Numerical solution of structured population models

Numerical methods are presented for a general mass-structured population model with demographic rates that depend on individual mass, time, and total population mass. Several types of numerical methods are described, Eulerian methods, implicit methods, and the method of characteristics. These methods are compared for a sample problem with an exact solution. The preferred numerical technique combines the method of characteristics with an adaptive grid. Numerical solution of model equations developed for mosquitofish illustrates this method and demonstrates how seasons can play a dominant role in shaping population development.

[1]  Alfred J. Lotka,et al.  A Problem in Age-Distribution , 1911 .

[2]  O. Diekmann,et al.  The Dynamics of Physiologically Structured Populations , 1986 .

[3]  R. D. Richtmyer,et al.  Difference methods for initial-value problems , 1959 .

[4]  A. Nicholson.  An outline of the dynamics of animal populations. , 1954 .

[5]  Louis W. Botsford,et al.  Optimal fishery policy for size-specific, density-dependent population models , 1981 .

[6]  James W. Sinko,et al.  A New Model For Age‐Size Structure of a Population , 1967 .

[7]  W. Hager.  Applied Numerical Linear Algebra , 1987 .

[8]  L. Botsford.  More Realistic Fishery Models: Cycles Collapse and Optimal Policy , 1981 .

[9]  J. Harper,et al.  The concept of population in modular organisms , 1981 .

[10]  A. Nicholson,et al.  The Self-Adjustment of Populations to Change , 1957 .

[11]  William Gurney,et al.  The systematic formulation of tractable single-species models incorporating age structure , 1983 .

[12]  T. Hughes,et al.  Population Dynamics Based on Individual Size Rather than Age: A General Model with a Reef Coral Example , 1984, The American Naturalist.

[13]  John VanSickle,et al.  Analysis of a distributed-parameter population model based on physiological age. , 1977 .

[14]  Louis A. Krumholz,et al.  Reproduction in the Western Mosquitofish, Gambusia affinis affinis (Baird & Girard), and Its Use in Mosquito Control , 1948 .

[15]  J. Lambert.  Computational Methods in Ordinary Differential Equations , 1973 .

[16]  E. Werner,et al.  THE ONTOGENETIC NICHE AND SPECIES INTERACTIONS IN SIZE-STRUCTURED POPULATIONS , 1984 .

[17]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[18]  J. Hyman,et al.  Using mathematical models to understand the AIDS epidemic , 1988 .

[19]  J. Brackbill,et al.  Adaptive zoning for singular problems in two dimensions , 1982 .

[20]  L. Murphy.  A nonlinear growth mechanism in size structured population dynamics , 1983 .

[21]  A. Nicholson,et al.  Compensatory reactions of populations to stresses, and their evolutionary significance. , 1954 .

[22]  S. P. Blythe,et al.  Nicholson's blowflies revisited , 1980, Nature.

[23]  G H Weiss,et al.  Equations for the age structure of growing populations. , 1968, The Bulletin of mathematical biophysics.

[24]  D. Sulsky,et al.  The demographic meanings of the classical population growth models of ecology , 1988 .

[25]  William Gurney,et al.  Modelling fluctuating populations , 1982 .

[26]  Morton E. Gurtin,et al.  Non-linear age-dependent population dynamics , 1974 .

[27]  G. Webb.  Theory of Nonlinear Age-Dependent Population Dynamics , 1985 .

[28]  D. Sulsky,et al.  Time delays in age-structured populations. , 1989, Journal of theoretical biology.

[29]  R. M. Nisbet,et al.  THE SYSTEMATIC FORMULATION OF TRACTABLE SINGLE-SPECIES POPULATION MODELS , 1983 .

[30]  Patrick J. Roache,et al.  Variational grid generation , 1986 .

[31]  R. B. Hamilton,et al.  The Vital Statistics of an Unexploited Gray Squirrel Population , 1970 .

[32]  Louis W. Botsford,et al.  The Effects of Increased Individual Growth Rates on Depressed Population Size , 1981, The American Naturalist.

[33]  James M. Hyman,et al.  The Effect of Social Mixing Patterns on the Spread of AIDS , 1989 .

[34]  P. Lax,et al.  Systems of conservation laws , 1960 .

[35]  A. M'Kendrick.  Applications of Mathematics to Medical Problems , 1925, Proceedings of the Edinburgh Mathematical Society.

[36]  S. Osher,et al.  Uniformly High-Order Accurate Nonoscillatory Schemes. I , 1987 .

[37]  J. Goudriaan.  Boxcartrain methods for modelling of ageing, development, delays and dispersion. , 1986 .