Messages
Todays topics: Exercises 3.18; 3.19; 3.21; 6.4 + those in dropbox folder (named Exercises week 19.pdf).
This was the final lecture before the exam.
Todays topics: Hodgkin-Huxley model and signal transmission in neurones.
On Monday I will present solutions to Exercises 3.18; 3.19; 3.21; 6.4 + those in dropbox folder (named Exercises week 19.pdf).
Todays topics: Sections 6.1-6.3 (biochemical reactions, enzymes & Michaelis-Minten kinetics, metabolism & activators/inhibitors). On Wednesday we will discuss the Hodgkin-Huxley model for signal transmission in neuronal cells.
New exercises: 6.4 + file in dropbox folder (named Exercises week 19.pdf).
Todays topics: Models for macroscopic parasites and SIR-type models reflecting evolutionary traits such as virulence.
On Monday next week we begin with Chapter 6.
New exercises: 3.21.
Todays topics: SIR-type models for Malaria and Exercises 3.15 a); 3.16; 3.17.
On Monday we will go through Sections 3.8 and 3.9.
New exercises: 3.18; 3.19.
Todays topics: Exercises: 3.1; 3.2; 3.5; 3.6; 3.8; 3.9; 3.13.
On Monday we will continue with Sections 3.6 and 3.7.
New exercises: 3.15 a); 3.16; 3.17.
Todays topics: Section 3.4 (SIR endemics) and Section 3.5 (vaccination against an epidemic).
The Wednesday lecture is devoted to exercises.
Todays topics: SIR endemics (no disease-related deaths). On Monday we will complete Section 3.4 (disease related deaths).In addition, the plan is to cover Sections 3.5 and 3.6.
New exercises: 3.1; 3.2; 3.5; 3.6; 3.8; 3.9; 3.13.
Todays topics: Chapter 3; SI, SIS, and SIR models for infectious diseases.
"Mandatory assignment with answers" will be posted in the dropbox folder tomorrow.
Todays topics: Exercises 2.7, 2.11, and 2.12.
Next lecture is Monday April 8. Happy Easter.
Todays topics: Exercises 1.19, 1.20, 1.21, 2.4, 2.6.
We will continue with exercises 2.7, 2.11, and 2.12 on Wednesday.
I have finished correcting the mandatory assignments. Those that did not get the assignment approved must hand in a new attempt by Monday April 8 (send me a notification on e-mail when you have done this). Most likely I will post FASIT (answers) on Tuesday April 9.
Todays topics: Completed Chapter 2, discussed several examples of systems of differential equations and applied previously established stability theory / phase plane analysis, systems with competition among species (2.5), limit cycles, the Poincare-Bendixson theorem (appendix B), Dulac/Bendixson criterion (appendix B), and bifurcation theory for systems of differential equations-Hopf bifurcation (appendix B).
Exercises (for the following weeks): 2.6, 2.7, 2.11, 2.12.
The lectures next week are devoted to exercises.
No lectures in weeks 13 and 14 (Easter). The first lecture after Easter is Monday April 8. After Easter the lecture plan is: Chapter 3 (Infectious diseases) and Chapter 6 (Molecular and cellular biology).
Todays topics: Extensions of Lotka-Volterra prey/predator model (logistic growth of prey and type-II functional response due to predator) --> the Rosenzweig-MacArthur prey/predator model (section 2.4 in the book), steady states and their stability properties (saddle point, stable node, unstable node, stable focus/spiral, unstable focus/spiral, neutral center), systems in Kolmogorov form, Routh-Hurwitch criteria for eigenvalues of 2x2 matrices (Appendix B), phase plane analysis and sketching of phase plane diagrams (Appendix B).
Exercises: 2.4 in the book + perform an analysis of the non-dimensionalized version of the Rosenzweig-MacArthur model (see p. 61 in the book, eqn. 2.4.13). Determine all steady states and their stability properties (saddle point, stable node, unstable node, stable focus/spiral, unstable focus/spiral, neutral center) by explicit computing the eigenvalues and / or using the Routh-Hurwitch criteria.
