An Analytical Study of Applied Mathematical Model towards Several Ecological Systems

This paper focuses on those parts of 20th century applied mathematics that have entered into the toolkit of ecology. Historically, the recent trend of applying mathematics to ecology is discussed. It is proposed that this new development can be seen in the extension of its application as the natural development of applied mathematics. There is no assumption that the mathematical concepts and methods employed would be significantly different from those used in mainstream applied mathematics. In terms of estimates for reference to empirical data, conventional concepts and methods of statistical physics can be successfully applied. A case study on the history of statistics and operational research discusses several ways to integrate the development of applied mathematics of the twentieth century into an ecological context.


Introduction
The applied mathematics to issues that arise in different areas, such as science, engineering, or other diverse sectors and/or the development of new or improved methods for the complexities of new problem areas are part of Applied Mathematics [1]. They consider applied mathematics as the application of mathematics to real problems in order to clarify phenomena and forecast new phenomena, which are still ignored. The emphasis is therefore on science, for example, the growth of fresh ways in order to address the issues of up-and-coming technologies as well as the actual world. The questions arise from different areas such as research of physics and biology, electronics and social sciences. The solutions require knowledge of different scientific fields, such as analytical methods, differential equations, and stochastic. Very often, the faculty and students interact directly with experimentalists to see the results of their research. The use of mathematical methods is applied to various fields such as physics, engineering, medicine, biology, business, CS with organizations. Applied Mathematics, then, may be described as an amalgamation of information with the mathematical science. The term "Applied Mathematics" frequently defines the skill of mathematical problems through the creation as well as the learning attribute to the prototypes of mathematics. The mathematical theories have been derived from the practical applications from the past, which were studied in core mathematics and the central ideas are discussed in their own context. The study in applied mathematics is thus closely linked to pure mathematics practice [2].
Mathematics is practically applied all over the world today and this number of activities cannot be found in one department. The emphasis of Waterloo's Department of Applied Mathematics is the application of mathematics to science, engineering and medicine problems. The undergraduate programs are based on courses with a sea mathematical and computer background and a variety of courses in application areas. These fields of application are quite diverse and serve the research interests of depart-mental members.
For example, complex behavior and its motions are fundamental to our very existence on this planet. Mind the oceans, the atmosphere, the earth's crust and fossil fuels in the deep. The fluid flow, including the waves travelling through it, is represented by the topic of fluid dynamics. This fascinating topic will be covered by a series of two courses. You might want to know of chaotic dynamics, the volatile nature of non -linear systems, or how engineers design control systems that are used in a variety of fields like robotics. In each of these fields, we offer senior courses [3].
Have you ever heard about Einstein's relativity theory, one of the twentieth century advances in physics? Alternatively, how does quantum mechanics-very small -scale sciencevary from traditional mechanics in everyday life? If so, a course in general relativity or quantum mechanics can attract you. Of course, all these applications require a strong mathematical background, starting with Calculus and linear algebra, developed in the first threeyears.

