Abstract
Biological systems can be subjected to short-term fluctuations in the environment or in the character of the system. This paper models such dynamics by evolution equations with impulses. An appropriate setting is a space of Banach space valued functions of bounded variation. Some general properties, such as existence and uniqueness, are considered, along with some dynamical system concepts of impulsive evolution equations. The study is motivated by, and refers to an example of a continuously age-distributed population subjected to impulses from time to time.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Z. Artstein, 1977. Topological dynamics of ordinary differential and Kurzweil euations, J. Differential Equations 23, pp. 224–243.
Z. Artstein, 1977. The limiting equations of nonautonomous ordinary differential equations, J. Differential Equations 25, pp. 184–20.
J.M. Ball, 1977. Strongly continuous semigroups, weak solutions and the variation of constants formula, Proc. Amer. Math. Soc. 63, pp. 370–373.
T.S. Chew, 1988. On Kurzweil generalized ordinary differential equations, J. Diff. Eqs. 76, pp. 286–293.
L.M. Graves, 1946. The Theory of Functions of Real Variables, McGraw-Hill, New York.
M.E. Gurtin and R.C. MacCamy, 1974. Nonlinear age-dependent population dynamics, Arch. Rat. Mech. Anal. 54, pp. 281–300.
J. Kurzweil, 1958. Generalized ordinary differential equations, Czechoslovak Math. J. 8, pp. 360–389.
G. Oster and J. Guckenheimer, 1975. Bifurcation Phenomena in population models, in J.E. Marsden and M. McCracken (Eds.), The Hopf Bifurcation and its Applications, Springer-Verlag, New York
G. Oster and Y. Takahashi, 1974. Models for age-specific interactions in a periodic environment, Ecol. Monogr. 44, pp. 483–501.
S.G. Pandit and S.G. Deo, 1982. Differential Systems Involving Impulses, Lecture Notes in Mathematics No. 954, Springer-Verlag, Berlin.
A. Pazy, 1983. Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York.
J. Pruss, 1983. On the qualitative behaviour of populations with age-specific interactions, Comp, and Maths, with Appls. 9, pp. 327–339.
G.F. Webb, 1985. Theory of nonlinear age-dependent population dynamics, Marcel Dekker, New York.
N.U. Ahmed, 1983. Properties of relaxed trajectories for a class of nonlinear evolution equations on a Banach space, SIAM J. Optim. Control 21, pp. 953–967.
S. Bochner and A.E. Taylor, 1983. Linear functional on certain spaces of abstractly-valued functions, Ann. Math. 39, pp. 913–944.
L. Cesari, 1988. Discontinuous optimal solutions and applications, in K.L. Teo et al (Eds), International Conference on Optimization Techniques and Applications, Singapore, 8–10 April, 1987, University of Singapore pp. 107–121.
J.A. Clarkson, 1936. Uniformly convex spaces, Trans. Amer. Math. Soc. 40, pp. 396–414.
K. Kuratowski. 1966. Topology. Academic Press, New York.
J. Kurzweil, 1957. Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 7, pp. 418–446.
J.P. LaSalle, 1976. Stability theory and invariance principles, in L. Cesari, J.K. Hale and J.P. LaSalle (Eds), Dynamical Systems, Vol. 1, Academic Press, New York pp. 211–222.
R.M. McLeod, 1980. The Generalized Riemann Integral, The Mathematical Association of America, Carus Mathematical Monographs.
E.J. McShane. 1983. Unified Integration. Academic Press, New York.
S.D. Milusheva and D.D. Bainov, 1982. Justification of the averaging method for a class of functional differential equations with impulses, J. Lond. Math. Soc. 25, pp. 309–331.
N.H. Pavel, 1987. Nonlinear Evolution Operators and Semigroups, Lecture Notes in Mathematics 1260. Springer-Verlag, Berlin.
S. Schwabik, 1985. Generalized Differential Equations, Rozpravy Ceskoslovenske Akademie Ved, Prague.
S.H. Saperstone, 1981. Semidynamical Systems in Infinite Dimensional Spaces, Springer-Verlag, New York.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Birkhäuser Boston
About this chapter
Cite this chapter
Diamond, P. (1990). Impulsive Evolution Equations and Population Models. In: Vincent, T.L., Mees, A.I., Jennings, L.S. (eds) Dynamics of Complex Interconnected Biological Systems. Mathematical Modelling, vol 6. Birkhäuser Boston. https://doi.org/10.1007/978-1-4684-6784-0_10
Download citation
DOI: https://doi.org/10.1007/978-1-4684-6784-0_10
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4684-6786-4
Online ISBN: 978-1-4684-6784-0
eBook Packages: Springer Book Archive