Todays topics: The Lotka-Volterra prey-predator equations, steady states, linearized analysis is not sufficient to determine the stability of the non-trivial steady state, the existence of periodic solutions deduced via the phase plane equation, the average population of prey-predator fish is the steady state population (Section 2.3 in the book).
Todays topics: Continuous age-structured models, McKendrick's partial differential equation and its solution (Section 1.11 in the book). In addition, I presented a solution to Exercise 1.4 in the book (see dropbox folder).
New exercises: 1.19, 1.20, and 1.21 from the book, and a separate exercise on the McKendrick equation found in the dropbox folder (download the file "Exercise (McKendrick's equation).pdf").
On Wednesday we continue with Chapter 2 (Population Dynamics of lnteracting Species).
Todays topics: Continued with discrete age-structured models. We used eigenvalues & eigenvectors of the Leslie matrix to determine the stability and long-time behavior of solutions to the matrix equation (Sections 1.9 and 1.10 in the book); Continuous age-structured models and the McKendrick partial differential equation (Section 1.11 in the book).
PS! Always go through the lecture notes (see dropbox folder) for a detailed account on what have been covered in the lectures. These notes give an indication of the most relevant parts of the book and supply additional material in order to facilitate the reading of the book.
Todays topics: Fibonacci's rabbit model, discrete age-structured models, matrix formulation, Leslie matrices (sections 1.8, 1.9, and 1.10 in the book). We continue with these topics on Wednesday.
Mandatory assignment can be downloaded [ HERE ]. Deadline is Thursday March 7.
Todays topics: Stability analysis for steady state solutions of 2. order systems of differential equations, with examples and an application to an evolutionary modifed logistic population model.
Exercises for next week: 1.5 and 1.6 from the book.
Todays topics: Derivation of an explicit solution formula for logistic and similar differential equations, bifurcation theory for differential equations (saddle-node, transcritical, and pitchfork bifurcations), systems of differential equations, and linear stability analysis for systems.
Next time we will continue with stability analysis of steady states for systems, applying it to the evolutionary example on page 16 in the book.
The lectures on Monday and Wednesday next week will be given by Nils Henrik Risebro.
NOTE: The mandatory assignment will most likely be handed out on Monday February 25. The deadline is Thursday March 7.
Tropics treated today include period-doubling bifurcation, population models based on differential equations, Malthus equation / exponential growth, logistic equation, steady state solutions of differential equations, and conditions for stability of steady states. Next week we will continue with (systems of) differential equations. Solutions to exercises 1.3 and 1.4 will be presented as well.?Scanned copies of lecture notes/solution to exercises will be eventually found in the Dropbox folder.?
Today we introduced and discussed some basic aspects of the general theory of bifurcation (and chaos) for first order discrete equations, including saddle-node, transcritical, and pitchfork bifurcations. On Wednesday we will continue with period-doubling bifurcations, and also introduce continuous-time models based on differential equations.
We continued with determining stability/instability of steady state solutions through some examples. A solution to Exercise 1.1 was presented. Scanned copies of lecture notes/solution to exercise can be found at [ More ].
Exercises for next week: 1.3 and 1.4 from the book. A scanned copy of the exercises can be found in the dropbox folder (linked above).
Student representative: Kristin Mathilde Drahus - kristmdr at student.matnat.uio.no.
Today we continued with cobwebbing, with emphasis on oscillatory solutions. We introduced and derived conditions for determining the stability of an steady state solution, distinguishing between monotonically stable, monotonically unstable, oscillatory stable, and oscillatory unstable. We illustrated these conditions on an example from the book.
Today we recapped material from last week, introduced the notions of (1) solution, (2) stability, and (3) equilibrium point (or steady state). We also introduced and demonstrated a graphical method for visualizing the evolution of the solution called cobwebbing.
Link to lecture notes (handwritten) [ More ]. This directory will be updated with additional notes throughout the semester.
There is no lecture on Wednesday 23.01. Use the available time to solve Exercise 1.1, page 9, from the book. A scanned copy of the exercise can be found here [ More ]; take the file "Book; exercises 1.1-1.4".