Lecture 2 in #MathematicalMethodsInPopulationBiology finishes scalar linear differenital equations & introduces nonlinear differential equations (x'=f(x,t)). We cover existence/uniqueness of solutions & examine provocative examples: desiccating crabs & doomsday predictions.
January 23, 2025 at 5:16 AM
Lecture 2 in #MathematicalMethodsInPopulationBiology finishes scalar linear differenital equations & introduces nonlinear differential equations (x'=f(x,t)). We cover existence/uniqueness of solutions & examine provocative examples: desiccating crabs & doomsday predictions.
Week 4 covered discrete-time matrix models, eigenvalues & eigenvectors, Perron Frobenius Theorem, and reproductive values.
#MathematicalMethodsInPopulationBiology
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#MathematicalMethodsInPopulationBiology
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March 17, 2024 at 5:01 PM
Week 4 covered discrete-time matrix models, eigenvalues & eigenvectors, Perron Frobenius Theorem, and reproductive values.
#MathematicalMethodsInPopulationBiology
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#MathematicalMethodsInPopulationBiology
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Teaching #MathematicalMethodsInPopulationBiology again. This course covers basic methods for deterministic & stochastic dynamical models in population biology. In this thread*, I'll highlight topics from all twenty 90-min lectures.
*Redoing this thread as I didn't complete it in 2024
*Redoing this thread as I didn't complete it in 2024
January 23, 2025 at 4:59 AM
Teaching #MathematicalMethodsInPopulationBiology again. This course covers basic methods for deterministic & stochastic dynamical models in population biology. In this thread*, I'll highlight topics from all twenty 90-min lectures.
*Redoing this thread as I didn't complete it in 2024
*Redoing this thread as I didn't complete it in 2024
Lecture 13 in #MathematicalMethodsInPopulationBiology discussed heteroclinic cycles, periodic orbits, Poincare-Bendixson, invasion analysis for higher dimensional models (e.g. SEIR model, R* rule), and chaos for ODEs
February 24, 2025 at 11:19 PM
Lecture 13 in #MathematicalMethodsInPopulationBiology discussed heteroclinic cycles, periodic orbits, Poincare-Bendixson, invasion analysis for higher dimensional models (e.g. SEIR model, R* rule), and chaos for ODEs
Lecture 3 in #MathematicalMethodsInPopulationBiology covers slope fields, phase lines, classifying equilibria, and introduces linearization. Applications to Carlson's yeast data (crazy good fit) and Gypsy moth outbreaks
January 24, 2025 at 5:28 PM
Lecture 3 in #MathematicalMethodsInPopulationBiology covers slope fields, phase lines, classifying equilibria, and introduces linearization. Applications to Carlson's yeast data (crazy good fit) and Gypsy moth outbreaks
Lecture 16 in #MathematicalMethodsInPopulationBiology covered hitting probabilities and hitting times for Markov chains, calculating mean times and probabilities using conditioning. Applications included life history questions, disease outbreaks, gambler's ruin, and the Wright-Fisher model
March 14, 2025 at 4:49 PM
Lecture 16 in #MathematicalMethodsInPopulationBiology covered hitting probabilities and hitting times for Markov chains, calculating mean times and probabilities using conditioning. Applications included life history questions, disease outbreaks, gambler's ruin, and the Wright-Fisher model
Lecture 17 in #MathematicalMethodsInPopulationBiology completed hitting probabilities for discrete-time Markov chains and introduced continuous-time chains, highlighting differences like the form of the Kolmogorov equations and how irreducibility suffices for both limit laws.
March 15, 2025 at 3:44 PM
Lecture 17 in #MathematicalMethodsInPopulationBiology completed hitting probabilities for discrete-time Markov chains and introduced continuous-time chains, highlighting differences like the form of the Kolmogorov equations and how irreducibility suffices for both limit laws.
Lecture 18 of #MathematicalMethodsInPopulationBiology reviewed random variables and their expectation, introduced random walks, and discussed the Law of Large Numbers
March 16, 2025 at 4:32 PM
Lecture 18 of #MathematicalMethodsInPopulationBiology reviewed random variables and their expectation, introduced random walks, and discussed the Law of Large Numbers
Lecture 8 of #MathematicalMethodsInPopulationBiology covered general solutions of x(n)=Ax(n−1) where A is a k×k matrix. We explored cases with k distinct eigenvalues, complex eigenvalues, and analyzed the stability of the origin.
