Mathematics AA SL

Sample Internal Assessment

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Table of content

Rationale

Aim

Research question

Background information

One - horned rhino population in india

Predation in india

Predator - prey model

Runge kutta method

Data collection

Collected data

Process of calculation

Predator prey modelling

Solution of differential equation using rk method

Analysis

Analysis

Conclusion

Reflection

Strenght

Weakness

Future scope

Bibliography

“The purpose of our lives is to be happy.” The famous words by Dalai Lama have not only inspired millions by simplifying the objective of a life, but also aspired a thousand lives to come out of mental illness. However, as interpretation is always based on one’s perspective, if there could be a positive aspect of a statement, there could be a negative one as well. Many people misinterpret the term ‘happiness’ with greed, lust, and exorbitant ambition. I firmly believe that, if one’s happiness is the reason behind anyone’s tear, then that is indeed not the purpose of life and that happiness is for a limited time span.

IB has always inspired me to be principled. It has taught me to evolve integrity and honesty in life. It has enabled me to differentiate between earning money in an honest way from a way that threatens others life. Poaching is one of such acts which is increasing, not only in India but also across the globe at an exponential rate. Despite of the strict laws that are implemented by the Government and each Forest Departmental authority, there is still some lack in co-ordination which should be an alarming call to one and all who loves this ecosystem. Poaching will initiate the imbalance in the population of each biotic component and hence, the environment which will indirectly affect human lives and abiotic world in many ways.

A statement from one of my favorite movies – “With great power, comes great responsibilities” as pushed me to use the knowledge that I have received so far to explore a process that could save the life of innocent wild animals from the heinous means of the poachers. As studied in Topic 4 of the curriculum, I have learnt modelling of data to predict a competitive relationship between any two species. It triggered a question in my mind. Can I use Predator Prey model to compute a relationship between an animal and poachers? Initially, I was confused as hunters are not wild creatures. Thus, I started my research on poaching of One – Horned Rhino’s as they are killed by the hunters for their horn which are further used to make artifacts. I read a few research journals and got a clear idea about the model and hence come to a conclusion that hunters can be considered as the other species in the model as Predator Prey Model deals with increase or decrease in population at the time of the interaction between the two species only. However, with further research, I found that the interaction or the increase or decrease in population of one species in presence of the other was represented as differential equation. It has again claimed a few questions in my mind. Are the differential equations of predator prey model be solved using the methods thought in curriculum? What is the result of solving the differential equations of population rate? As the differential equations involve modelling of data, I found that solving the equations in an analytical way will be more beneficial and will serve the purpose also. Then I found a way of solving differential equations as mentioned in our guide as – Runge Kutta Method. After watching a few tutorials, I came to the research question which will not only help the forest departmental authorities to take strict action or preventive measures against poaching of wild animals but also save the innocent lives in the wild.

The prime objective of this exploration is to prepare a Predator Prey Model to analyze the relationship between One – Horned Rhino and Poachers in three different wildlife sanctuaries and National Parks and deduce a relationship between their population so that the Government can take strict action to prevent poaching in the wild.

How does the population of One Horned Rhino in India is affected by the action of poaching, determined using Predator Prey Model, applying Runge Kutta Method of Solving Differential Equation?

Patrick Jones

1.2M+ YouTube subscribers, 30+ years' experience, Ivy League visiting faculty!

One – Horned Rhino or popularly known as Greater One – Horned Rhinoceros is mostly found in the state of Assam, India. More than 90% of the countries Greater One – Horned Rhinoceros population resides in Kaziranga National Park. Each rhino lives about 35 years to 45 years. Their horn ranges between 8 inches to 25 inches in length. It is often used to make artifacts. The skin of the rhinos is also useful in various commercial purposes. However, they are usually taken from a dead rhinoceros. It is currently at a conservation status of Vulnerable in the Red Data Book.

India is one of the countries which leads in the rate of poaching of wild animals. Tiger, Leopard, Rhinoceros, Elephant are the most affected wild animals. Moreover, in the pandemic situations, due to lack of administration, poaching has been doubled in the country. Poaching of Greater One Horned Rhinos is one of the most common cases of poaching that has been observed last year. They are killed (or hunted) for their horn and skin. As rhinos are slow moving animals, they are comparatively easy targets for the poachers. Furthermore, their horn is sold at very high rates in the illegal markets and their skin is collected by some collectors at a very high price.

