Although not required, it is highly recommended that honors degree candidates take the intermediate honors courses Math and during the freshman or sophomore year. Students interested in the departmental honors program should consult with the undergraduate director or department chairperson as early as possible, preferably during the freshman or sophomore year, even if they have not yet declared a major. The department's Undergraduate Committee is responsible for certifying students who graduate with departmental honors in mathematical biology.
The Undergraduate Committee is empowered to make minor changes in requirements based on individual circumstances. Notes 1.
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University of Pittsburgh. Departmental Honors in Mathematical Biology The Department also offers an opportunity for students to graduate with departmental honors in mathematical biology. To qualify for departmental honors in mathematical biology, a student must: Fulfill all requirements for a degree in mathematical biology.
About this book
The spread of disease in a large population of susceptibles may be thought of as an invasion process generated by independent contacts between a huge pool of susceptibles and a few infectious individuals. In this chapter we shall consider populations with a fixed interval between generations or possibly a fixed interval between measurements. The underlying assumption will always be that population size at each stage is determined by the population sizes in past generations, but that intermediate population sizes between generations are not needed.
Usually the time interval between generations is taken to be a constant. However, there are situations in which the growth rate does not respond instantaneously to changes in population size.
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Volterra constructed the model that has become known as the Lotka-Volterra model because A. Lotka constructed a similar model in a different context about the same time , based on the assumptions that fish and sharks were in a predator—prey relationship. The topics in this chapter are part of the subject of natural resource management and bioeconomics.
This is an important subject that is developing rapidly.
The classical reference is the book by Clark , where additional references may be found. In the preceding chapters we studied mainly models in which all members were alike, so that birth and death rates depended on total population size. However, we gave a few examples of populations with two classes of members and a birth rate that depended on the size of only one of the two classes, for discrete models in Section 2. These are examples of structured populations.
In this chapter we shall study models for populations structured by age. In practice, animal populations are often measured by size with age structure used as an approximation to size structure. The study of age-structured models is considerably simpler than the study of general size-structured models, primarily because age increases linearly with the passage of time while the linkage of size with time may be less predictable.
Age-structured models may be either discrete or continuous. We begin with linear models, for which total population size generally either increases or decreases exponentially over time. Populations may be structured by spatial location. There are two common different ways to include spatial location in a population.
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One way is by means of metapopulations , that is, populations of populations, with links between them such as a collection of towns and cities connected by a transportation network. The air transport subnetwork includes connecting links between distant communities, and we may study the dynamics of populations of different cities as a function of the flow of people between them and their own local dynamics in this framework.
A metapopulation may be divided into patches , with each patch corresponding to a separate location. The corresponding models may be systems of ordinary differential equations, with the population size of each species in each patch as a variable. Thus metapopulation models are often systems of ordinary differential equations of high dimension.
Communicable diseases such as measles, influenza, and tuberculosis are a fact of life. We will be concerned with both epidemics, which are sudden outbreaks of a disease, and endemic situations, in which a disease is always present. The AIDS epidemic, the recent SARS epidemic, recurring influenza pandemics, and outbursts of diseases such as the Ebola virus are events of concern and interest to many people.