and Theoretical Computer Science

Morgan State University

July 9 - 15, 2006

(Sunday evening through Saturday afternoon)

Abdul-Aziz Yakubu, Howard University (ayakubu@Howard.edu)

The study of population dynamics raises questions involving the four basic demographic processes: immigration, emigration, birth and death. The inclusion of population structure, spatial heterogeneity or mating systems, is dictated by the specifics of each question being asked. Mathematical models are a useful tool and a model's details and level of complexity depend on the nature of the questions being asked. The overall abstract structure of a mathematical model can be simple, but the consequences for the population dynamics can be rich and have captured the imagination of many generations of scientists and mathematicians.

In some biological populations, such as most animal and plant species, population growth is a discrete-process, and discrete-time models are more appropriate in studying their population dynamics. However, simple discrete-time population models are capable of generating complex dynamics such as period-doubling bifurcations route to chaos, multiple attractors with fractal basin boundaries, strange or chaotic attractors, and so on. This workshop will use population models in ecology and epidemiology to motivate discrete-dynamical system concepts. The following topics will be covered during the workshop.

1. Linear population models · Introduction to Population Models · Linear Population Models: Geometric Growth · A Simple Death or Extinction Process · Simple Population Models With Migration · Leslie Matrix Model 2. Nonlinear Population Models · Reproduction Function & Life-History Dynamics · Equilibrium Population Sizes · Local Stability of Equilibrium Population Sizes · Graphical Study of Population Life-History Dynamics: Cobwebbing · Population Cycles · Global Stability 3. Compensatory and Overcompensatory Dynamics · Intraspecific Competition with migration · Ricker's model · Period-Doubling Bifurcations Route to Chaos · Period Three Population Cycles · Chaos · Allee effect in population models 4. Connections to Epidemics · S-I-S Epidemic models · Asymptotically bounded growth · Geometric growth · Bistability · Epidemics on attractors 5. Interactions · Nicholson-Bailey's Model · Stability of Equilibrium of Systems of Two Equations · Density Dependence In Nicholson-Bailey's Population Model · S-E-I-S Epidemic Model 6. Competition models · Lotka-Volterra Model · Coexistence Versus Extinction · Models With Mixed Competitive Local Regimes · Hierarchical Models 7. Age-structured models with Complex life-history of evolution · LPA model · Size-structured models 8. Metapopulation Dynamics · Local Patch Dynamics · Metapopulation Model · Compensatory Local Dynamics Connected Via Dispersal · Overcompensatory Local Dynamics Connected Via Dispersal · Nature of Basins of Attraction · Mixed Compensatory-Overcompensatory Systems 9. Interspecific and Intraspecific Competition in Patchy Environment · Exclusion Principles · Persistence 10. Population Models in Periodic Environments · Periodically forced Beverton-Holt model · Periodically forced Ricker Model · Attenuant and resonant cycles · Periodically forced epidemic models

[1] Hastings, A., "Complex interactions between dispersal and dynamics: Lessons from coupled logistic equations," *Ecology*, **75** (1993), 1362-1372.

[2] Levin, S.A., Grenfell, B., Hastings, A., and Perelson, A.S., "Mathematical and computational challenges in population biology and ecosystems science," *Science*, **17** (1997), 334-343.

[3] May, R.M., *Stability and Complexity in Model Ecosystems*, Princeton University Press, Princeton, NJ, (1974).

[4] May, R.M., "Simple mathematical models with very complicated dynamics," *Nature*, **261** (1977), 459-469.

[5] May, R.M., Hassell, M.P., Anderson, R.M., and Tonkyn, D.W., "Density dependence in host-parasitoid models," *J. Anim. Ecol.*, **50** (1981), 855-865.

[6] May, R.M. and Oster, G.F., "Bifurcations and dynamic complexity in simple ecological models," *Amer. Naturalist*, **110** (1976), 573-579.

[7] Nicholson, A.J., "Compensatory reactions of populations to stresses, and their evolutionary significance," *Aust. J. Zool.*, **2** (1954), 1-65.

[8]Nusse, H.E., and Yorke, J.A., *Dynamics: Numerical Explorations*, Springer-Verlag, New York, (1997).

[9] Ricker, W.E., "Stock and recruitment," *Journal of Fisheries Research Board of Canada II*, **5** (1954), 559-623.

Back to Main Page