One of the widely used ecological and mathematical model, enables us to determine the extent of competitive relationship between two species is the Predator Prey Model. Competition signifies a relationship where one species is benefitted at the cost other’s life. As a result, population of one species increases and the other species decreases. Interaction between two species leading to the increase or decrease in population of each of the species can only be determined using differential calculus.

Predator Prey Model comprises two variables. Each variable represents each organism. Let, *x* represents the population of One – Horned Rhinoceros, and *y* represents the population of poachers. The variation in population with respect to time 't' is represented using a differential equation.

\(\frac{dx}{dt}= - ax + bxy\) .........(equation - 1)

\(\frac{dx}{dt}= - ax - bxy\) .........(equation - 2)

\(\frac{dx}{dt}= - ax + dxn - bxy\) .........(equation - 3)

\(\frac{dx}{dt}= - ax - dxn - bxy\) .........(equation - 4)

In the above four equation (1, 2, 3, 4), the variables denoting the physical parameters are shown below:

\(\frac{dx}{dt}\)= Rate of change of population of one horned rhino with respect to time 't'

*a* = coefficient of impact of environmental factors on the population of one horned rhino

*b* = coefficient of impact of poaching on population of one horned rhino

*d* = coefficient of impact on population of one horned rhino by other rhinos of same species

*x* = population of one horned rhino

*y* = population of poachers

\(\frac{dy}{dt}= - ay + bxy\) .........(equation - 5)

\(\frac{dx}{dt}= - ay - bxy\) .........(equation - 6)

\(\frac{dy}{dt}= - ay + dyn - bxy\) .........(equation - 7)

\(\frac{dy}{dt}= - ay - dyn - bxy\).........(equation - 8)

In the above four equation (5, 6, 7, 8), the variables denoting the physical parameters are shown below

\(\frac{dy}{dt}\)= Rate of change of population of poachers with respect to time 't'

*a* = coefficient of impact of environmental factors on the population of poachers

*b* = coefficient of impact of attack of rhinos on poachers

*d* = coefficient of impact on population of poachers by other poachers

*x* = population of one horned rhino

y = population of poachers

Runge Kutta method is a method of numerically integrating ordinary differential equations by using a trial step at the midpoint of an interval to cancel out lower-order error terms. The fourth-order formula is shown below:

*k*_{1} = *h* × *f* (*x* * _{n}*,

\(k_2 = h × f \biggl(x_n+\frac{1}{2}h, \, y_{n}+\frac{1}{2}k_{1}\biggl{)}\)

\(k_3 = h × f\biggl(x_n+\frac{1}{2}h, \, y_{n}+\frac{1}{2}k_{2}\biggl{)}\)

\(k_4 = h × f\biggl(x_n+h, \, y_{n}+k_{3}\biggl{)}\)

\(y_n+1 = yn +\frac{1}{6}\biggl[k_{1}+2k_{2}+2k_{3}+k_{4}\biggl{]}\)

In the above formula, the variables denoting the physical parameters are shown below:

*k*_{1}, *k*_{2}, *k*_{3}, *k*_{4} = constant

*h* = step length

*f*(*x _{n}*,

Patrick Jones

1.2M+ YouTube subscribers, 30+ years' experience, Ivy League visiting faculty!

In this exploration, three different forests (two national park and one wildlife sanctuary) have been chosen where the population of rhinoceros are mostly found in India. The rate of variation of population of One – Horned Rhinoceros as well as poachers in the above three forests are shown. The data has been collected from the official webpages of each of the forest departments and some research journals.

- Kaziranga National Park:

\(\frac{dx}{dt} = - 0. 008x - 0. 002xy\) ………(equation - 9)

\(\frac{dy}{dt} = - 0. 004y + 0. 004xy\)………(equation - 10)

x_{2020 }= 2413

y_{2020} = 1243

- Jaldapara National Park:

\(\frac{dx}{dt}= - 0. 9x - 0. 0009x2 - 0. 012xy\)………(equation - 13)

\(\frac{dy}{dt}= - 0. 049y + 0. 007xy\)………(equation - 14)

x_{2020} = 231

y_{2020} = 28

- Gorumara National Park:

\(\frac{dx}{dt}= 0. 1x - 0. 0008x2 - 0. 004xy\)………(equation - 17)

\(\frac{dy}{dt}= 0. 2y - 0. 004y2 - 0. 001xy\)………(equation - 18)

x_{2020} = 52

y_{2020} = 87

In the above collected data, x_{2020} represents the population of One – Horned Rhino in 2020 and y_{2020 } represents the population of poachers in 2020. The step length of each of the forest department has been assigned to *h* = 0.1

Patrick Jones

1.2M+ YouTube subscribers, 30+ years' experience, Ivy League visiting faculty!

