International ejournals

Transcription

1 ISSN Available online at International ejournals International Journal of Mathematical Sciences, Technology and Humanities 19 (2012) MATHEMATICAL MODELLING & ECOLOGY-A BRIEF SURVEY B. Bhaskara Rama Sarma1 & N.Ch.Pattabhiramacharyulu2 1. Faculty of Mathematics, BRS Classes, #2-284,Vivekanandastree, Hanumannagar, Ramavarappadu,Vijayawada,A.P ; Mobile: ; bbramasarma@ yahoo.co.in 2. Faculty of Mathematics (Retd.), NIT, Warangal, A.P ABSTRACT Modeling may be treated as a science concerning with the interaction between mathematics and other subjects an academic discipline on some aspects of the everyday world.mathematical models have become important tools in biological investigations and iterative procedures of information collection.. In this paper, brief summary on specific topics of modeling in life sciences is provided for ready reference. Ecology relates to the study of living beings (animals and plants) in relation to their habits and habitats. This discipline of knowledge is a branch of evolutionary biology purported to explain how or to what extent the living beings are regulated in nature. As models in any branch of science and technology, mathematical models in theoretical ecology are of great importance and utility because they both answer and raise questions related to natural phenomena. It is natural that two or more species living in a common habitat interact in different ways with each other. Some possible ecological interactions that exist among species are also illustrated in this paper. Key words: Modeling, Ecology, Competition, Prey-predation, Amensalism, Mutualism AMS Classification: 92 D 25, 92 D Introduction Mathematical Modeling is a rich interdisciplinary activity involving the study of some aspects of diverse disciplines such as Biology, Bio economics, Epidemiolgy, Genetics, Pharmaco kinetics, Physiology, Ecology, Immunology, Sociology and even Politics apart from Physical Sciences, Engineering & Technology. It is an endeavor as old as the first human being and as modern as tomorrow s newspaper. Mathematical Modelling in Bio Medical sciences is an attempt to identify and describe some instances of day to day life in the language of Mathematics. In recent years, mathematical modeling has grown so vast that it found its due place almost in every walk of life, attracting the attention of even a common man. This subject constantly endeavors to widen the areas to which, techniques of mathematics can be applied for gaining a better insight and help in deepening our understanding various phenomena

2 194 that occur in nature. Real life-situations are quiet complex and we should have some insight into the situation before an attempt is made to formulate a new mathematical model. Employing proper mathematical techniques, the consequences of the model so formed can be deducted and the results compared with observations. The discrepancies between theoretical conclusions and the actual observations would suggest further improvements in the model from time to time. While widening and deepening the scope and utility of mathematical modeling in real life, one is not just restricted to the use of mathematical techniques already known. It is very interesting to note that one of the important roles of mathematician working in diverse areas of knowledge is to evolve new mathematical techniques dealing with complex situations that arise in nature and in general, in day-to-day life. There are instances of evolving new techniques to supplement the existing ones in the process of understanding the problems in such fields. Instances in real life are quiet complex and so one should have some insight into that situation before attempting the formulation of a meaningful mathematical model. When once a model is formulated, its consequences can be noticed by employing suitable mathematical techniques and the results can be compared with observations. The discrepancy between theoretical conclusions and observations may suggest further improvement in the model taking into consideration, other parameters related to the problem which were not given proper attention earlier. The modified model is to be put to test likewise. This process is to be repeated till a satisfactory model is evolved with the certain tolerable limits for the discrepancies. Mathematical Modeling essentially consists of transliterating (using Mathematical alphabets) real world problems, solving the mathematical problems and interpreting the solutions in the language of the real world. This can be shown in the following block diagram : REAL PROBLEM MATHEMATICAL PROBLEM MATHEMATICAL SOLUTION INTERPRETATION A real world problem, in all its generality can seldom be translated into a mathematical problem and even if it can be so translated, it may not be possible to solve the resulting mathematical problem. Hence, it is necessary to simplfiy or idealize or approximate the problem and at the same time can be translated and solved mathematically. In this idealization process, we try to retain all the essentail features of the problem, giivng up those features which are neither inevitable nor so relevant to the situation we are investigating. So we can modify the above block diagram on mathematical modelling as follows : REAL WORLD Idealisation and Approximations Based on the experience and understanding of the solution REAL WORLD MODEL Comparison Abstract symbolic representation based on Mathematical experience CONCLUSIONS MATHEMATICAL MODEL

