I am a graduate student trying to write a survey paper on population models. I am new to writing mathematical papers as you guessed. My professor did not give me much direction, but from what I surmised can I just go over the different types of models (I.e. Continuous, discrete, models with age structure) in my abstract and then go further into detail about the equations and results in my paper?
An example of a paper on the subject or another survey article would be extremely helpful.
Thanks
On
The books [1,2] would be helpful. For instance in [1], you may find the following models of mathematical biology.
But these references mostly cover the subjects of oscillation, nonoscillation and stability for delay differential equations.
References
[1] R.P. Agarwal, L. Berezansky, E. Braverman, A. Domoshnitsky, Nonoscillation Theory of Functional Differential Equations with Applications, Springer, 2012.
[2] I. Gyori, G. Ladas, Oscillation Theory of Delay Differential Equations: With Applications, Oxford University Press, 1991
You are fortunate in that the subject is so vast. Much has been written, and is being written, on the topic of population modeling. Once you read into the literature, I think you will find a paper that you can call your own. You could easily make this a very large book if you are not careful.
Start by thoroughly saturating your feet with the obvious soft writings on the web. For a very small sample of reasonably light reading, read the -pedias: Population Model, Malthus, Verhulst, Logistic function, Nurgaliev's Law, Predator Prey, Lokta-Volterra, and another very interesting read: Demographic Transition. Many more here, just whet your appetite.
You might take a brief interlude into the thought of how governments actually try to predict populations by briefly digging into soft papers written for the general public like census of India. Good luck there. You will basically find that actual human population predictions involve many dynamics, and statistical inclusions, and rarely see these boiled down into some system of ODE's. Fungi in a petri dish behave much better.
After that, I would spend a few hours tapping the scholarly citable resources that are exclusive to your institution. Just go to the library. Your school pays for access to online scholarly resources, and some schools have better access than others, so that is on you. Any librarian will be able to point you in the right direction. You are basically looking for citable scholarly articles and papers that read as being understandable and useful, while ignoring the glut of papers which read more like someone's attempt to get published and confound the world with their intelligence. Dig through the weeds for a few hours and collect some gems. I definitely advise the paid resources of your institution versus google scholar, as if you hit the right tap you will get much more exclusive content. That being said, scholar is useful. You may find that some of the non-free articles are actually free to your institution. I found these citable scholarly sources to be interesting to me and accessible to anyone. This is just a start:
Malthus: A seminal work. This is a must read for an early historical perspective.
Verhulst: Another important historical seminal work, but unfortunately it is in French. Most people writing a paper like yours would cite this and find a way to reference it in their paper, and I suspect many who do did not truthfully read it. I am not familiar with your language skills.
Lokta-Volterra
Matrix Population Models
...nonlinear age dependent... (I like this)
Basic Properties of... (good paper)
Analysis of... (not too shabby!)
Predator-Prey (basic but citable if necessary)
You just have to collect what interests you.
Papers like the one you are about to write usually read best with a really great introduction, and I think historical perspective and skeletal framework is appropriate here. If you just get all the historical perspective, tap all of the bigger models, mention the different types of modellable and modeled populations, and get enough heavy math into the situation to make your professor happy, you will walk away with a great paper and some very valuable knowledge.
Try to have fun with it, the more you research the more authoritative you will feel, and when you feel that you are authoritative enough on the title of your paper, just write it out, keep it interesting, keep rewriting, and try to be conscious of the fact that eventually you actually need to stop writing. The topic is vast. Enjoy.