zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The dynamics of hierarchical age-structured populations. (English) Zbl 0823.92018

An age hierarchical population was defined as one in which the birth and death rates of an individual of age a are dependent upon the size of the class of individuals younger than a and/or the size of the class of individuals older than a. Under this assumption a proof of the existence and uniqueness of a solution of the McKendrick model equations was given which yields a decoupled ordinary differential equation of the total population size. Thus, for this class of model populations our results provide a means by which the population level dynamics can be related to individual (age specific) vital rates.

Moreover, because the equation for total population size is a scalar ordinary differential equation, these results also provide the possibility that the global asymptotic dynamics of the population can be determined. For example, in the autonomous case one of our theorems implies that only equilibrium dynamics are possible and shows how the asymptotic age distributions can be determined. Some examples illustrate how these kinds of models can be used to address certain interesting problems concerning intra-specific competition and prediction.

MSC:
92D25Population dynamics (general)
34D05Asymptotic stability of ODE
34E99Asymptotic theory of ODE
References:
[1]Botsford, L. W.: The effects of increased individual growth rates on depressed population size. Am. Nat. 117, 38-63 (1981) · doi:10.1086/283685
[2]Busenberg, S. N., Iannelli, M.: Separable models in age-dependent population dynamics. J. Math. Biol. 22, 145-173 (1985) · Zbl 0593.92010 · doi:10.1007/BF00276489
[3]Costantino, R. F., Desharnais, R. A.: Population Dynamics and the Tribolium Model: Genetics and Demography, Monographs on Theoretical and Applied Genetics 13. Berlin, Heidelberg, New York: Springer 1991
[4]Cushing, J. M.: A simple model of cannibalism. Math. Biosci. 107(1), 47-71 (1991) · Zbl 0738.92015 · doi:10.1016/0025-5564(91)90071-P
[5]Cushing, J. M.: A size-structured model for cannibalism. Theor. Popul. Biol. 42(3), 347-361 (1992) · Zbl 0768.92020 · doi:10.1016/0040-5809(92)90020-T
[6]Cushing, J. M., Li, J.: On Ebenman’s model for the dynamics of a population with competing juveniles and adults. Bull. Math. Biol. 51, 687-713 (1989)
[7]Cushing, J. M., Li, J.: Juvenile versus adult competition. J. Math. Biol. 29, 457-473 (1991) · Zbl 0727.92022 · doi:10.1007/BF00160472
[8]Cushing, J. M., Li, J.: Intra-specific competition and density dependent juvenile growth. Bull. Math. Biol. 54(4), 503-519 (1992)
[9]Ebenman, B.: Niche differences between age classes and intraspecific competition in age-structured populations. J. Theor. Biol. 124, 25-33 (1987) · doi:10.1016/S0022-5193(87)80249-7
[10]Ebenman, B.: Competition between age classes and population dynamics. J. Theor. Biol. 131, 389-400 (1988) · doi:10.1016/S0022-5193(88)80036-5
[11]Ebenman, B.: Dynamics of age- and size-structured populations: intraspecific competition, Size-Structured Populations, pp. 127-139 Ebenman and Persson (eds.). Berlin, Heidelberg, New York: Springer 1988
[12]Ebenman, Bo, Persson, L: Size-Structured Populations: Ecology and Evolution. Berlin, Heidelberg, New York: Springer 1988
[13]Elgar, M. A., Crespi, B. J.: Cannibalism: Ecology and Evolution among Diverse Taxa. Oxford: Oxford University Press 1992
[14]Fox, Laurel, R.: Cannibalism in natural populations. Ann. Rev. Ecol. Syst. 6, 87-106 (1975) · doi:10.1146/annurev.es.06.110175.000511
[15]Gurtin, M., MacCamy R. C.: Nonlinear age-dependent population dynamics. Arch. Rat. Mech. 54, 281-300 (1974) · Zbl 0286.92005 · doi:10.1007/BF00250793
[16]Gurtin, M., MacCamy, R. C.: Some simple models for nonlinear age-dependent population dynamics. Math. Biosci. 43, 199-211; 213-237 (1979) · Zbl 0397.92025 · doi:10.1016/0025-5564(79)90049-X
[17]Gurney, W. S. C., Nisbet, R. M.: Ecological stability and social hierarchy. Theor. Popul. Biol. 16, 48-80 (1979) · Zbl 0417.92024 · doi:10.1016/0040-5809(79)90006-6
[18]Hastings, A.: Cycles in cannibalistic egg-larval interactions. J. Math. Biol. 24, 651-666 (1987)
[19]Hale, J., Kocak, H.: Dynamics and Bifurcations, Texts in Applied Mathematics 3, Berlin, Heidelberg, New York: Springer 1991
[20]Hoppensteadt, Frank: Mathematical Theories of Populations: Demographics, Genetics and Epidemics, Reg. Conf. Series in Appl. Math., SIAM, Philadelphia, 1975
[21]Lomnicki, A: Population Ecology of Individuals, Monographs in Population Biology 25, Princeton University Press, Princeton, New Jersey, 1988
[22]Lomnicki, A., Ombach, J.: Resource partitioning within a single species population and population stability: a theoretical model. Theor. Popul. Biol. 25, 21-28 (1984) · Zbl 0533.92020 · doi:10.1016/0040-5809(84)90003-0
[23]Loreau, M.: Competition between age classes, and the stability of stage-structured populations: a re-examination of Ebenman’s model, J. Theor. Biol. 144, 567-571 (1990) · doi:10.1016/S0022-5193(05)80090-6
[24]May, R. M., Conway, G. R., Hassell, M. P., Southwood, T. R. E.: Time delays, density-dependence and single-species oscillations. J. Anim. Ecol. 43, 747-770 (1974) · Zbl 02319030 · doi:10.2307/3535
[25]Metz, J. A. J., Diekmann, O.: The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics 68. Berlin, Heidelberg, New York: Springer 1986
[26]Polis, Gary, A.: The evolution and dynamics of intraspecific predation, Ann. Rev. Ecol. Syst. 12, 225-251 (1981) · doi:10.1146/annurev.es.12.110181.001301
[27]Simmes, Stephen, D.: Age dependent population dynamics with non-linear interactions, Ph.D. dissertation, Carnegi-Mellon University Pittsburgh, 1978
[28]Tschumy, W. O.: Competition between juveniles and adults in age-structured populations. Theor. Pop. Biol. 21, 255-268 (1982) · Zbl 0519.92022 · doi:10.1016/0040-5809(82)90017-X
[29]Tucker, S. L., Zimmerman, S. O.: A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables, SIAM J. Appl. Math. 48(3), 549-591 (1988) · Zbl 0657.92011 · doi:10.1137/0148032
[30]van den Bosch, F., de Roos, A. M., Gabriel, W.: Cannibalism as a life boat mechanism. J. Math. Biol. 26, 619-633 (1988)
[31]Webb, G. F.: Theory of Nonlinear Age-dependent Population Dynamics. New York: Marcel Dekker, Inc. 1985
[32]Werner, E. E., Gilliam, J. F.: The onogenetic niche and species interactions in size-structured populations. Ann. Rev. Ecol. Syst. 15, 393-425 (1984) · doi:10.1146/annurev.es.15.110184.002141