Model Diatom Population by Branching Stochastic Processes
Antoanela Terzieva,
Georgi Terziev
Issue:
Volume 8, Issue 1, February 2019
Pages:
3-8
Received:
14 December 2018
Accepted:
11 January 2019
Published:
21 February 2019
Abstract: The phytoplankton is one of the most ancient inhabitants of our planet. It consists of mostly unicellular aquatic species, both fresh water and marine. The purpose of this work is to model the dynamics of a diatoms population because it is a predominant phytoplankton kind and plays a key role at the base of the food chains, climate regulation and ecology. The formulated mathematical model would give a better idea about the expected population size in the near and further future. As a modelling tool we propose the branching stochastic process of Bellman-Harris (BPBH) Z (t). In general, the generating function (g.f.) F (t) for non Markov multidimensional BPBH is difficult for explicit expression. Impossibility for simultaneous birth and death of the BPBH-particle together with producing offspring would correspond to the biological side. Only after completion of the whole cycle the cell is capable of dividing and every particle is of zero age at birth, which corresponds to the condition of right continuity at the zero point of the distribution function (d.f.) G (t). It makes the multidimensional g.f. F (t) more suitable for research and analytical expression, allowing the use of basic theorems. The matrix U (t) of means meets the requirements and satisfies the basic matrix equation for a multidimensional non Markov branching processes. The matrix equation, corresponding to the system of sixteen integral equations is determined. The moments of Z (t) are expressed. The most characteristic feature of the diatoms is their cell wall - the cause of mitosis to result in one of the two daughters decreasing in size. This again directs the authors to determine the particle's type by its initial size and model by suggesting a decrease in the offspring size. The diatom's cell stops dividing when their size drops below the minimum. Accumulating sufficient critical mass, cells that have ceased to divide begin to merge with each other, generating a new cell. In contrast to the determined models the stochastic processes assess the probable future development. A certain fact is that the diatoms number is influenced by many factors of random nature in the environment.
Abstract: The phytoplankton is one of the most ancient inhabitants of our planet. It consists of mostly unicellular aquatic species, both fresh water and marine. The purpose of this work is to model the dynamics of a diatoms population because it is a predominant phytoplankton kind and plays a key role at the base of the food chains, climate regulation and e...
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Transient Mixed Convection Boundary Layer Flow of an Incompressible Fluid Past a Wedge in Presence of Magnetic Field
Shayma Joya Saha,
Litan Kumar Saha
Issue:
Volume 8, Issue 1, February 2019
Pages:
9-20
Received:
1 February 2019
Accepted:
11 March 2019
Published:
25 March 2019
Abstract: In this paper, an analysis is performed to explorethe transient, laminar two-dimensional, mixed convection boundary layer flow of a viscous and incompressible fluid past a vertical wedge taking into account the effect of magnetic field. With appropriate transformations the boundary layer equations are reduced to a local nonsimilarity equations and the solutions are obtained employing three distinct methods, namely, (i) perturbation method for small time; (ii) asymptotic solution method for large time; (iii) straight forward finite difference method for any time. The agreement between the solutions obtained from prescribed methods is found to be excellent. In this study the evaluation of skin-friction coefficient and the local Nusselt number with the effects of different governing parameters such as different time, τ, the exponent, m (= 0.4, 0.5, 1.0), mixed convection parameter, λ (= 0.0, 0.2, 0.4) and magnetic field parameter, M (=0.0, 1.0) for fluids having Prandtl number, Pr= 0.72, 1.0 and 7.0have been discussed. It is observed that both the local skin friction and local Nusseltnumber decreases due to an increase in the value of M. It is also found that an increase in the value of Prandtl number, Pr, leads to a decrease in the value of local skin friction coefficient and the value of local Nusselt number coefficient increases with the increasing values of Prandtl number.
Abstract: In this paper, an analysis is performed to explorethe transient, laminar two-dimensional, mixed convection boundary layer flow of a viscous and incompressible fluid past a vertical wedge taking into account the effect of magnetic field. With appropriate transformations the boundary layer equations are reduced to a local nonsimilarity equations and ...
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Search Directions in Infeasible Newton’s Method for Computing Weighted Analytic Center for Linear Matrix Inequalities
Shafiu Jibrin,
Ibrahim Abdullahi
Issue:
Volume 8, Issue 1, February 2019
Pages:
21-28
Received:
11 February 2019
Accepted:
22 March 2019
Published:
22 April 2019
Abstract: Four different search directions for Infeasible Newton’s method for computing the weighted analytic center defined by a system of linear matrix inequality constraints are studied. Newton’s method is applied to find the weighted analytic center and the starting point can be infeasible, that is, outside the feasible region determined by the linear matrix inequality constraints. More precisely, Newton’s method is used to solve system of equations given by the KKT optimality conditions for the weighted analytic center. The search directions for the Newton’s method considered are the ZY, ZY+YZ, Z−1 and NT methods that have been used in semidefinite programming. Backtracking line search is used for the Newton’s method. Numerical experiments are used to compare these search direction methods on randomly generated test problems by looking at the iterations and time taken to compute the weighted analytic center. The starting points are picked randomly outside the feasible region. Our numerical results indicate that ZY+YZ and ZY are the best methods. The ZY+YZ method took the least number of iterations on average while ZY took the least time on average and they handle weights better than the other methods when some of the weights are very large relative to the other weights. These are followed by NT method and then Z−1 method.
Abstract: Four different search directions for Infeasible Newton’s method for computing the weighted analytic center defined by a system of linear matrix inequality constraints are studied. Newton’s method is applied to find the weighted analytic center and the starting point can be infeasible, that is, outside the feasible region determined by the linear ma...
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