3 195 If no satisfactory comparison is noted between the conclusions based on the model and the real world problems, we modify the assumptions in model-making or try for another form/structure for the mathematical model. A variety of mathematical techniques are being employed in analyzing the model equations. The mathematical and biological models complement one another in the process of understanding the reality. The mathematical model, structured on a biological model, would enable in the prediction/quantitative estimation of population strength at subsequent instants of time. In the absence of a biological model, the mathematical treatment would become more and more abstract and in general, making the whole treatment difficult to analyze in drawing meaningful conclusions. The absence of mathematical treatment makes difficult to identify the general relevance of a particular biological model. Certain principles to be followed to formulate a mathematical model and the diversed techniques can be adopted in mathematical modeling have been mentioned/illustrated exhaustively in the treatise of Kapur [ 20] on mathematical modeling. Another detailed monograph by Kapur deals exhaustively with diverse topics on mathematical modeling in biological and medical sciences [19 ]. These two references provide a vast information and the basic ideas of modeling in diverse areas of knowledge and function as the source/reference materials for the initial stages of the investigations in the areas of knowledge of bio-mathematics. Modeling may be treated as a science concerning with the interaction between mathematics and other subjects an academic discipline on some aspects of the everyday world. The interaction process can be viewed as consisting of the following phases : i) Identification of the problem in the scientific, in particular biological / medical / social setting. ii) iii) iv) Formulation of the mathematical model. Solution of the mathematical problems that arise in the study of the model selected. Developments of algorithms and associated computer programs for the relevant computations. v) Explanation and interpretation of the results in the context of the original problem and the communication of this information to all those interested. This amounts to the evaluation of the results. Mathematical models have become important tools in biological investigations and iterative procedure of information collection. If such models are properly developed and used, they can provide insight into the relations between the physical variables influencing the system under study. The resulting interplay between the experiment and the theoretical model investigated could be an essential factor in data interpretation. There are various types of mathematical modeling. Since real-life systems are complex, mathematical formulations have been developed to reproduce the experimental results irrespective of the underlying mechanisms. Such models can be extremely useful in highlighting

4 196 the performance of the biological systems, although the components of the model are not identifiable with the components and mechanisms of the real system. However, experimental result can be reproduced in such circumstances by arbitrarily adjusting the models to explore the relation among various systems. The insight obtained from studies of such models has been proved to be of immense use in understanding complex real-life systems. In general, there can be two types of mathematical models; deterministic and stochastic. In the deterministic models, the formulation of the systems depend upon the various axioms/assumptions to be considered due to the system-biology involved and these may be presented in the forms of autonomous or non-autonomous ordinary / partial (linear or nonlinear) differential or integro-differential equations or difference equations etc., The stochastic models are in general, probabilistic models leading to difference or differential-difference equations etc., 1.2 Introduction to Ecology : Ecology relates to the study of living beings (animals and plants) in relation to their habits and habitats. This discipline of knowledge is a branch of evolutionary biology purported to explain how or to what extent the living beings are regulated in nature. Allied to the problem of population regulation there are problems of species distribution-prey-predator, competition and so on. The subject of ecology can be broadly sub-divided as auto-ecology (the study of single species populations) and synecology (the study of two or more communities). Synecological studies outcome of the intensive work of several life scientists/biologists and botanists of many generations. An eco-system may be considered as a unit that includes animals, plants and the physical environment in which they live. This area of knowledge seeks to explain how many different kinds of plants and animals can live together in the same environment for many generations. Animals and plants share habitat. Sometimes they can only share for so long before some locally go extinct, but there are other circumstances when many different kinds persist coexist in a habitat indefinitely. As such, ecology may also be referred to as the study of distribution and abundance of species under habitat availing the same resources. In this connection, it is important and necessary to mention some references on specific topics of modeling in life sciences for an immediate reference. Mathematics, the queen of sciences is not properly employed as a tool in understanding phenomenon in life sciences compared to the extent it is put to frequent use in physical and engineering sciences. Several centuries have lapsed in employing mathematics to formulate biological principles. In 1202, Leonardo of Pisa (also known as Fibonacci) proposed a population model for the rabbit growth, employing simple arithmetic, to estimate the number of pairs of rabbits after each successive month having started with on pair reproduction following a particular pattern [21 ]. The credit of presenting a more serious and systematic attempt in formulating mathematical problems of life sciences goes ot Giovanni Borelli [15 ] in 1680 who discussed the mechanism of animal motion quantitatively by a geometric approach. The nineteenth century has witnessed an upsurge of interest in interdisciplinary investigation via natural philosophers-mathematicians, engineers, physicists, chemists and life scientist.

