Yanjmaa Jutmaan Department of Applied mathematics Some mathematical models related to physics and...
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Yanjmaa Jutmaan Yanjmaa Jutmaan Department of Applied Department of Applied mathematicsmathematics
Some mathematical Some mathematical models related to models related to
physics and biology physics and biology
22
The Effect of oil spills on The Effect of oil spills on the temporal and spatial the temporal and spatial
distribution of fish distribution of fish
populationpopulation
Model 1 Model 1
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Contents Contents Introduction Introduction
Mathematical modelling of oil spilling on fish populationMathematical modelling of oil spilling on fish population
• Simple model of fish growth Simple model of fish growth • Density dependent growth modelDensity dependent growth model• Dependence of density birth and death rates on the pollution levelDependence of density birth and death rates on the pollution level
Solution methodology Solution methodology
Experimental result Experimental result
Conclusion Conclusion
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IntroductionIntroduction
Mathematics vs. BiologyMathematics vs. Biology• Arguments are like patterns and mathematical modelingArguments are like patterns and mathematical modeling
Creatures most affected by oil spills in river ?Creatures most affected by oil spills in river ?
FISHESFISHES
Study the effect of oil spilling on fish populationStudy the effect of oil spilling on fish population• Provide reasonable mathematical model under liable assumptionsProvide reasonable mathematical model under liable assumptions• Provide effective solutions after having a model with less error Provide effective solutions after having a model with less error
raterate
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2.Mathematical modeling of effect of 2.Mathematical modeling of effect of oil spilling on fish populationoil spilling on fish population
N(t)- number of fish at a particular instant of time tN(t)- number of fish at a particular instant of time t
0 0
( ) ( ) ( ) ( )lim limt t
N t N t t N t dN t
t t dt
),,,content mineral,),(()(
pollutionionconcentratoxygenfoodTtNfdt
tdN
The growth rate of fish is dependent on many parametersThe growth rate of fish is dependent on many parameters
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2.1 Simple model of fish 2.1 Simple model of fish growth growth
RR00 : dependent on the constant birth rate B : dependent on the constant birth rate B00 and death rate D and death rate D00
)()(
0 tNRdt
tdN
000 DBR BB00 instantaneous birth rate, births per individual per time period (t). instantaneous birth rate, births per individual per time period (t).
DD00 instantaneous death rate, death per individual per time period (t).instantaneous death rate, death per individual per time period (t).
• Assumption : Constant environment conditions and no spatial limitations
Analytical solutionAnalytical solution
Exponential growth !!!! (Does the fish population keep on incExponential growth !!!! (Does the fish population keep on increasing)reasing)
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2.2 Density dependent growth 2.2 Density dependent growth modelmodel
))(()( 00 tNkDtNkBR db
)()(0 tNkkRR db
kkb : b : density dependent birth rate density dependent birth rate k kd : d : density dependent death density dependent death
ratesrates
Charles Darwin theory : Survival of fittestCharles Darwin theory : Survival of fittestgrowth rate of species effected !!!growth rate of species effected !!!
)()(
tRNdt
tdN )(.)()(
)(0 tNtNkkR
dt
tdNdb
),( oilofondistributitemporalspatialfkandk db
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0R0 DB
Kkk
RN
dbsteady
0Steady state found as :Steady state found as :
At steady state solution of density dependent growth model :
K is called carrying capacity of the environmentK is called carrying capacity of the environment
K has inverse relation with kb and kd
)()()()(
tNKtNkkdt
tdNdb
K
tNtNR
dt
tdN )(1)(
)(0
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Pollution ModelsPollution Models
,3
)2(7.0)exp( 2
oilxP 52 oil
7.011
)15(7.0)exp( 2
oilxP
205 oil
Linear time model of oil spillingLinear time model of oil spilling
1010
))5(7.0exp( 22 oilxP 52 oil
7
)15(7.0exp
22 oilxP
205 oil
Gaussian time model of oil spillingGaussian time model of oil spilling
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3. Solution methodology3. Solution methodology
)1(
d
d
1)exp(
1)exp(1
2
1
0
0
1log
Make the growth rate equation dimensionless
Obtain analytic solution using initial condition:
where β is dependent on the initial normalized population of fishes, 0
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1)exp(
1)exp(1
2
1
10
2( ( ) ( )) ( )sin( ( )).dI t t qI t t
dt
Change in carrying capacity effects temporal dependence of growth curves
Obtain analytic solution using initial condition, due to oil
spilling
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1
db kk
Making database for different density dependent rates
Value of varied from [0, 1] in steps of 0.01
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Experimental resultExperimental result
Analysis at impact space (point of onset of oil spilling) using linear pollution model
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Analysis at impact space (point of onset of oil spilling) using Gaussian
pollution model
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Analysis at half distance from impact space using linear pollution model
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Analysis at half distance from impact space using Gaussian pollution
model
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Analysis at different distances from impact space using linear pollution
model
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Analysis at different distances from impact space using Gaussian pollution
model
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Benefits of the model
Predefined database
Choice of pollution model (user can choose the model)
Linear interpolation is used, instead of closest value
Considers both temporal and spatial distribution
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Conclusion Conclusion
A simple mathematical model of growth of fishes is A simple mathematical model of growth of fishes is developed. developed.
