Models for biological processes based on ordinary and partial differential equations. Topics selected from models of population growth, predator-prey dynamics, biological oscillators, reaction-diffusion systems, pattern formation, neuronal and blood flow physiology, neural networks, biomechanics.
(MATH 244 OR MATH 252 OR MATH 292) AND (MATH 250). In laymen’s terms, you should have a good understanding of the standard engineering calculus sequence through ordinary differential equations, as well as linear algebra. Most of the analysis of models studied in this course will rely heavily on techniques from the listed previous math courses, so it is very important to have a good understanding of the prerequisites. Of course, I will review theory as we encounter it, but it most likely won’t be sufficient as a stand-alone treatment. For a good self-check, see the material (and more importantly, problem sets) appearing in Appendix A of the textbook (“Review of ordinary differential equations.”) Also, see the review homework in the Course Calendar. If you have concerns about the prerequisites, please feel free to contact me.
Formally, grades will be determined by homework and three exams. The breakdown is as follows:
| Homework | 40% |
| Midterm I | 20% |
| Midterm II | 20% |
| Final Exam | 20% |
Homework Homework will be assigned weekly, and will typically be due one week after assignment. All assignments will be distributed on the webpage, and should be turned in by the end of class on the due date. The two lowest grades will be dropped. Late homework will not be graded. Assignments will consist of a combination of theoretical and computational problems. Some assignments will require the use of computer software (in particular ODE solvers/phase portrait plotters). I recommend using MATLAB, although any programming software is acceptable. You will be expected to write your own code, but I will provide samples, and am very happy to talk about MATLAB programming in general. For a basic tutorial, see the Resources section below. Based on past experience, I strongly recommend learning MATLAB immediatly, as to avoid becoming overwhlemed later in the course; in general programming expectations will increase as the course progresses. For accessing MATLAB, see Resources below. Problems requiring MATLAB will be explicitly denoted by an M; otherwise the problem is expected to be worked by hand.
MATLAB projects I may also assign a few MATLAB projects, which will involve reading a short chapter on a specific biological phenonmena, and studying a mathematical model of the biological system using MATLAB. For these assignments, template code will be provided. In general, you will have a few (2-3) weeks to complete these projects, and your grade will be included in the homework portion for the course.
Note on collaboration You are encouraged to work together on HW assignments, but problem sets should be both written up and turned in individually. Furthermore, only submit work that is your own. Plagiarism is a serious offense, and will result in a score of zero for the assignment, as well as possible punishment from the University. You must also cite any reference you use and clearly mark any quotation or close paraphrase that you include. Such citation will not lower your grade, although extensive quotation might.
Exams The Final Exam will take place during the Exam week; it appears we are scheduled for May 6th, from 8-11 am. Note that it will be cumulative, but possibly more heavily weighted to the material since Midterm II. The other two exams will take place in class, and will be announced at least two weeks prior to their administration, both in class and on the webpage (Course Calendar). In general they will be closed book, and no calculators or other electronic devices will be permitted. You will be allowed one sheet of notes (back and front), where you can summarize anything you want. Material there has to be written by you; you are not permitted to paste material directly from the course notes.
This class will follow Prof. Sontag’s notes fairly carefully, although material will at times be supplemented from other sources. I will keep an updated class schedule on the webpage (Course Calendar) during the semester, which will include sections covered, homework postings with due dates, and the exam schedule. Below you will find a tentative listing of model frameworks for the course, along with the approximate number of weeks spent on the particular subject:
| Scalar difference equations | 1-2 weeks |
| Ordinary differential equations (ODEs) | 8-10 weeks |
| Partial differential equations (PDEs) | 3-4 weeks |
| Stochastic processes | 0-3 weeks |
You are expected to attend class. The material will not always overlap the lecture notes. If you miss class, it is your responsibility to catch up, and you should please use the University absence reporting website to indicate the date and reason for your absence. An email is automatically sent to me. There will be no make-ups for exams. In case of a medical or family emergency, please contact me by email before the exam. In cases of a justified and documented absence (for a medical or family emergency - and I was contacted before the exam) the weight of the missed exam will be shifted to the Final Exam.
I will use Sakai for email contact, as well as to post homework solutions. All enrolled students should have automatic access to the site after logging in to Sakai. Make sure to frequently check your email associated to your Sakai account.