N0ekt
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Properties of the Model
• Doubling Time/Half life:
• Steady state– Ne = 0
• Stability– Ne = 0 is stable if k < 0– Ne = 0 is unstable if k > 0
€
ln2k
€
dNdt
= 0
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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
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Modified Model
• New assumptions– Population growth rate increases linearly
with nutrient concentration
units of nutrient are consumed in producing one new unit of bacteria€
k(C) = κC
€
dCdt
= −α dNdt
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Modified Model
• We now have two equations
• Upon integration we see
• So any initial nutrient concentration can only
support a fixed amount of bacteria
€
dCdt
= −α dNdt
€
dNdt
= κCN
€
C(t) = −αN(t) + C0€
C
€
N€
C0
€
C0
α
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The Logistic Growth Model
• Substitute to find
• where€
dNdt
= κ C0 −αN( )
€
dNdt
= rN 1− NK
⎛ ⎝ ⎜
⎞ ⎠ ⎟
€
r = κC0
€
K = C0
αIntrinsic
growth rateEnvironmental
Carrying capacity
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The Logistic Growth Model
• Model• Solution
• Note: as N K, N/K 1 and 1-N/K 0• As the population size approaches K, the
population growth rate approaches zero€
N(t) = N0KN0 + (K − N0)e−rt€
dNdt
= rN 1− NK
⎛ ⎝ ⎜
⎞ ⎠ ⎟
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Breakdown• In general, single species population
growth models can all be written in the following form
where g(N) is the actual growth rate.
€
dNdt
= f (N) = Ng(N)
Actual growth rate
Actual birth rate
Actual death rate =
€
g(N) = b(N) − d(N)
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Breakdown• The logistic equations makes
certain assumptions about the relationship between population size and the actual birth and death rates.
• The actual death rate of the population is assumed to increase linearly with population size
€
d(N) = d0 + δN€
d0
€
N
€
d(N)
Intrinsic death rate
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Breakdown
• The actual birth rate of the population is assumed to decrease linearly with population size
€
b(N) = b0 − βN€
b0
€
N
€
b(N)
Intrinsic birth rate
€
b0
β
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Breakdown
• Rearrange to get:€
dNdt
= g(N)N = b(N) − d(N)[ ]N
€
dNdt
= b0 − βN( ) − d0 −δN( )[ ]N
€
dNdt
= b0 − d0( )N 1− β −δb0 − d0
N ⎡ ⎣ ⎢
⎤ ⎦ ⎥
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Breakdown
• Now let
€
r = b0 − d0 = κC0
Intrinsic growth rate
Intrinsic birth rate
Intrinsic death rate =
€
K = b0 − d0
β −δ= C0
α
Carrying capacity
Sensitivity of birth and death rate to population size
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Plot of Actual Birth and Death Rates
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are needed to see this picture.
K
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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
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Solution Profiles
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General Single Species Models
• 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
• Steady states are N = 0 and N = K, both always exist.
€
dNdt
= f (N) = Ng(N)
€
dNdt
= rN 1− NK
⎛ ⎝ ⎜
⎞ ⎠ ⎟
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General Single Species Models
• Stability– How do small perturbations away from
steady state behave?
1. Let N = Ne + n where |n| << 12. Substitute into model equation3. Expand RHS in a Taylor series and
simplify4. Drop all nonlinear terms
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General Single Species Models
• Stability– Once steps 1 - 4 are preformed, you’ll arrive at
an equation for the behavior of the small perturbations
– n(t) grows if • Therefore N = Ne is unstable
– N(t) decays if • Therefore N = Ne is stable
€
dndt
= ′ f (Ne )n
€
n(t) = e ′ f (Ne )t
€
′ f (Ne ) > 0
€
′ f (Ne ) < 0
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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 state.
• Example
€
dNdt
= rN 1− NK
⎛ ⎝ ⎜
⎞ ⎠ ⎟€
dNdt
€
N
€
0
€
K
unstable
stable
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General Single Species Models
• Stability
€
dNdt
= rN 1− NK
⎛ ⎝ ⎜
⎞ ⎠ ⎟
€
f (N) = rN 1− NK
⎛ ⎝ ⎜
⎞ ⎠ ⎟
€
′ f (N) = r 1− 2NK
⎛ ⎝ ⎜
⎞ ⎠ ⎟
€
′ f (0) = r > 0
€
′ f (K) = −r < 0
Ne = 0 is always unstable
Ne = K is always stable
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Compare Continuous and Discrete Logistic ModelDiscrete Continuous
€
dNdt
= rN
€
dNdt
= rN 1− NK
⎛ ⎝ ⎜
⎞ ⎠ ⎟
€
N t +1 = rN t
Solutions grow or decay -- possible oscillations Solutions grow or decay
--no oscillations
Solutions approach N = 0 or N = K or undergo period doubling bifurcations to chaos All solutions approach N = K€
N t +1 = rN t 1− N t
K ⎛ ⎝ ⎜
⎞ ⎠ ⎟
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Nondimensionalization
• Definition: Nondimensionalization is an informed rescaling of the model equations that replaces dimensional model variables and parameters with nondimensional counterparts
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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!!
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How to Nondimensionalize• Perform a dimensional analysis
€
dNdt
= rN
Variables/Dimension Parameters/Dimension
N density
t time
r 1/time
N0 density
€
N(0) = N0
€
r > 0
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How to Nondimensionalize• Introduce an arbitrary scaling of all
variables
• Substitute into the model equation
€
u = NA
€
τ =Bt
€
AB dudτ
= rAu
€
dNdt
= rN
€
N(0) = N0
€
Au(0) = N0
Original Model Scaled Model
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How to Nondimensionalize
• Choose meaning scales
• Let€
AB dudτ
= rAu
€
Au(0) = N0
€
dudτ
= rB
u
€
u(0) = N0
A
€
A = N0
€
B = r
€
dudτ
= u
€
u(0) =1
Time is scaled by the intrinsic growth rate
Population size is scaled by the initial size
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How to Nondimensionalize
• Note: The parameters of the system are reduced from 2 to 0!!
• There are no changes in initial conditions or growth rate that can qualitatively change the behavior of the solutions-- ie no bifurcations!!
€
dudτ
= u
€
u(0) =1
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Nondimensionalize the Logistic Equation