**Case 1: Kaziranga National Park**

\(\frac{dx}{dt}= - 0. 008x - 0. 002xy\)………(equation - 9)

\(\frac{dy}{dt}= - 0. 004y + 0. 004xy\)………(equation - 10)

From equation (9):

**\(\frac{dx}{dt}= 0\)**

=> - 0. 008*x* - 0. 002*xy* = 0

=> 0. 002x - 4 - *y* = 0………(equation - 11)

From equation (10):

**\(\frac{dy}{dt}= 0\)**

=> - 0. 004*y* + 0. 004*xy* = 0

=> 0. 004*y* -1+ *x* = 0……………(equation - 12)

From the above two equations (11) and (12), we can write four solution sets, such as:

Set 1:

0. 002*x* = 0

=> *x* = 0

0. 004*y* = 0

=> *y* = 0

**(0, 0) represents zero population.**

Set 2:

(- 4 - *y*) = 0

=> *y* = - 4

(- 1 + *x*) = 0

=> *x* = 1

**(1, - 4) represents stable population.**

**Real Life Interpretation:**

In the Predator Prey Model of Kaziranga National Park, two different cases have been obtained. First of all, if the population of both the predator and the prey are removed from the ecosystem, then there will not be any growth or decline of either of the two populations in the national park. It will be a stable relationship. On the other case, it has been observed that if the present situation continues then eventually, the poacher’s population will decline and reach a stable equilibrium state with a population of rhinos only in the wild. In the process, the population of rhinos will decline initially; however, once the population of poacher becomes extinct, then the population of rhinos will increase again as the population of rhinos is positive at the time of extinction of poacher population.

**Case 2 - Jaldapara Wildlife Sanctuary**

\(\frac{dx}{dt}= - 0. 9x - 0. 0009x^2 - 0. 012xy\)………(equation - 13)

\(\frac{dy}{dt}= - 0. 049y + 0. 007xy\) ………(equation - 14)

From equation (13), we can write,

**\(\frac{dx}{dt}= 0\)**

=> - 0. 9x - 0. 0009*x*^{2} - 0. 012*xy* = 0

=> 0. 012*x* (- 75 - 0. 075*x* - *y*) = 0………(equation - 15)

From equation (14), we can write,

**\(\frac{dy}{dt}= 0\)**

=> - 0. 049*y* + 0. 007*xy* = 0

=> 0. 007*y* (- 7 + *x*) = 0………(equation - 16)

From the above two equations (15) and (16) -

Set 1 -

0. 012*x* = 0

=> *x* = 0

0. 007*y* = 0

=> *y *= 0

**(0, 0) represents zero population.**

Set 2 -

( - 7 + *x*) = 0

=> *x* = 7

( - 75 - 0. 075*x* - *y*) = 0

=> - 75 - 0. 075 × 7 - *y* = 0

=> *y* = - 75 - 0. 525

=> *y* = - 75. 525

**(7, -75.525) represents stable population.**

**Real Life Interpretation:**

In the Predator Prey Model of Jaldapara National Park, two different cases have been obtained. First of all, if the population of both the predator and the prey are removed from the ecosystem, then there will not be any growth or decline of either of the two populations in the national park. It will be a stable relationship.

On the other case, it has been observed that if the present situation continues then eventually, the poacher’s population will decline and reach a stable equilibrium state with a population of rhinos only in the wild. In the process, the population of rhinos will decline initially; however, once the population of poacher becomes extinct, then the population of rhinos will increase again as the population of rhinos is positive at the time of extinction of poacher population.