Historical Background
Mathematics is a topic that requires a great deal of reflection, logic, comprehension and resolution. Typically, mathematics students are considered smart people because of their skills. Even, we have all studied and learned maths every year during school, sometimes even cursing the complex calculations and procedures wondering what they have to do with our lives. Good news for you is that we have explored how maths and the real world are connected. Come, read on! Calculus is a concept used to evaluate the minimum difference in a number. It is divided into differentiation and integration. You might have studied the concept deeper in your school or college. However, have you ever wondered what it actually means and what it is used for? An architect uses calculus to compute the resources required to make curved structures such as domes and arches. Electrical engineers use it to estimate the measurement of electric cables between places that are miles apart. Biologists use it to calculate the rate of growth in microbes [4].
Algebra "I know everybody's exes, other than algebra's 'x'." If that was one of your statements while you were wondering what that 'x' had to do in your life, then here's the answer to that burning question [5]. The report of DNA and genes can be enhance described in form of equations. Their features can be simply deduced with the support of x and ychromosomes. In economics, x and y are expended to determine the supply and demand of goods. In CG (computer graphics), x and y are expended as organizes to design objects and object movements. The famous game Mario was designed using 3D animation on CG. Trigonometry Sine, cos, tan and cosec. What were all those names anyway? Each one denotes an angular measurement. Trigonometric angles are used in the construction of marine slopes. They are used to spot a position for map reading. They are used to determine the depth of algae underwater and sonarsystems.
Logarithms we all can write numbers up to 10 million. Beyond that, can you read the numbers at one go without actually re-reading and counting the zeros? Logarithms simplify complicated numbers through exponents and coefficients. Log tables help you to refer the corresponding numbers and make calculations less laborious. It is largely used in computing memory spaces and in computer languages. Probability is a concept we use in everyday life. 'Should I take a car or bus to office?' is something we ask ourselves on a regular basis, and we make the decision using probability. Probability tells you the chances of a specific occurrence of an event. The meticulous calculations can be used in determining big events such as the probability of you winning a jackpot or a lottery! Statistics, like probability, is a more familiar concept and like probability, we use statistics in our everyday lives. Population census is an epitome of statistics. To get the demographics of a sample size, statistics plays an important role with its various functions, like while determining average and mean estimates.
Matrices Matrix is an array of elements processed according to specific needs. In robotic systems, matrices are used to control the movement of robots. Data structures and databases make use of this concept for an organized algorithm. Mensuration is nothing but measurement of different units. From buying groceries to checking our weight, we use mensuration day in and day out. From determining the body temperature using a thermometer and a barometer to determining the weather, mensuration plays an integral role.
Mathematics can be summed up as a study of concepts like quantity (numbers), shape, area as well as variation. As far as the proper definition and scope of mathematics is concerned, there is multiple varying approaches among the mathematicians as well asphilosophers.
The professionals, who are working in the study of Mathematics, are searching in similarities and making new conjectures. Through mathematical proof, mathematicians solve validity or fallacy of conjectures. Mathematical reasoning may suggest concepts or intuitions into the nature, where mathematical models are good model of real phenomena. The mathematics emerged from counting, estimation, analysis, and organized learning of the shape as well as movement of substantial objects, using intuition and logic. For as far as written records exist, practical mathematics was human activity. The analysis required to resolve issues related to mathematics might last from years to decades or up to centuries of ongoing research.
Precisely in the Euclid's Elements, initially in Greek mathematics, had some hard logics. Pioneers like Giuseppe Peano (1858-1932), David Hilbert (1862-1943) as well as other mathematicians, on axiomatic systems, towards the last phase of the 19th century, mathematical study turned out to be normative to interpret the facts by rigorously de-ducting axioms and definitions correctly chosen. Mathematics progressed relatively slowly until the Renaissance, when mathematical advancements, combined with new scientific findings, caused the rate of mathematical discovery to accelerate quickly and continuously.
Galileo Galilei (1564-1642) said that the world could not be understood until the characters in which it is written are known and recognizable. It is written in mathematical language with its characters are triangles, circles plus other geometric shapes, where it is difficult for man to understand anything. With 1809-1880, Benjamin Peircedescribed mathematics as the science that brings the significant outcomes. In the words of David Hilbert, assuming mathematics as: "We are not speaking here of arbitrariness in any sense‖. Mathematics is not like a game where functions remain arbitrarily determined by rules. Rather, this is a logical framework with an intrinsic requirement and cannot be otherwise. "We are uncertain about the laws of mathematics and they do not apply to truth as far as they are positive," Albert Einstein (1879-1955) said. French mathematician Claire Voisin narrates, -There is creative drive in mathematics; it's all about movement trying to express itself." Across the globe, Mathematics is considered a key tool and used in various sectors like engineering, social science, economics, healthcare, environmental science, engineering, and science. The domain of mathematics is associated with the implementation of mathematical information across the wide array of applied mathematics, stimulates and uses newer mathematic developments that have resulted into the growing sectors like game theory viz-a-viz statistics. Those who work in the domain of Mathematicians are experts in pure mathematics, or mathematics for individual interests. It is not specified on which lines pure mathematics can be distinguished from applied mathematics with the conceptual implementations are often found started in the form of puremathematics.

