Dynamic System In Biology - Coursescs6404/DynamicSystem.pdf• Phase Plot and Dynamic System •...

Computational Science and Engineering

Dynamic System In Biology

Yang Cao

Department of Computer Science

http://courses.cs.vt.edu/~cs6404


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


Malthus Model

“I SAID that population, when unchecked, increased in a geometrical ratio, and subsistence for man in an

arithmetical ratio. “

---- Thomas Malthus


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


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


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


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


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

&

&



Example: from different initial value, the trajectory follow the direction of the arrows and reaches to its equilibrium state



+−=

+−=

)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



+−=

+−=

)90

1(1.0)(

)100

1(1.0)(

yxyty

yxxtx

&

&


If , X species will win.

The sign of the derivatives are decided by two values

and


+−=

+−=

)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


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.


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

)(

)(

&

&


There are two corresponding equilibrium points:

or

Phase Plot Analysis

+−=

−=

ydxcy

xbyax

)(

)(

&

&

)0,0( ),( ba

d

c

)0,0(

b

a

d

c

),( −+

),( −− ),( +−

),( ++


Matlab Simulation Result

+−=

−=

yxy

xyx

)4.03(

)2.01(

&

&Based on example:


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?


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 −→+→ ,


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


Different Dynamic Behavior


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


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


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


Thanks! Questions? Plato is my friend, Aristotle is my friend, but my best

friend is truth --- Newton

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