**Case 3 - Gorumara Wildlife Sanctuary -**

\(\frac{dx}{dt}= 0. 1x - 0. 0008x^2 - 0. 004xy\) ………(equation - 17)

\(\frac{dy}{dt}= 0. 2y - 0. 004y^2 - 0. 001xy\) ………(equation - 18)

From equation (17) -

\(\frac{dx}{dt}= 0\)

=> 0. 004*x*(25 - 2*x* - *y*) = 0………(equation - 19)

From equation (18) -

\(\frac{dy}{dt}= 0\)

=> 0. 001*y*(200 - 4*y* - *x*) = 0 ………(equation - 20)

From the above two equations (19) and (20) -

Set 1 -

0. 004 *x* = 0

=> *x* = 0

0. 001 *y* = 0

=> *y* = 0

**(0,0) represents zero population.**

Set 2 -

(25 - 2*x* - *y*) = 0

=> *y* = 2*x* - 25………(equation - 21)

(200 - 4*y* - *x*) = 0

=> 200 - 4 (2*x* - 25) - *x* = 0

=> 200 - 8*x* + 100 - *x* = 0

=> - 9*x* + 300 = 0

=> 9*x* = 300

\(=> x =\frac{300}{9}= 33. 33\)

From equation (21) -

*y* = 2*x* - 25

=> *y* = 2 × 33. 33 - 25

=> *y* = 66. 66 - 25

=> *y* = 41. 66

**(33.33,41.66) represents stable population.**

Set 3 -

0. 004*x* = 0

=> *x* = 0

(200 - 4*y* - *x*) = 0

=> (200 - 4*y* - 0) = 0

=> 4*y* = 200

=> *y* = 50

(0,50) represents Zero X population.

Set 4 -

0. 001*y* = 0

=> *y* = 0

(25 - 2*x* - *y*) = 0

=> (25 - 2*x* - 0) = 0

=> 2*x* = 25

=> *x* = 12. 5

**(12.5,0) represents Zero Y population.**

**Real Life Interpretation:**

In the Predator Prey Model of Gorumara Wildlife Sanctuary, four different cases have been obtained. Firstly, if the population of both the predator and the prey are removed from the ecosystem, then there will not be any growth or decline of either of the two populations in the wildlife sanctuary. It will be a stable relationship.

Secondly, it has been observed that if the present situation continues then eventually, both the populations will decline and reach a stable equilibrium state with both the species present in the wild. In the process, the population of rhinos will be invariably affected and ultimately, the poachers will get an upper hand over the population of rhinos. This is because, at equilibrium state, the population of poacher is more than that of rhinos.

Thirdly, if the population of only the prey (rhinos) is removed from the ecosystem, then there will be a significant increase in the population of poachers which will sustain its growth rate.

Fourthly, if the population of only the predator (poachers) is removed from the ecosystem, then there will be a significant increase in the population of rhinos which will sustain its growth rate.

**Case 1 - Kaziranga National Park -**

\(\frac{dx}{dt}=f(x, y) = - 0. 008x - 0. 002xy\)………(equation - 9)

\(\frac{dy}{dt}=f(x, y) = - 0. 004y + 0. 004xy\)………(equation - 10)

For calculation of population of One – Horned Rhino in 2021:

\(k1 = h × f\biggl(x_{2020}, y_{2020}\biggl)\)

=> k_{1} = 0. 1 × [ - 0. 008 × 2413 - 0. 002 × 2413 × 1243]

= 0. 1 × [ - 19. 304 - 5998. 71z8] = 0. 1 × - 6018. 022 = - 601. 8022

\(k2 = h × f\biggl(x_{2020}+\frac{1}{2}h, y_{2020}+\frac{1}{2}k_{1}\biggl{)}\)

\(=> k2 = 0. 1 × f\biggl(2413 +\frac{1}{2}×0.1, 1246 +\frac{1}{2}×-\,601.\,8022\biggl) \)

= 0. 1 × f (2413. 05, 942. 0989) = 0. 1 × [ - 0. 008 × 2413. 05 - 0. 002 × 2413. 05 × 942. 0989]

= 0. 1× [ - 19. 3044 - 4546. 663501] = 0. 1 × - 4565. 967901 = - 456. 5967901

\(k3 = 0. 1 × f (\biggl(x_{2020}+\frac{1}{2}h, y_{2020}+\frac{1}{2}k_{2}\biggl)\)

\(k3 = 0. 1 × f \biggl(2413+\frac{1}{2}×0. 1,\, 1243+\frac{1}{2}×-\,456.\,5967901\biggl)\)

k_{3} = 0. 1 × f (2413. 05, 1014. 701605) = 0. 1 × [ - 0. 008 × 2413. 05 - 0. 002 × 2413. 05 × 1014. 701605]

