You need to get to grips with the basics of dynamics

So watch this series of lectures which will give you an introduction.

https://www.youtube.com/watch?v=wJGR5Y-Qqac&list=PLqOZ6FD_RQ7lnGZ7fkn503y_7U4rrJ-Se

Video 5 onwards is what you want, but earlier videos are very useful too.

Another good course with a book:

https://www.youtube.com/watch?v=ycJEoqmQvwg&list=PLbN57C5Zdl6j_qJA-pARJnKsmROzPnO9V

https://www.biodyn.ro/course/literatura/Nonlinear_Dynamics_and_Chaos_2018_Steven_H._Strogatz.pdf

To solve the differential equations you could use xppaut, which works on Linux (used it myself in Linux) and windows

Download it here

https://services.math.duke.edu/ode-book/xppaut/installation.html

Videos on xppaut:

https://www.youtube.com/watch?v=tjLB3JDEXGs

https://www.youtube.com/watch?v=SQZGlJUsKTg

https://www.youtube.com/watch?v=DzjIs3ry40o

https://www.youtube.com/watch?v=tjLB3JDEXGs

If you use matlab you can use an m file called pplane

https://uk.mathworks.com/matlabcentral/fileexchange/61636-pplane

How to use pplane:

https://www.cfm.brown.edu/people/dobrush/am33/Matlab/ch1/pplane.html

https://www.youtube.com/watch?v=8XILeDBwXys

https://www.youtube.com/watch?v=ta9BfrkknEI

https://services.math.duke.edu/~ciocanel/teaching/Tutorial.pdf

https://www.math.purdue.edu/~park633/pplane.pdf

Some useful books articles and tutorials

https://web.mst.edu/~huwen/teaching_Predator_Prey_Tyler_Blaszak.pdf

https://www.math.colostate.edu/~shipman/47/volume3b2011/M640_MunozAlicea.pdf

https://www.ucl.ac.uk/~ucesgvd/hopf.pdf

http://mitran-lab.amath.unc.edu/courses/MATH564/biblio/ShonkwilerHerod09.pdf

https://www.pdfdrive.com/mathematical-biology-e41984249.html

https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2011/06/MA221-4.1.2.pdf

https://www.cs.bham.ac.uk/~szh/teaching/matlabmodeling/Lecture6_body.pdf

https://www.academia.edu/43192876/Lotka_Volterra_Predator_Prey_Models

https://www.math.uci.edu/~ndonalds/math3d/predator.pdf

https://tbb.bio.uu.nl/rdb/books/tb.pdf

Dynamical systems Programming in python / Matlab

https://www.pdfdrive.com/dynamical-systems-with-applications-using-python-e195281976.html

Thank you for this, Loopindor Sood. This will help me a lot in my undergraduate studies. I just want to add a question, is XXPAUT free to use?

With regard to Numerical Simulation, I have something to offer. But, what type of Differential Equations are included in the model ? is it ODE, stiff ODE or PDE and of what order?, what is the form of the DE? is it delay , fussy, Wave/Heat etc?. Let see the model [email protected]

Re Xppaut, it is free. Python is also free. Matlab is not free, but the m file pplane is free.

Re Thank you very much Sir for your help. I truly appreciate it.

This book is also very useful

https://b-ok.cc/s/mathematical%20models%20in%20biology%20edelstein%20keshet%20solutions

To go with the course

https://www.youtube.com/watch?v=wJGR5Y-Qqac&list=PLqOZ6FD_RQ7lnGZ7fkn503y_7U4rrJ-Se

And this book uses xppaut to model dynamical systems

A primer on mathematical models in biology

Society for Industrial and Applied Mathematics

Lee A. Segel, Leah Edelstein-Keshet

see:

https://b-ok.cc/s/Leah%20Edelstein-Keshet/?e=1

Since this is a quasilinear system of ODEs, you could simply use an implicit Euler scheme so you end up with an algebraic system of two equations and two unknowns that can be easily solved with Newton's method.

Let me know if you need more details!

As Loopinder Sood has noted, Matlab is not free; but there is the open source (and free) Scilab which provides a very similar functionality.

See http://www.scilab.org/