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Computational Science and Engineering
Dynamic System In Biology
Yang Cao
Department of Computer Science
http://courses.cs.vt.edu/~cs6404
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Computational Science and Engineering
Outline
• Single Species Population Model
– Malthus Model
– Logistic Model
• Two species Model
– Competition Model
– Predator and Prey Model
• Phase Plot and Dynamic System
• Stochastic Model and Simulation
– Lotka Model
– Brusselator Model
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Computational Science and Engineering
Malthus Model
“I SAID that population, when unchecked, increased in a geometrical ratio, and subsistence for man in an
arithmetical ratio. “
---- Thomas Malthus
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Computational Science and Engineering
Malthus Model
ttkPtPttP ∆+=∆+ )()()(
The reproduction rate is proportional to the population
Solve it we have
The population in the United States in year 1790 is .
The corresponding population in year 1800 is .
With a data fitting, we obtain:
)(
00)(
ttkePtP
−=
6109.3 ×
6103.5 ×
)1790(0307.06109.3)( −×= tetP
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Computational Science and Engineering
Logistic Population Model
• Developed by Belgian mathematician Pierre Verhulst (1838) in 1838
• The rate of population increase may be limited, i.e., it may depend on population density
ttPktPttP ∆+=∆+ ))(()()(where
)()(
1))(( 0 tPP
tPktPk
m
−=
( )( )
( )0000
00
)1(1)(
)(
0
00
0ttkm
m
m
ttk
mttk
eP
P
P
PPeP
PePtP
−−−
−
−+
=−+
=
The solution is
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Computational Science and Engineering
Logistic Population Model
( )( )
( )0000
00
)1(1)(
)(
0
00
0ttkm
m
m
ttk
mttk
eP
P
P
PPeP
PePtP
−−−
−
−+
=−+
=
The solution of the Logistic model
With a data fitting
03134.0 ,10197 06 =×= kPm
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Computational Science and Engineering
Model of two species (Competition)
Let the population of two species be and , and they compete
in the same environment. If there is no competition, the population of X
will satisfy
With the competition,
For another species, there is a similar equation
The physical meaning of and can be understood as:
Thus we have
)(tx
)1)(()(1
1N
xtxrtx −=&
)(ty
)1)(()(1
1N
yxtxrtx
α+−=&
)1)(()(2
2N
yytyrty
β+−=&
α β
.consume species Yeach resource the
consume species X each resource the=α
1=αβ
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Computational Science and Engineering
State Dynamics Plot vs Phase Plot
State Dynamics Plot: state vs time,
Phase Plot: the state space, use arrow to represent the tangent vector
The phase plot reveals the geometric property of a dynamic system represented by a pair of ODEs.
+−=
+−=
)100
1(1.0)(
)100
1(1.0)(
yxyty
yxxtx
&
&
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Computational Science and Engineering
State Dynamics Plot vs Phase Plot
Example: from different initial value, the trajectory follow the direction of the arrows and reaches to its equilibrium state
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Computational Science and Engineering
State Dynamics Plot vs Phase Plot
+−=
+−=
)90
1(1.0)(
)100
1(1.0)(
yxyty
yxxtx
&
&
However, a slight change of parameters make a big difference in phase plot and lead to a different conclusion
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Computational Science and Engineering
State Dynamics Plot vs Phase Plot
+−=
+−=
)90
1(1.0)(
)100
1(1.0)(
yxyty
yxxtx
&
&
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Computational Science and Engineering
If , X species will win.
The sign of the derivatives are decided by two values
and
State Dynamics Plot vs Phase Plot
+−=
+−=
)1()(
)1()(
2
2
1
1
N
yxyrty
N
yxxrtx
β
α
&
&
)(1 yxN α+− )(2 yxN αα +−
A direct analysis through the phase plot
21 NN α>
If , Y species will win. 21 NN α
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Computational Science and Engineering
Model of two species (Predator and Prey)
• Lotka-Volterra Model
• The simplest model of predator-prey interactions developed independently
by Lotka (1925) and Volterra (1926)
• Ancona’s observation on Shark’s population during world war I.
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Computational Science and Engineering
Model of two species (Predator and Prey)
Assumption:
• The predator species is totally dependent on a single prey species as its only food supply,
• The prey species has an unlimited food supply, and there is no threat to the prey other than the specific predator.
Let X represent the prey and Y represent the predator, without the predator, the Malthus model can be applied
However, because of the predator, r has to be modified
For the predator, the situation is just the opposite.
Thus we get the ODEs for this model
axx =&
xbyax )( −=&
ydxcy )( +−=&
+−=
−=
ydxcy
xbyax
)(
)(
&
&
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Computational Science and Engineering
There are two corresponding equilibrium points:
or
Phase Plot Analysis
+−=
−=
ydxcy
xbyax
)(
)(
&
&
)0,0( ),( ba
d
c
)0,0(
b
a
d
c
),( −+
),( −− ),( +−
),( ++
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Computational Science and Engineering
Matlab Simulation Result
+−=
−=
yxy
xyx
)4.03(
)2.01(
&
&Based on example:
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Computational Science and Engineering
Effect of Parameters
b
ay
d
cx == ,
prey for the rate onreproducti natural the:a
predator theof because rate killing the:b
The solution of the LV predator-prey model is
where
predator for the rate death natural the:cprey theof because rate onreproducti the:d
Question: Why the shark ratio increases during world war I?
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Computational Science and Engineering
Parameter Analysis
)0,0(
b
ay
d
cx == ** ,
When fishing is introduced in the model, their effect will be increase the death rate of the predator and decrease the reproduction rate for the prey. Thus
eaaecc −→+→ ,
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Computational Science and Engineering
Stochastic Modeling
Lotka reactions:
ZY
YYX
XXA
c
c
c
→
→+
→+
3
2
1
2
2
Lead to ODEs
+−=
−=
yxccy
xycAcx
)(
)(
23
21
&
&
The stochastic simulation generates interesting trajectories.
10
,01.0
,10
3
2
1
=
=
=
c
c
Ac
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Computational Science and Engineering
Different Dynamic Behavior
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Computational Science and Engineering
Brusselator
DY
YYX
CYXB
XA
c
c
c
c
→
→+
+→+
→
4
3
2
1
32
−=
−+−=
yxBycy
xcyxBycAcx
c
c
2
22
4
2
221
3
3
&
&
.5
,00005.0
,50
,5000
4
3
2
1
=
=
=
=
c
c
Bc
Ac
Lead to ODEs
.5
,0001.0
,50
,5000
4
3
2
1
=
=
=
=
c
c
Bc
Ac
Bifurcation happens around the condition:
( )
3
4
2
4
2
12 23
2
c
c
c
Acc
Bc +=
J. Tyson’s 1973, 1974 paper
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Computational Science and Engineering
Oregonator
XZE
DY
ZYYC
BYX
YXA
c
c
c
c
c
→+
→
+→+
→+
→+
5
4
3
2
1
2
2
−=
−+−=
+−−=
EzcCycz
ycCycxycAxcy
EzcxycAxcx
53
2
4321
521
&
&
&
26c ,016.0c ,104c ,1.0c ,2 54321 ===== ECAc
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Computational Science and Engineering
Oregonator
XZE
DY
ZYYC
BYX
YXA
c
c
c
c
c
→+
→
+ →+
→+
→+
5
4
3
2
1
2
2
26c ,016.0c ,104c ,1.0c ,2 54321 ===== ECAc
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Computational Science and Engineering
Thanks! Questions? Plato is my friend, Aristotle is my friend, but my best
friend is truth --- Newton
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