= 0. 1 × [ - 19. 3044 - 4897. 051416] = 0. 1 × - 4916. 355816 = - 491. 6355816

\(k_4 = h × f \biggl(x_{2020}+h, y_{2020}+k_{3}\biggl)\)

k_{4 }= 0. 1 × f (2413 + 0. 1, 1243 - 491. 6355816)

k_{4} = 0. 1 × f (2413. 1, 751. 3644184) = 0. 1 [ - 0. 008 × 2413. 1 - 0. 002 × 2413. 1 × 751. 3644184]

= 0. 1 × [ - 19. 3048 - 3626. 234956] = 0. 1 × - 3645. 539756 = - 364. 5539756

\(x_{2021} = x_{2020} +\frac{1}{6}\biggl[k1 + 2k2 + 2k3 + k4\biggl]\) \(\)

\(x_{2021} = 2413 +\frac{1}{6}[ - 601. 8022 + 2× ( - 456. 5967901) + 2×( - 491. 6355816) + ( - 364. 5539756)]\)

\(= 2413 +\frac{1}{6}[ - 601. 8022 - 913. 1935802 - 983. 2711632 - 364. 5539756]\)

\(= 2413 +\frac{1}{6}[ - 2862. 820919] = 2413 - 477. 1368198 = 1935. 86318\)

In Figure 2, the variation in the population of Greater One – Horned Rhino at Kaziranga National Park has been shown. The population count of 2020 is collected from sources and that of 2021 has been predicted from the Predator Prey Model. It has been observed that, there is a decrease in the number of Greater One – Horned Rhinos from 2413 in the year 2020 to 1935 (predicted) in the year 2021. It signifies that the effective decrease in population of rhinos at Kaziranga National Park in the year 2020 should be 478. This could have happened due to predation by the poachers as well as other environmental conditions.

**Case 2: Jaldapara National Park:**

\(\frac{dx}{dt}= - 0. 9x - 0. 0009x^2 - 0. 012xy\) ………(equation - 13)

**\(\frac{dy}{dt} = - 0. 049y + 0. 007xy\) ** ………(equation - 14)

For calculation of population of One – Horned Rhino in 2021:

\(k_1 = h × f\biggl(x\,_{2020},y\,_{2020}\biggl)\)

k_{1 }= h × f (49, 87)

=> k_{1} = 0. 1 × [ - 0. 9 × 231 - 0. 0009 × 231^{2} - 0. 012 × 231 × 28]

= 0. 1 [ × - 207. 9 - 48. 0249 - 77. 616] = 0. 1 × - 333. 5409 = - 33. 35409

\(k_2 = h × f \biggl(x\,_{2020}+\frac{1}{2}h, y\,_{2020}+\frac{1}{2}k_{1}\biggl)\)

= 0. 1 × 0. 1 × [ - 0. 9 × 231. 5 - 0. 0009 × 231. 5^{2} - 0. 012 × 231. 5 × 11. 322955]

= 0. 1 × [ - 208. 35 - 48. 233025 - 31. 45516899] = 0. 1 × - 288. 038194 = - 28. 8038194

\(k3 = h × f\biggl(x\,_{2020}+\frac{1}{2}h,y\,_{2020}+\frac{1}{2}k_{2}\biggl)\)

\(=> k3 = 0. 1 × f \biggl(231+\frac{1}{2}×0.\,1,28+\frac{1}{2}×-28.\,8038194\biggl) = 0. 1 × f (231. 5, 13. 5980903)\)

= 0. 1 × 0. 1 × [ - 0.9 × 231. 5 - 0. 0009 × 231. 5^{2} - 0. 012 × 231. 5 × 13. 5980903]

= 0. 1 × [ - 208. 35 - 48. 233025 - 37. 77549485] = 0. 1 × - 294. 3585199 = - 29. 43585199

\(k_4 = h × f\biggl(x\,_{2020}+h,y\,_{2020}+k_{3}\biggl)\)

k_{4 }= 0. 1 × f [231 + 0. 1, 28 + (- 29. 43585199)]

k_{4} = 0. 1 × f (231. 1, - 1. 43585199)

