Yanjmaa Jutmaan  Department of Applied mathematics Some mathematical models related to physics and biology

2 The Effect of oil spills on the temporal and spatial distribution of fish population Model 1

3 Contents  Introduction  Mathematical modelling of oil spilling on fish population Simple model of fish growthSimple 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  Experimental result  Conclusion

4 Introduction  Mathematics vs. Biology Arguments are like patterns and mathematical modelingArguments are like patterns and mathematical modeling  Creatures most affected by oil spills in river ? FISHES  Study 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 rateProvide effective solutions after having a model with less error rate

5 2.Mathematical modeling of effect of oil spilling on fish population N(t)- number of fish at a particular instant of time t The growth rate of fish is dependent on many parameters

6 2.1 Simple model of fish growth R 0 : dependent on the constant birth rate B 0 and death rate D 0 B 0 instantaneous birth rate, births per individual per time period (t). D 0 instantaneous death rate, death per individual per time period (t). Assumption : Constant environment conditions and no spatial limitations  Analytical solution Exponential growth !!!! (Does the fish population keep on inc reasing) Exponential growth !!!! (Does the fish population keep on inc reasing)

7 2.2 Density dependent growth model k b : density dependent birth rate k d : density dependent death rates  Charles Darwin theory : Survival of fittest growth rate of species effected !!!

8 Steady state found as : At steady state solution of density dependent growth model : K is called carrying capacity of the environment K has inverse relation with k b and k d

9 Pollution Models Linear time model of oil spilling

10 Gaussian time model of oil spilling

11 3. Solution methodology  Make the growth rate equation dimensionless  Obtain analytic solution using initial condition: where β is dependent on the initial normalized population of fishes,  0

12  Change in carrying capacity effects temporal dependence of gr owth curves  Obtain analytic solution using initial condition, due to oil spilling

13 Making database for different density dependent rates Value of  varied from [0, 1] in steps of 0.01

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

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

16 Analysis at half distance from impact space using linear pollution model

17 Analysis at half distance from impact space using Gaussian pollution model

18 Analysis at different distances from impact space using linear pollution model

19 Analysis at different distances from impact space using Gaussian pollution model

20 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

21 Conclusion  A simple mathematical model of growth of fishes is developed.  The density dependent growth rate of fishes is also dependent on the amount of pollution due to oils spills.  Oil concentration, and The density dependent growth rates are function of time and distance  We solve the equation analytically to find the solution to the first order logistic growth model

22 We find out the variation caused to the normal growth conditions due to pollution by oil spills. This is done by plotting the curve for different steady sate values and then finding the value of fish population from the values of density dependent growth curves obtained from pollution model Summary

23 Second model: Swing high

24 Contents  Introduction  Mathematical modeling of pumping the swing by changing the center of mass with the knees Condition on rate of change of effective length of pendulum for swing pumpingCondition on rate of change of effective length of pendulum for swing pumping Position of maximum energy transferPosition of maximum energy transfer  Theory  Phase plane and asymptotic analysis of different swing trajectory Phase plane analysisPhase plane analysis Asymptotic analysisAsymptotic analysis  Conclusion

25 Introduction This is a study of the mechanism of pumping a swing (from a standing position) Certain action of an individual on a swing tak es them higher and higher without actually t ouching the ground, This is also called as s wing pumping. Certain action of an individual on a swing tak es them higher and higher without actually t ouching the ground, This is also called as s wing pumping.

26 2. Mathematical modeling of pumping the swing by changing the center of mass with the knees 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 net torque about the fixed support due to all forces acting on the point mass.

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

28 Pumping strategy to increase amplitude

Condition on rate of change of effective length of pendulum for swing pumping Multiplying to equation (3) gives

30 Fig. 2. Forces acting on a point mass m

31   Rate of change of energy is given as

Position of maximum energy transfer Maximum energy transfer occurs at

33 Assume that initially swing has a total energy E given by 3 Theory

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36 4 Phase plane and asymptotic analysis of different swing trajectory 4 Phase plane and asymptotic analysis of different swing trajectory

Asymptotic analysis

38 Linear swing trajectory for energy pumping

39 Cosine swinging trajectory for energy pumping

40 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 energy pumped is a function of change of effective length of the pendulum.

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