5 197 Influenced by these developments, Dr. Arey Went Worth, Thompson, in 1917 published his work on growth and form, that marks the beginning of modern theoretical biology employing mathematics as an essential tool. Since then, more and more mathematicians are getting interest in this area of knowledge. The general concept of mathematical modeling can be noted in the book written by Meyer (26 ). A detailed study on ecological species is given in the treatises of the authors such as Cushing ( 8), Freedman (9,13 ), and Paul Collinvaux (28 ).Exhaustive information on modeling of infectious and epidemic diseases was referred in a monograph written by Bailey ( 1) Modelling in immunology has been discussed by Marchuk (24 ), Modelling of the communicable diseases like gonorrhea, in the population can be noted in an exhaustive monograph written by Hethcote and Yorke ( 7) Modelling of some more epidemic diseases / helminthes infections can be referred in a lecture material of Frauenthal ( 11). Some applications of ordinary differential equations in the forms of modeling in life and social sciences have been obtained by referring the monograph written by the authors such as Braun et al (4 ), Heberman(17 ), Lucas et al,roberts et al. Significant researches in the area of theoretical ecology have been initiated in 1925 by Lotka (23 ) and in 1931 by Volterra.Since then, several mathematicians and ecologists contributed to the growth of this area of knowledge. Mathematical ecology can be broadly divided into two sub-areas; Static-ecology and Dynamic ecology, which are described in the treatises of Pielogu (29 ), May (22 ), Smith (32 ), Paul Colinvaux etc. As is already noted, ecology is a branch of evolutionary biology a science that explains how different kinds of living beings (species) can live together in the same environment for generations, sharing the same habitat, interact with each other in diverse ways. Some typical interactions between the species are given as follows. All the complex interactions within tow or more species in a certain environment may be thought of the composition of two or more of these processes. As models in any branch of science and technology, mathematical models in theoretical ecology are of great importance and utility because they both answer and raise questions related to natural phenomena. The formulations of problems in ecology are quiet complex as one should have a deeper insight into a situation before we attempt to formulate a mathematical model. It is natural that two or more species living in a common habitat interact in different ways with each other. Some possible interactions between species is shown in the following table. Interaction Neutralism Commensalisms Mutualism Definition Absence of any interaction between members of a mixed population. One member of a mixed population benefits from another member which is unaffected itself. Both members of a mixed population benefit from the presence of each other.

6 198 Syntrophism Competition Ammansalism Parasitism Predation Interaction Neutralism Commensalisms Mutualism Syntrophism Competition This is a special case of mutualism. This occur when an organism is growing in an environment that can grow only if it has a close association with second organism which grows on metabolic products extracted by the first one. A conflict for nutrients, space or some other factor which results in all members of the mixed populations growing when compared with the growth characteristic alone. One population adversely effects the growth of another population which itself being unaffected by it. One organism consumes another, often insubtle, nondelitating relationship. One organism ingests another organism and consumes it, often in a violent destructive relationship. Examples Certain plant species, bacteria, animal species which do not effect each others. Sucker Fish is found attached sharks, whales & turtles. Oyster crab is found in mantle cavity of the oyster.epiphytes & liaxas are commensal plants. Lichens which are associations of an algae with a fungus, insects & flowers, wild bees (Nomia) & Lucerne plants Photosynthetic bacterial cultures,chlorobium & cyano bacteria Honey bees & butterflies competing for honey,rabbits & deer competing for carrots, paramecium Aurelia & P. caudatum Beet & Mustard, Potato & Pumpkin, Tomato & Cucumber, Amensalism Thiobacillus & lactic acid bacteria Measles & humanbody, ox tapeworm & cattle, Peach potato aphid & Peach tree, ants& wasp species