= 0. 1 × 0. 1 × [ - 0. 9 × 231. 1 - 0. 0009 × 231. 1^{2} - 0. 012 × 231. 1 × ( - 1. 43585199)]

= 0. 1 × [ - 207. 99 - 48. 066489 + 3. 981904739] = 0. 1 × - 252. 0745843 = - 25. 20745843

\(x_{2021} = x_{2020} +\frac{1}{6}\biggl[k_{1}+2k_{2}+2k_3+k_{4}\biggl]\)

\(x_{2021} = 231 +\frac{1}{6}[ - 33. 35409 + 2 × ( - 28. 8038194) + 2 × ( - 29. 43585199) + ( - 25. 20745843)]\)

\(= 231 + \frac{1}{6}[ - 33. 35409 - 57. 6076388 - 58. 87170398 - 25. 20745843]\)

\(= 231 +\frac{1}{6}[ - 175. 0408912]\)

= 231 - 29. 17348187

= 201. 8265181

In Figure 3, the variation in the population of Greater One – Horned Rhino at Jaldapara National Park has been shown. The population count of 2020 is collected from sources and that of 2021 has been predicted from the Predator Prey Model. It has been observed that, there is a decrease in the number of Greater One – Horned Rhinos from 231 in the year 2020 to 201 (predicted) in the year 2021. It signifies that the effective decrease in population of rhinos at Jaldapara National Park in the year 2020 should be 30. This could have happened due to predation by the poachers as well as other environmental conditions. However, similar to the predator prey model of rhinos at Jaldapara, it also shows that the population of rhino at Jaldapara is very stable.

**Case 3 - Gorumara Wildlife Sanctuary -**

\(\frac{dx}{dt}= 0. 1 x - 0. 0008x2 - 0. 004xy………(equation - 17)\)** **

\(\frac{dy}{dt}= 0. 2y - 0. 004y2 - 0. 001xy………(equation - 18)\)

For calculation of population of One – Horned Rhino in 2021 -

\(k_1 = h × f\biggl(x\,_{2020},y\,_{2020}\biggl)\)

*k*_{1 }= *h* × *f* (52, 87)

=> *k*_{1} = 0. 1 × [0. 1 × 52 - 0. 0008 × 52^{2} - 0. 004 × 52 × 87]

= 0. 1 × - 15. 052 = - 1. 5052

\(k_2 = h × f\biggl(x\,_{2020}+\frac{1}{2}h, y\,_{2020}+\frac{1}{2}k_{1}\biggl)\)

\(=> k2 = 0. 1 × f\biggl(52 +\frac{1}{2}× 0. 1, 87 +\frac{1}{2} × - 1. 5052\biggl) = 0. 1 × f (52. 05, 86. 2474)\)

= 0. 1 × [0. 1 × 52. 05 - 0. 0008 × 52. 05^{2} - 0. 004 × 52. 05 × 86. 2474]

= 0. 1 × - 14. 919 = - 1. 4191

\(k_3 = h × f\biggl(x\,_{2020}+\frac{1}{2}h, y\,_{2020}+\frac{1}{2}k_{2}\biggl)\)

\(=> k_3 = 0. 1 × f \biggl(52 +\frac{1}{2}× 0. 1, 87 +\frac{1}{2}× - 1. 4191\biggl) = 0. 1 × f (52. 05, 86. 29045)\)

*x* = 0. 1 × [ 0. 1 × 52. 05 - 0. 0008 × 52. 05^{2} - 0. 004 × 52. 05 × 86. 29045]

= 0. 1 × - 14. 928 = - 1. 4928

\(k_4 = h × f\biggl(x\,_{2020}+h,y\,_{2020}+k_3\biggl)\)

*k*_{4 }= 0. 1 × f [52 + 0. 1, 87 + (- 1. 4928)] = *k*_{4} = 0. 1 × f (52. 1, 85. 5072)

= 0. 1 × [0. 1 × 52. 1 - 0. 0008 × 52. 1^{2} - 0. 004 × 52. 1 × 85. 5072]

= 0. 1 × - 14. 781 = - 1. 4871

\(x_{2021} = x_{2020} +\frac{1}{6}\biggl[k_{1}+2k_{2}+2k_{3}+k_4\biggl]\)

\(x_{2021} = 52 +\frac{1}{6}[- 1. 5052 + 2 × (- 1. 4191) + 2 × ( - 1. 4928) + (- 1. 4871)]\)

\(= 52 +\frac{1}{6} [- 8. 8161] = 52 - 1. 46935 = 50. 53\)

Patrick Jones

1.2M+ YouTube subscribers, 30+ years' experience, Ivy League visiting faculty!

