# Math and Programming

**Wildlife Ecology: Animal Population Sampling & Estimation: Brief Introduction**

**Mohammed Ashraf**

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Ecology is not the kind of science that takes people by storm hence I am not expecting that it is just what the doctor ordered. But we at Species Ecology are pretty ‘gung ho’ about the motion and rolled up our sleeves and buckled down to do our part to ensure science bound ecological sustainability find its niche in the face of anthropogenic development across the chessboard. I am not going to beat around the bush hence one of the main purposes of reaching out to people neatly rooted into the fact that collaborative and collective actions are fundamental to reinforce the conservation pillars in which wildlife science and ecology are basic ingredients. Therefore, I am at the crossroad reaching out potential academic scholars so that collectively we could go back to the drawing board and crank out rudiments of common language (mathematics) to preserve mosaic of heterogeneous pristine ecological units from Baluchistan in Pakistan to Yosemite in California and anything in between. I like to keep the ball rolling and I am twisting arms to get scholars on board depending on what variety of fresh food (ecology) and ingredients (mathematical tools) they can bring on the table.

Lot of ecological inquires can be modeled into finding priority action measures and to predict scenarios hence for example looking into fish population (denoted with ) which can be modeled into quadratic equation to predict future population size. Here I have modeled the fish population below and solved the equation to determine the time (in days) when fish population will reach 500. This is just an example of some of the works I am pretty ‘gung ho’ about.

Therefore for fish population to reach 500 it would require 88.5 days or roughly 12 weeks. Refer to the population model curve.

Its critically important to develop an algorithm so that we can generalize quadratic model and in this example I have used Python programming language to model the equation into square-root function for the purpose of fish population prediction.

The other example I would like to draw attention to is sampling size and the determination of sampling size based on simple (or stratified) random sampling. Animal population estimation is function of two critical parameters. 1. Occupancy 2. Detectability. Here the probability of animal occupancy is one of the statistical factors that need to be taken into account before carrying out animal population survey. In other words, statistically valid survey design is at paramount importance. Generally speaking, at one given time, our chance to detect any animal depends on whether our sampling units are true representation of the population size. For example, If I ought to find out the Florida panther (subspecies of mountain lion) population in any given area of 100 sq km, my primary objective is design a survey unit based on proportional and true representation of all the units. It simply means, if we conduct animal detection survey of roughly 2 sq km that I can cover in a day on foot, then I need to ensure that each 2 sq km I choose is a true representation or have the equal probability of selection among my fifty 2-sq-km panther survey unit (50 times 2 equates our total 100 sq km). Surely 100 sq km is a relatively big area for me to survey on my own but I still need to conduct the survey hence I could survey 40 sq km out of my total 100 sq km potential survey area to estimate the Florida panther population. Since my survey units are all 2 sq km each, hence 40 sq km translates to total 20 blocks which I would need to randomly select out of total 50 blocks or 100 sq km. Here I have used R programming language to write up a function that will allow me to randomly select 20 blocks out of 100 or any numbers of blocks depending on how many blocks you wish to include into your survey as random sample. I have provided below the block matrix of 100 in which twenty 2-sq-km block are randomly selected. Blocks are highlighted for the purpose of clarity. Also note, this is not an algebraic matrix that you may often utilize to solve problems in linear algebra. This is just a block sample that some folks may simply present in a grid block as oppose to block matrix.

These 20 blocks are true representation of my sampling survey area and if survey is carried out in these blocks, even if I can detect only few panther from my survey unit, the sampling size would still be true representation of the population size hence it would allow me to estimate the detection probability of panther population from the entire 100 sq.km. ecological unit. As an example, if I manage to detect only 3 panther out of my 40 sq km survey unit and my detection probability stands out 0.1, it then translates to undetected panther population size of 30 which in turn give me the total population size of 33 in that particular Everglade mangrove habitat.

This is just a short article providing some very brief understanding with regards to ecological study focusing animal population survey design and estimation techniques. The article deduced hard core mathematical rigor and modeling techniques to produce succinct easy-to-understand ecological piece without compromising the statistical rigor. The primary objective of this short essay is to publicize these rather mathematically challenging models in simplistic coherent format so that average people from non scientific background yet avid conservationist can able to digest the rudiment of population ecology and its conservation implications.

*This draft is prepared in – the brainchild of Donald Knuth, developed by American Mathematical Society (AMS) and created by George Gratzar from University of Manitoba Department of Mathematics. I have also utilized both Python and R Programming Language to develop quadratic population model and for designing random sampling matrix. No commercial software under capitalistic market share is used in preparation of this draft. UNIX variant GNU-Debian Linux is used throughout as core to run all software packages.*

**Properties, Equations and Formulas of Basic Algebra, Trigonometry, Calculus & Geometry**

Consumption of natural resources can be modeled by where is the annual consumption of natural resource in year and consumption is growing exponentially at growth rate

DeMoivre’s Theorem:

The distance d between any two points and is given by

The standard form of the equation of a circle with center and radius is

Slope–Intercept Form of an Equation or where the slope of the line is m and y-intercept of the line is

The difference quotient for a function is the ratio

The solution of where can be found using the quadratic formula

Polynomial Function

Sum-to-Product Trigonometric Identities

Half Angel Trigonometric Identities

Angle between two vectors

Standard Equation of Hyperbola with Center at Transverse Axis Horizontal

For any binomial and any natural number ,

The Quotient Rule