DIMACS
The Center for Discrete Mathematics
and Theoretical Computer Science

Reconnect Satellite Conference 2006:
Morgan State University

Reconnecting Teaching Faculty to the Mathematical Sciences Enterprise

July 9 - 15, 2006
(Sunday evening through Saturday afternoon)


Simple and Complex Discrete-time Population Models in Ecology and Epidemiology
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.


Tentative Outline of Topics to be Covered
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
References:

[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.


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