Mathematics and applied Mathematics Evolution
The easiest way to think about it is to mathematics simply by itself, while applied mathematics is mathematics with a practical application. It is not that simple, however, because even the most complicated mathematics can have unforeseen applications. The mathematics division called "number theory," for instance, was once thought to be one of the most "useless," but is now critical in computer encryption systems.

Evolution Conclusion
History of maths may be considered as a continuously growing number of abstractions or, extension of the subject [6]. Initial interpretation that various animals retain was certainly the one of numbers: the concept that a group of two oranges with a group of two apples possesses something mutual in between (for example), as the sum of its parts [7].
Adding to it, the knowledge of the way substantial materials is measured, old people also have learned the way of calculating conceptual numbers, such as days ‗time, periods, years. It was not until 3000 BC that abstract mathematics appeared, the times in which the Egyptians as well as the Babylonians initiated using arithmetic for taxation as well as other financial calculations, to construct as well as in astronomy with algebra and geometry.
Mathematics used most early in trade, the measurement of land, the weaving and painting, with the recording of the time.
Simple algebra (multiplication, subtraction, addition and division) is used in Babylonian mathematics, for the first time. Various Pre-date numeracy and numerical systems were available in abundance and distinct, and Egyptians made initially known numerals in texts like Rhind Mathematical Papyrus in the Middle Kingdom. The ancient Greeks started with Greek mathematics an organized learning of mathematics itself around 600-300 BC. Since then, maths has been widely expanded and the relationship around mathematics and science was fruitful to both. Yet, scientific developments continue. According to Mikhail B. Sevryuk in the American Mathematical Society Bulletin in his January 2006 issue, from 1940 database (the first year in which MR has operated) the quantity of books and papers has been over 1.9 million with 75,000 plus objects get in the databank everyyear.