February 5, 2025 at 9:02 PM
Lecture 8 of #MathematicalMethodsInPopulationBiology covered general solutions of x(n)=Ax(n−1) where A is a k×k matrix. We explored cases with k distinct eigenvalues, complex eigenvalues, and analyzed the stability of the origin.
The @scifri.bsky.social interview with @stevenstrogatz.com to celebrate Pi is a fun listen. While not mentioned, my favorite magical appearance of pi: the Central Limit Theorem - the focus of my final week of lectures in #MathematicalMethodsInPopulationBiology
www.sciencefriday.com/segments/pi-...
www.sciencefriday.com/segments/pi-...
March 19, 2025 at 8:40 PM
The @scifri.bsky.social interview with @stevenstrogatz.com to celebrate Pi is a fun listen. While not mentioned, my favorite magical appearance of pi: the Central Limit Theorem - the focus of my final week of lectures in #MathematicalMethodsInPopulationBiology
www.sciencefriday.com/segments/pi-...
www.sciencefriday.com/segments/pi-...
Week 2 (part 1) covered existence & uniqueness of solutions for scalar ordinary differential equations x'=f(x), equilibria, phase lines, linearization and stability
#MathematicalMethodsInPopulationBiology
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#MathematicalMethodsInPopulationBiology
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March 2, 2024 at 6:21 PM
Week 2 (part 1) covered existence & uniqueness of solutions for scalar ordinary differential equations x'=f(x), equilibria, phase lines, linearization and stability
#MathematicalMethodsInPopulationBiology
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#MathematicalMethodsInPopulationBiology
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Lecture 5 in #MathematicalMethodsInPopulationBiology discusses scalar difference equations, stability of equilibrium & periodic orbits, Lyapunov exponents, and chaos.
January 27, 2025 at 9:56 PM
Lecture 5 in #MathematicalMethodsInPopulationBiology discusses scalar difference equations, stability of equilibrium & periodic orbits, Lyapunov exponents, and chaos.
Day 15 of #MathematicalMethodsInPopulationBiology covered classification of states and the two Markov chain limit laws, with applications to Wright-Fisher with mutation, reflecting random walks, human emotions, gambling, and rain in Davis.
March 12, 2025 at 6:48 PM
Day 15 of #MathematicalMethodsInPopulationBiology covered classification of states and the two Markov chain limit laws, with applications to Wright-Fisher with mutation, reflecting random walks, human emotions, gambling, and rain in Davis.
Lecture 19 in #MathematicalMethodsInPopulationBiology discussed additional applications of the Law of Large Numbers by uncovering random walks hidden in a mask of nonlinearity
March 18, 2025 at 9:15 PM
Lecture 19 in #MathematicalMethodsInPopulationBiology discussed additional applications of the Law of Large Numbers by uncovering random walks hidden in a mask of nonlinearity
Teaching Mathematical Methods in Population Biology again this Winter. Thought it would be fun to provide a thread about the weekly content here.
#MathematicalMethodsInPopulationBiology 1/n
#MathematicalMethodsInPopulationBiology 1/n
February 19, 2024 at 9:29 PM
Teaching Mathematical Methods in Population Biology again this Winter. Thought it would be fun to provide a thread about the weekly content here.
#MathematicalMethodsInPopulationBiology 1/n
#MathematicalMethodsInPopulationBiology 1/n
Lecture 14 in #MathematicalMethodsInPopulationBiology segued into stochastic models. Introduced discrete-time Markov chains, discussed conditional probabilities, and had the students derive the Chapman-Kolmogorov equations via an example in our daily breakout session.
March 2, 2025 at 2:18 AM
Lecture 14 in #MathematicalMethodsInPopulationBiology segued into stochastic models. Introduced discrete-time Markov chains, discussed conditional probabilities, and had the students derive the Chapman-Kolmogorov equations via an example in our daily breakout session.
