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Some mathematical models related to physics and biology. Yanjmaa Jutmaan. Department of Applied mathematics. Model 1. The Effect of oil spills on the temporal and spatial distribution of fish population. Contents. Introduction Mathematical modelling of oil spilling on fish population
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Some mathematical models related to physics and biology Yanjmaa Jutmaan • Department of Applied mathematics
Model 1 The Effect of oil spills on the temporal and spatial distribution of fish population
Contents • Introduction • Mathematical modelling of oil spilling on fish population • Simple model of fish growth • Density dependent growth model • Dependence of density birth and death rates on the pollution level • Solution methodology • Experimental result • Conclusion
Introduction • Mathematics vs. Biology • Arguments 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 assumptions • Provide effective solutions after having a model with less error rate
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
2.1 Simple model of fish growth • Assumption : Constant environment conditions and no spatial limitations R0 : dependent on the constant birth rate B0 and death rate D0 B0instantaneous birth rate, births per individual per time period (t). D0instantaneous death rate, death per individual per time period (t). • Analytical solution Exponential growth !!!! (Does the fish population keep on increasing)
2.2 Density dependent growth model • Charles Darwin theory : Survival of fittestgrowth rate of species effected !!! kb : density dependent birth rate kd : density dependent death rates
At steady state solution of density dependent growth model : Steady state found as : K is called carrying capacity of the environment K has inverse relation with kb and kd
Pollution Models Linear time model of oil spilling
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
Change in carrying capacity effects temporal dependence of growth curves • Obtain analytic solution using initial condition, due to oil spilling
Making database for different density dependent rates Value of varied from [0, 1] in steps of 0.01
Experimental result Analysis at impact space (point of onset of oil spilling) using linear pollution model
Analysis at impact space (point of onset of oil spilling) using Gaussian pollution model
Analysis at half distance from impact space using linear pollution model
Analysis at half distance from impact space using Gaussian pollution model
Analysis at different distances from impact space using linear pollution model
Analysis at different distances from impact space using Gaussian pollution model
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
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
Summary 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
Second model: Swing high
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 pumping • Position of maximum energy transfer • Theory • Phase plane and asymptotic analysis of different swing trajectory • Phase plane analysis • Asymptotic analysis • Conclusion
Introduction This is a study of the mechanism of pumping a swing(from a standing position) Certain action of an individual on a swing takes them higher and higherwithout actually touching the ground, This is also called as swing pumping.
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 His 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.
After differentiating and rearranging the terms we get, • (3)
2.1 Condition on rate of change of effective length of pendulum for swing pumping Multiplying to equation (3) gives
2.2 Position of maximum energy transfer Maximum energy transfer occurs at
3 Theory Assume that initially swing has a total energy E given by
4 Phase plane and asymptotic analysis of different swingtrajectory
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.
