BSHS Calculus 2


BSHS Calculus 2: 01. Stochastic Processes The following data set from the CBC was used in Van Bael and Pruett-Jones (1996) as a proxy for population size. We will use this data set to build a model that describes the population growth of this species.
BSHS Calculus 2: 02. Linear Transformations and Linear Regression In this module, we will learn what functions are used to describe linear relationships after the data is transformed logarithmically.
BSHS Calculus 2: 03. Discrete Time, One-dimensional Dynamical Systems We will explore a simple example of population growth with density dependence. (Gause, 1934)
BSHS Calculus 2: 04. Ordinary Differential Equations Numerical analysis is a field in mathematics that is concerned with developing approximate numerical methods and assessing their accuracy, for instance for solving differential equations. We will discuss the most basic method, the Euler method, explain the basic ideas behind this numerical method, and how to implement it on a spreadsheet using the example of exponential and logistic growth.
BSHS Calculus 2: 05. Age-structured models – the Leslie matrix In many species, reproduction is highly age-dependent. For instance, periodical cicadas spend 13-17 years in the nymphal stage; they only reproduce once in their lifetime. Many animals, such as humans, elephants, etc., do not reproduce during their first years and then their reproductive success is age-dependent. To model such situations, age-dependent population models are appropriate. Patrick Leslie introduced matrix models that have discrete age classes with synchronous reproduction (Leslie 1945). The models are parameterized by age-specific survival probabilities and average number of female offspring.
BSHS Calculus 2: 06. Systems of Difference Equations – Host Parasitoid Models We will find point equilibria and explain how to analyze their stability, first by investigating linear systems and then nonlinear systems.
BSHS Calculus 2: 07. Linear Systems of Differential Equations Numerical analysis is a field in mathematics that is concerned with developing approximate numerical methods and assessing their accuracy, for instance for solving differential equations. We will discuss the simplest method, the Euler method, explain the basic idea behind this numerical method, and how to implement it on a spreadsheet.
BSHS Calculus 2: 08. Nonlinear Systems of Differential Equations – Consumer-Resource Models There are a large number of models that deal with consumer and resource interactions. The oldest such model is the Lotka-Volterra model, which describes the interaction between a predator and its prey.
BSHS Calculus 2: 09. Nonlinear Differential Equations – Bifurcations In 1969, Richard Levins introduced the concept of metapopulations (Levins, 1969). This was a very influential paper that is highly cited even today. A metapopulation is a collection of subpopulations. Each subpopulation occupies a patch, and different patches are linked via migration of individuals between patches.
BSHS Calculus 2: 10. Modeling Chemical Reactions The law of mass action has its origin in modeling chemical reactions. Michaelis and Menten (1913) simplified assumptions. A more realistic model was developed by Briggs and Haldane (1925). Other models including gene regulatory networks will be examined.
BSHS Calculus 2: 11. Modeling the Cell Cycle A phenomenological model can be used to describe the alternation between interphase and mitosis. Additional models related to cell cycle will be considered.