Applied Mathematics in Ecology
Historically, biology mathematical models have primarily been used for contextual interpretations of nature phenomena [8] [9] [10]. The attempts in order to utilize rivalry prototypes to describe genre diversity (Diamond and Case, 1986) [11] are a textbook example of this approach. Simple models of competition have shown that under the right conditions' organisms using the same resource will co -exist (Begon et al., 1996) [12]. Nevertheless, this approach observation results in considerable argument on overall question of clash influences substantial societies.
That sort of a normal argument on growth is sometimes of very few benefits considering the resource management issues. As a result, modelling efforts in many applications, particularly pesticide control, been rejected several times simplistic mathematic prototypes in favoring big imitation proto-types (Onsted 1988). A process of Simulation proto-types possesses multiple factors with static variables that needs years for building with sometimes detailed as pages may be taken. These prototypes are opposite to the simplistic prototypes that are being considered to utilize in academic research in at-tempting to substitute comprehensibility, considering ecological factualism.
Values of scientific ecologists, however, have in-creased in the last few decades and the literature that has arisen has started to propose alternative uses for basic mathematical models (Hilborn and Mangel, 1996) [13]. Over addition, the normal equations of mathematics may get utilized in predictive assumptions, as well as linear prototypes utilized in traditional computational arm of mathematics. To fact, research of current times indicates the numerous collections of ecological information does not clarify complex models mathematically. While design may seem sophisticated, actual information cannot show additionally advanced prototypes are represented superior as compared to simple proto-types (Hilborn and Mangel, 1996). Though it is due to the environment is actually normal or the results are so loud, because of the various actual reasons is meaningless. Reality seems as if one need gainful countable existence specifications, we usually need less than tenfactors.
The present assignment on the ecological modelling also illustrates relationships between hypothesis and information as a statistical hypothesis about nature and the use of mathematical models. Consequently, models once seen to be of just scholarly intrigue can also become valuable in the administration of pests. In making this particular point, I mustexamine my own analysis of a forest vector virus disease, the gypsy moth Lymantriadispar.
Environmental models of bug infections started with the basic Anderson and May (1981) model [14], starting with the human plague model and rising crawlies and microbes populace elements. Anderson and May have utilized the sample for showing that pathogens may influence forest insect dynamics that can lead to significant outbreaks such as the larch bulge, Zeiraphera diniana. More work on these with distinct crawlies have shown initial single -variable accounts of the population of forest in-sects are most certainly incomplete, and secondly, pathogenic are not always important players in forest insect population dynamics (Hunter and Dwyer, 1998) [15]. Nonetheless, although the initial generalization is too broad, the Anderson and May models were useful for understanding the diseases of insects.
In addition, the Anderson and May prototype thought of the horizontal transmission rate ofthe virus enhances linearly alongside the microbes number. That hypothesis was a valuable quantitative assumption, although data indicates that it is often inaccurate. For example, GM Virus transmission data neglect a linear sample yet are unable to neglect a nonlinear prototype (Dwyer et al., 1997) [16]. Nonetheless, more experiments have proposed this non -linearity due to the variation of the vulnerability of host insects to the virus and a model that makes this heterogeneity may forecast precisely the timeframe with severity of viral epidemics or epizootic diseases as organically growing GMV populace. Shockingly, only four parameters are required for the resulting model [17].
While this definition originated from attempts to address normal research questions, it has tended to be applied in practice. e.g., attempts used to get evolve it with distinct microbes genetically. The issue of ecological issues is therefore, "Can engineered virus strains overtake wild strains and change the ecological balance between the host and the pathogen?" Since the prototype is able to anticipate the epidemics by test transmission information, that may be utilized before such strains are released to assess the risk of adding engineered strains " (Dwyer et al. in press). The Precursor task anticipated as it is not likely, which, minimum one deletion mutant of the GM Virus is a better contender and is progressing than utilize the model to evaluate commercially produced strains of the trichoplusian pest nuclear polyhedrosis virus. More concretely, gypsum moth colonies continue to be very patchy, making the detection of species [18] that must be managed and that are likely to collapse a major challenge for managers. Since this prototype may be utilized to forecast the populace, which is susceptible against relentless viral disease, which may help identify which communities will supposed to fall. Such experiments show other benefits related to the usage of general mathematical prototypes. Initially, costs for modelling, simulating and analyzing models are extremely low compared to the logistic costs of performing experiments and gathering data. Secondly, models would help us to extrapolate the complexities of populations between smallscale field and laboratory interventions. For example, the GM-Virus prototype utilizes the recurrence as well as density of infection by GM Virus at the starting phase, in the field as an input as well as calculates the spread of this epidemic with mortality rates of miniature-size laboratory with fieldstudies. Furthermore, this prototype may anticipate the frequency with severity of disease by the virus in natural GM populace on 3-10 ha in a wide variety of densities with great precision (Dwyer et al., 1997; Dwyeret al. in press). The capability in order to extrapolate in the parameters helps a prototype for anticipation of results of bigreleases of morphed mi-crobes by prediction prior to these discharges. Thirdly, by focusing at general expression of that seem to be superficially hard organic approach, simple mathematic prototypes offer valuable studies. In addition, the development of the model GMV, that requires just 4 factors, shows certain general events used to be normal in contrary to which seemed in the starting. Theoretically, such benefits by basic prototypes would be much better at the control of pesticides. That is because environmental science can also be presented qualitatively whereas pesticide research is essentially cost-effective and thus necessarily quantitative. I will also suggest that an overemphasis on complicated modelling methods is due to the unusual use of statistical models in the administration of crawlies. As well as being easier for getting it clear, these prototypes are significantly more complex than the simple models that are recommended by me in this piece of content. Consequently, sophisticated simulation ones used to be lesser likely for being evaluated with lesser chances to be rejected in favour of simpler models. In the end, ideally simple mathematical models are as effective in pesticide management as they are in eco-logical research [19].

Conclusion
This paper describes the concept of Mathematics with its historical background and focus on the applied mathematics evolution and modelling specially in the field of ecology. Now a day, the concept "applied mathematics" is utilized more generally and in a broader understanding. This comprises of the traditional field aforementioned and different domain which, being progressively significant with implementations.