**Analysis**

In graph 4, the variation in the population of Greater One – Horned Rhino at Gorumara Wildlife Sanctuary has been shown. The population count of 2020 is collected from sources and that of 2021 has been predicted from the Predator Prey Model. It has been observed that, there is a decrease in the number of Greater One – Horned Rhinos from 52 in the year 2020 to 50 (predicted) in the year 2021. It signifies that the effective decrease in population of rhinos at Gorumara Wildlife Sancuary in the year 2020 should be 2. This could have happened due to predation by the poachers as well as other environmental conditions.

**How does the population of One Horned Rhino in India is affected by the action of poaching, determined using Predator Prey Model, applying Runge Kutta Method of Solving Differential Equation?**

- According to the Predator Prey Model, (1,0) signifies the stable population between the One – Horned Rhino and Poachers respectively in Kaziranga National Park.
- The population of rhinos is dominant over the poachers at Kaziranga National Park.
- There is a decrease in the number of Greater One – Horned Rhinos from 2413 in the year 2020 to 1935 (predicted) in the year 2021 at Kaziranga National Park.
- It signifies that the effective decrease in population of rhinos at Kaziranga National Park in the year 2020 should be 478.
- According to the Predator Prey Model, (7, -77.575) signifies the stable population between the One – Horned Rhino and Poachers respectively at Jaldapara National Park.
- The population of rhinos is dominant over the poachers at Jaldapara National Park.
- There is a decrease in the number of Greater One – Horned Rhinos from 231 in the year 2020 to 201 (predicted) in the year 2021.
- It signifies that the effective decrease in population of rhinos at Jaldapara National Park in the year 2020 should be 30.
- According to the Predator Prey Model, (33.33,41.66) signifies the stable population between the One – Horned Rhino and Poachers respectively at Jaldapara National Park.
- The population of poachers is dominant over the rhinos at Gorumara Wildlife Sanctuary.
- There is a decrease in the number of Greater One – Horned Rhinos from 52 in the year 2020 to 50 (predicted) in the year 2021.
- It signifies that the effective decrease in population of rhinos at Gorumara Wildlife Sanctuary in the year 2020 should be 2.

Patrick Jones

1.2M+ YouTube subscribers, 30+ years' experience, Ivy League visiting faculty!

- All the data required for the construction of predator prey model at each forest area has been collected from official websites with Government domain. It makes the collected data much reliable for use and strength the existence and stability of the exploration.
- In the exploration, not only the predator prey model has been prepared, but also its application by predicting the effective count of Greater One – Horned Rhino at each forest has been shown.

- The required data has been collected from a single source only. If the method of triangulation could have been followed, then in case of any discrepancy in the collected data, the effect on the exploration would have been minimized. However, it wasn’t followed in the exploration. Thus, it makes the exploration questionable on reliability.
- Runge Kutta Method is a numeric analytical method of solving ordinary differential equation and it gives an approximated value. As a result, it incurs some error during the approximation. Thus, the prediction cannot be accurately made.

Patrick Jones

1.2M+ YouTube subscribers, 30+ years' experience, Ivy League visiting faculty!

India, being a subtropical country, is a home to various wild animals. Unfortunately, the wild animals are not only threatened by the action of poaching. Covering almost 22% of the India’s total geographical area by forests, the animals often come in close proximity to human habitats. Out of several issues, one of the most alarming situations that the concerned forest departments are facing nowadays is death of elephants by trains.

A part of Indian Railway has paved its route through the forests. Elephants often come on the rail tracks and they are run away by the trains causing serious injury and even death of the elephant. Thus, by applying the same concepts of Predator Prey Model, a mathematical model should be created to estimate the life threat on elephants by the running trains to help the concerned authorities to take definite step to prevent the innocent lives of elephants from the running mechanical carriages. The research question of the above-stated exploration is mentioned as follows: **How does the population of Elephants in India is affected by the trains, determined using Predator Prey Model?**

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Patrick Jones

1.2M+ YouTube subscribers, 30+ years' experience, Ivy League visiting faculty!