Mathematical Models in the Life and Social Sciences

MA 432, Spring 2005

Learning Objectives

After completing the course, you should be able to do the following. (Items that are marked with an asterisk are worthy aims, but will not appear on a test or exam.)

1. Chapter 7: Population Dynamics
   • Write down and solve Malthus's model in discrete time (bacterial growth model for synchronous cell division): geometric growth.
   • Derive and solve Malthus's model in continuous time (bacterial growth model for asynchronous cell division): exponential growth.
   • Relate the two forms of the model and their parameters, but appreciate the importance of the asynchronous division assumption in the continuous time model.
   • Relate the growth rates in these models to the doubling time (cell division cycle length or generation time).
   • Explain why a linear relationship can be obtained between logged bacterial densities and time.
   • Calculate the best fitting straight line through a given set of data.
   • Use the best fit line to estimate growth rate or doubling time.
   • Draw the main features of growth curves of real bacteria, explaining the limitations of the exponential growth model.
   • Explain Fibonacci's model for yeast cells, both in terms of the number of newly born cells and in terms of the age distribution of cells (Leslie matrix formulation).
   • Solve the Fibonacci model in terms of geometrically growing terms, including the quadratic equation that determines the growth rates.
   • Explain the renewal equation model, both for the formulation in terms of births (B_n) and in terms of age distributions.
   • Find the solution of the renewal model, and explain the theorem that discusses the long-term behavior, including the maternity parameter and stable age distribution. You should be able to write down the expression for the stable age distribution.
   • Appreciate the limitations of the above models (namely that they are all linear and hence predict exponential growth or decay).
   • Explain the assumptions that lead to the logistic growth model (per-capita growth rate declines linearly with N).
   • Outline the ideas behind fitting a logistic growth model (or some other model) to a given set of data. You are NOT expected to produce your own MATLAB code.
   • Explain the limitations of the logistic growth model. (Are the parameters r and K constant over time? Why not?)
   • Produce a direction field plot (from a graph of dN/dt versus N) and use it to explain the dynamics of the logistic growth (or some other) model.
   • Explain what is meant by an equilibrium and its stability, and use the direction field plot (or dN/dt versus N plot) to find equilibria and their stability.
   • Use the direction field plot to indicate the behavior of N as a function of time.
   • Write down and explain the idea behind the Jacob-Monod model of bacterial growth. You should be able to give some motivation for the basic properties of the saturating function b(S), but need not justify its precise form.
   • Analyze the dynamics of the Jacob-Monod model, including the existence of a conserved quantity and the long-term behavior. You should be able to explain the biological interpretation of the conserved quantity.
   • Write down and explain the basis of the growth and quiescence model (SPQ model). Again, you should know the broad features of the functions alpha and beta, but need not justify their particular form.
   • Appreciate the dynamics of the SPQ model in broad terms. (Detailed analysis is not required.)
   • Write down and explain the continuous flow model.
   • Analyze the dynamics of the continuous flow model, including the equilibria and determination of whether washout occurs or not.
   • Write down and analyze simple models for populations that are being harvested, calculating equilibria and their stability and the maximum sustainable yield. Explain the deficiencies of the constant harvesting (E(N)=E) model, both in mathematical terms and in biological terms.
   • Explain what is meant by a transcritical bifurcation and a saddle-node bifurcation.
   • Produce a bifurcation diagram for simple one dimensional ODE models, finding equilibria and their stability using the graphical approach.