Lecture 9 in #MathematicalMethodsInPopulationBiology covered continuous-time matrix models x'(t) = Ax(t). Highlighted key differences from discrete-time models in writing general solutions, characterizing stability, and applying Perron-Frobenius theory e.g. non-negative vs quasi-positive matrix
February 8, 2025 at 6:21 PM
Lecture 9 in #MathematicalMethodsInPopulationBiology covered continuous-time matrix models x'(t) = Ax(t). Highlighted key differences from discrete-time models in writing general solutions, characterizing stability, and applying Perron-Frobenius theory e.g. non-negative vs quasi-positive matrix
Week 1 covered univariate, affine difference & differential equations (i.e. x(n+1)=ax(n)+b, x'(t)=alpha x(t)+beta), their solutions via a change of variables, the stability of their equilibria, and various applications.
#MathematicalMethodsInPopulationBiology 2/n
#MathematicalMethodsInPopulationBiology 2/n
February 21, 2024 at 3:00 AM
Week 1 covered univariate, affine difference & differential equations (i.e. x(n+1)=ax(n)+b, x'(t)=alpha x(t)+beta), their solutions via a change of variables, the stability of their equilibria, and various applications.
#MathematicalMethodsInPopulationBiology 2/n
#MathematicalMethodsInPopulationBiology 2/n
Students in #MathematicalMethodsInPopulationBiology enjoyed the Sheldon & Karen variants of linear love stories from @stevenstrogatz.com 's book "Nonlinear Dynamics & Chaos". Stories with and without emotional delays (i.e. discrete & continuous time) were discussed.
February 12, 2025 at 5:00 PM
Students in #MathematicalMethodsInPopulationBiology enjoyed the Sheldon & Karen variants of linear love stories from @stevenstrogatz.com 's book "Nonlinear Dynamics & Chaos". Stories with and without emotional delays (i.e. discrete & continuous time) were discussed.
Lecture 4 in #MathematicalMethodsInPopulationBiology examines bifurcation diagrams for scalar ODEs, with applications to metapopulations, evolutionary games, and harvesting.
January 25, 2025 at 9:28 PM
Lecture 4 in #MathematicalMethodsInPopulationBiology examines bifurcation diagrams for scalar ODEs, with applications to metapopulations, evolutionary games, and harvesting.
The course covers nonlinear dynamics, linear dynamics, Markov chains, and random walks in discrete and continuous time.
#MathematicalMethodsInPopulationBiology 2/n
#MathematicalMethodsInPopulationBiology 2/n
February 19, 2024 at 9:32 PM
The course covers nonlinear dynamics, linear dynamics, Markov chains, and random walks in discrete and continuous time.
#MathematicalMethodsInPopulationBiology 2/n
#MathematicalMethodsInPopulationBiology 2/n
Week 2 (part 2) covered bifurcations of scalar ordinary differential equations x'=f(x)
#MathematicalMethodsInPopulationBiology
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#MathematicalMethodsInPopulationBiology
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March 3, 2024 at 5:22 PM
Week 2 (part 2) covered bifurcations of scalar ordinary differential equations x'=f(x)
#MathematicalMethodsInPopulationBiology
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#MathematicalMethodsInPopulationBiology
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I start each lecture in #MathematicalMethodsInPopulationBiology with a fun quote. Lecture 10's quote is by Vladimir Arnold whose graduate text of dynamical systems (the yellow Springer book in the image) was one of my favorite reads during grad school
February 12, 2025 at 4:41 PM
I start each lecture in #MathematicalMethodsInPopulationBiology with a fun quote. Lecture 10's quote is by Vladimir Arnold whose graduate text of dynamical systems (the yellow Springer book in the image) was one of my favorite reads during grad school
Week 5 covered complex eigenvalues for discrete-time matrix models and all the parallel results for continuous-time matrix models including general solutions using eigenvalues & eigenvectors and Perron-Frobenius Theorem
#MathematicalMethodsInPopulationBiology
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#MathematicalMethodsInPopulationBiology
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March 18, 2024 at 4:54 PM
Week 5 covered complex eigenvalues for discrete-time matrix models and all the parallel results for continuous-time matrix models including general solutions using eigenvalues & eigenvectors and Perron-Frobenius Theorem
#MathematicalMethodsInPopulationBiology
7/n
#MathematicalMethodsInPopulationBiology
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