The density dependent growth rate of fishes is also The density dependent growth rate of fishes is also dependent on the amount of pollution due to oils spills.dependent on the amount of pollution due to oils spills.
Oil concentration , and The density dependent growth Oil concentration , and The density dependent growth rates are function of time and distance rates are function of time and distance
We solve the equation analytically to find the solution We solve the equation analytically to find the solution to the first order logistic growth modelto the first order logistic growth model
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We find out the variation caused to the normal growth We find out the variation caused to the normal growth conditions due to pollution by oil spills.conditions due to pollution by oil spills. This is done by plotting the curve for different steady sate This is done by plotting the curve for different steady sate values and then finding the value of fish population from values and then finding the value of fish population from the values of density dependent growth curves obtained the values of density dependent growth curves obtained from pollution modelfrom pollution model
SummarySummary
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Second model:Second model:
Swing highSwing high
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Contents Contents Introduction Introduction
Mathematical modeling of pumping the swing by changing the center of Mathematical modeling of pumping the swing by changing the center of mass with the kneesmass with the knees
• Condition on rate of change of effective length of pendulum for swing Condition on rate of change of effective length of pendulum for swing pumpingpumping
• Position of maximum energy transferPosition of maximum energy transfer Theory Theory
Phase plane and asymptotic analysis of different Phase plane and asymptotic analysis of different swing trajectoryswing trajectory
• Phase plane analysisPhase plane analysis• Asymptotic analysisAsymptotic analysis
Conclusion Conclusion
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IntroductionIntroduction
This is a study of the mechanism of This is a study of the mechanism of pumping a swing (from a standing pumping a swing (from a standing position)position)
Certain action of an individual on a swing Certain action of an individual on a swing takes them higher and higher without takes them higher and higher without actually touching the ground, This is actually touching the ground, This is also called as swing pumping.also called as swing pumping.
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2. Mathematical modeling of 2. Mathematical modeling of pumping the swing by changing pumping the swing by changing
the center of mass with the kneesthe center of mass with the knees
dH
dt
Conservation of angular momentum for a point mass undergoing planar motion gives
H is the angular momentum of the point mass about the fixed support
2( ( ) ( )) ( )sin( ( )).dl t t ql t t
dt
net torque about the fixed support due to all forces acting on the point mass.
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2 sin 0l l q
After differentiating and rearranging the terms we get, (3)(3)
2( ( ) ( )) ( )sin( ( )).dl t t ql t t
dt
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Pumping strategy to increase Pumping strategy to increase amplitudeamplitude
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2.1 2.1 Condition on rate of change of effective Condition on rate of change of effective length of pendulum for swing pumpinglength of pendulum for swing pumping
22 sin 0l l g
Multiplying to equation (3) gives
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Fig. 2. Forces acting on a point mass m
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Rate of change of energy is given as
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2.2 2.2 Position of maximum Position of maximum energy transferenergy transfer
Maximum energy transfer Maximum energy transfer occurs atoccurs at
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Assume that initially swing has Assume that initially swing has a total energy E given by a total energy E given by
3 Theory
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3535
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4 4 Phase plane and asymptotic Phase plane and asymptotic analysis of different swinganalysis of different swing
trajectorytrajectory
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4.2 Asymptotic analysis4.2 Asymptotic analysis
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Linear swing trajectory for Linear swing trajectory for energy pumping energy pumping
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Cosine swinging trajectory for Cosine swinging trajectory for energy pumping energy pumping
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5 Conclusion
The swing reaches higher amplitudes in every half cycle because of this gain in the energy.
The maximum energy is pumped at the center (theta = 0) and the rate of energypumped is a function of change of effective length of the pendulum.
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