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

33

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

44

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

55

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

66

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)

77

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

88

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

99

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

1111

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

1212

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

1313

1

db kk

Making database for different density dependent rates

Value of varied from [0, 1] in steps of 0.01

1414

Experimental resultExperimental result

Analysis at impact space (point of onset of oil spilling) using linear pollution model

1515

Analysis at impact space (point of onset of oil spilling) using Gaussian

pollution model

1616

Analysis at half distance from impact space using linear pollution model

1717

Analysis at half distance from impact space using Gaussian pollution

model

1818

Analysis at different distances from impact space using linear pollution

model

1919

Analysis at different distances from impact space using Gaussian pollution

model

2020

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

2121

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

2222

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

2323

Second model:Second model:

Swing highSwing high

2424

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

2525

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.

2626

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.

2727

2 sin 0l l q

After differentiating and rearranging the terms we get, (3)(3)

2( ( ) ( )) ( )sin( ( )).dl t t ql t t

dt

2828

Pumping strategy to increase Pumping strategy to increase amplitudeamplitude

2929

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

3030

Fig. 2. Forces acting on a point mass m

3131

Rate of change of energy is given as

3232

2.2 2.2 Position of maximum Position of maximum energy transferenergy transfer

Maximum energy transfer Maximum energy transfer occurs atoccurs at

3333

Assume that initially swing has Assume that initially swing has a total energy E given by a total energy E given by

3 Theory

3434

3535

3636

4 4 Phase plane and asymptotic Phase plane and asymptotic analysis of different swinganalysis of different swing

trajectorytrajectory

3737

4.2 Asymptotic analysis4.2 Asymptotic analysis

3838

Linear swing trajectory for Linear swing trajectory for energy pumping energy pumping

3939

Cosine swinging trajectory for Cosine swinging trajectory for energy pumping energy pumping

4040

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.

4141

THANK YOU FOR LISTENING