7 199 Parasitism Predation Foxes & Rabits, Lions & Deer, Cats & Rats, Herbivores & Plants, Didinium nasutum & Paramecium caudatum, snakes & frogs. REFERENCES 1. Bailey,N.T.J.: The mathematical theory of Epidemics, Griffin, London, Bounds, J. M. and Cushing, J.M: On the behavior of solutions with hereditary terms, SIAM J.Appl.Math., Vol.30, 1976, pp Brauer, F.: Periodic solutions of some ecological models. J.Theor.Biol. Vol , pp Braun, M.: Differential Equations and their Applications-Applied Mathematical Sciences(15), Springer, New York, Braun.M, Courtney.S.C., and Donald. A.D.: Differential equations models,vol-i Springer- Verlag,Newyork, Butler, G., Freedman, H.I. and Waltman, P.: Uniformly persistent systems. Proceedings of the American Mathematical Society.Vol. 96. No.3, 1986, pp Cooke, K. L. and Yorke, J.A.: Some Equations Modeling growth process and gonorrhea epidemics, Math. Biosci., Vol.16,1973, pp Cushing, J.M.: Integro-differential equations and Delay Models in Population Dynamics, Lect. Notes in Biomathematics. Vol. 20. Springer- Verlag, Heidelberg, Erbe, L. H. and Freedman, H. I.: Modelling persistence and mutual interference among subpopulations of ecological communities. Bull. Math. Biol., Vol. 47, 1985, pp Ernst Mayr : Populations, Species and evaluation,cambridge,harvard University Press, Frauenthal,J.C. : Mathematical Modeling in Epidemiology,Springer-Verlag, Newyork Freedman, H. I.: Stability analysis of a Predator-Prey model with mutual interference and time-lags in Predator-Prey systems, Bull. Math. Biol., Vol. 41, 1979,pp Freedman, H. I.: Deterministic Mathematical Models in Population Ecology, Marcel-Decker, New York, Gauss G.F.: The struggle for existence Newyork: Hafna(1934) 15.Giovanni Borelli: De motu animalium( On animal motion ),Vol. I, Benabo, Rome, 1680.

8 Hassell, M. P. : Mutual interference between searching insect parasites, J. Anim. Ecol.,Vol. 40, 1971, pp Heberman, R. : Mathematical Models, Prentice-Hall Inc., New Jerrsey, Kapur, Heberman, R. : Mathematical Models, Prentice-Hall Inc., New Jerrsey, Kapur, J. N. : Mathematical Models in biology and Medicine, Affiliated East-West, Kapur, J. N.: Mathematical Modelling,Wiley-Eastern, Leonardo of Pisa (Fibonacci): Liberabaci(Book of Counting Board), Levin, S. A. and May, R. M. : A note on difference-delay equations, Theor. Pop. Biol., Vol.9,1976,pp Lotka, A. J.: Elements of Physical biology, Williams and Wilkins, Baltimore, Marchuk, G. I. : Mathematical Models in Immunology, optimization Software Inc., Newyork, May, R. M.: Stability and Complexity in Model Eco-systems, Princeton University Press, Princeton, Meyer, W. J.: Concepts of mathematical modeling, Mc-Grawhill, Michael Olinck: An introduction to Mathematical models in the Social and Life Sciences, 1978,Adderson-Wesley 28. Paul Colinvaux: Ecology, John Wiley and Sons Inc., New York, Pielou, E. C.: Mathematical Ecology, John Wiley and Sons, New York, Rogers, D. J. and Hassell, M. P.: General models for insect parasite and predator searching behavior : Interference.J.anim.Ecol.,Vol.43, 1974,pp Simmons.G.F.Differential equations with applications and historic notes McGrawHill, Inc:Newyork, Smith, J. M.: Models in Ecology,Cambridge University Press,Cambridge, Svirezhev, Yu. M. and Logofet, D.O. : Stability of Biological Community, MIR, Moscow, 1983