zbMATH — the first resource for mathematics

A non-local PDE model for population dynamics with state-selective delay: local theory and global attractors. (English) Zbl 1082.92039
Summary: We propose a non-local PDE model for the evolution of a single species population that involves delayed feedback, where the delay such as the maturation time in the delayed birth rate, is selective and the selection depends on the status of the system. This delay selection, in contrast with the usual state-dependent delay widely used in ordinary delay differential equation, ensures the Lipschitz continuity of the nonlinear functional in the classical phase space. We also develop the local theory, and the existence and upper semi-continuity of the global attractor with respect to parameters.

92D25 Population dynamics (general)
35B40 Asymptotic behavior of solutions to PDEs
37N25 Dynamical systems in biology
47N20 Applications of operator theory to differential and integral equations
47D03 Groups and semigroups of linear operators
Full Text: DOI
[1] Aiello, W.G.; Freedmand, H.I.; Wu, J.H., Analysis of a model representing state-structured population growth with state-dependent time delay, SIAM J. appl. math., 52, 855-869, (1992) · Zbl 0760.92018
[2] Arino, O.; Hadeler, K.P.; Hbid, M.L., Existence of periodic solutions for delay differential equations with state dependent delay, J. differential equations, 144, 263-301, (1998) · Zbl 0913.34057
[3] Arino, O.; Hbid, M.L.; Bravo de la Parra, R., A mathematical model of growth of population of fish in the larval stage: density-dependence effects, Math. biosci., 150, 1-20, (1998) · Zbl 0938.92028
[4] Babin, A.V.; Vishik, M.I., Attractors of evolutionary equations, (1992), Amsterdam North-Holland, Amsterdam · Zbl 0778.58002
[5] Bélair, J.; Mackey, M., Consumer memory and price fluctuation in commodity markets: an integrodifferential model, J. dynamics differential equations, 19, 299-325, (1989) · Zbl 0682.34050
[6] Boutet de Monvel, L.; Chueshov, I.D.; Rezounenko, A.V., Inertial manifolds for retarded semilinear parabolic equations, Nonlinear anal., 34, 907-925, (1998) · Zbl 0954.34064
[7] Britton, N.F., Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. appl. math., 50, 1663-1688, (1990) · Zbl 0723.92019
[8] M. Büger, M.R.W. Martin, Stabilizing control for an unbounded state-dependent delay differential equation, in: Dynamical Systems and Differential Equations, Kennesaw (GA), 2000, Discret. Contin. Dyn. S. (Added Volume) 2001, 56-65.
[9] Cao, Y.; Wu, J., Projective ART for clustering data sets in high dimensional spaces, Neural networks, 15, 105-120, (2002)
[10] Y. Cao, J. Wu, Dynamics of projective adaptive resonance theory model: the foundation of PART algorithm, IEEE Trans. Neural Networks, 15 (2004) 245-260.
[11] I.D. Chueshov, A problem on non-linear oscillations of shallow shell in a quasistatic formulation, Math. Notes, vol. 47, 1990, pp. 401-407. · Zbl 0721.73028
[12] I.D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999 (in Russian) (English transl. Acta, Kharkov (2002)). · Zbl 1100.37046
[13] Chueshov, I.D.; Rezounenko, A.V., Global attractors for a class of retarded quasilinear partial differential equations, C. R. acad. sci. Paris, ser. I, 321, 607-612, (1995), (detailed version: Math. Phys. Anal. Geom. 2(3) (1995) 363-383) · Zbl 0845.35129
[14] Cooke, K.; Huang, W., On the problem of linearization for state-dependent delay differential equations, Proc. amer. math. soc., 124, 1417-1426, (1996) · Zbl 0844.34075
[15] de Roos, A.M.; Persson, L., Competition in size-structured populations: mechanisms inducing cohort formation and population cycles, Theoret. population biol., 63, 9, 1-16, (2003) · Zbl 1101.92322
[16] Diekmann, O.; van Gils, S.; Verduyn Lunel, S.; Walther, H.-O., Delay equations: functional, complex, and nonlinear analysis, (1995), Springer New York · Zbl 0826.34002
[17] Driver, R.D., A two-body problem of classical electrodynamics: the one-dimensional case, Ann. phys., 21, 122-142, (1963) · Zbl 0108.40705
[18] Eurich, C.W.; Ernst, U.; Pawelzik, K.; Cowan, J.D.; Milton, J.G., Dynamics of self-organized delay adaptation, Phys. rev. lett., 82, 1594-1597, (1999)
[19] Eurich, C.W.; Mackey, M.C.; Schwegler, H., Recurrent inhibitory dynamics: the role of state-dependent distributions of conduction delay times, J. theoret. biol., 216, 31-50, (2002)
[20] Gatica, J.A.; Waltman, P., A threshold model of antigen dynamics with fading memory, (), 425-439 · Zbl 0529.92005
[21] Gourley, S.A.; Bartuccelli, M.V., Parameter domains for instability of uniform states in systems with many delays, J. math. biol., 35, 843-867, (1997) · Zbl 0878.92001
[22] Gourley, S.A.; Britton, N.F., A predator prey reaction diffusion system with nonlocal effects, J. math. biol., 34, 297-333, (1996) · Zbl 0840.92018
[23] S.A. Gourley, Y. Kuang, Wavefronts and global stability in a time-delayed population model with stage structure, Proc. Roy. Soc. London Ser. A., 459 (2003) 1563-1579. · Zbl 1047.92037
[24] Gourley, S.A.; Ruan, S., Dynamics of the diffusive Nicholson’s blowflies equation with distributed delays, Proc. roy. soc. Edinburgh A, 130, 1275-1291, (2000) · Zbl 0973.34064
[25] Gourley, S.; So, J.; Wu, J., Non-locality of reaction – diffusion equations induced by delay: biological modeling and nonlinear dynamics, (), 84-120
[26] Halany, H.; Yorke, J.A., Some new results and problems in the theory of differential-delay equations, SIAM rev., 13, 1, 55-80, (1971) · Zbl 0216.11902
[27] Hale, J.K., Theory of functional differential equations, (1977), Springer Berlin, Heidelberg, New York · Zbl 0425.34048
[28] Hale, J.K.; Verduyn Lunel, S.M., Theory of functional differential equations, (1993), Springer New York · Zbl 1052.93028
[29] F.C. Hoppensteadt, P. Waltman, A flow mediated control model or respiration, Lectures on Mathematics in the Life Sciences, vol. 12, Amer. Math. Soc., Providence, RI, 1979. · Zbl 0415.92006
[30] J. Hunter, J.G. Milton, J. Wu, 2004, in progress.
[31] Kapitanskii, L.V.; Kostin, I.N., Attractors of nonlinear evolution equations and their approximations, Leningrad math. J., 2, 97-117, (1991) · Zbl 0724.35049
[32] Krisztin, T., An unstable manifold near a hyperbolic equilibrium for a class of differential equations with state-dependent delay, Discret. contin. dyn. S., 9, 993-1028, (2003) · Zbl 1048.34123
[33] Krisztin, T.; Arino, O., The 2-dimensional attractor of a differential equation with state-dependent delay, J. dynamics differential equations, 13, 453-522, (2001) · Zbl 1016.34075
[34] T. Krisztin, H.-O. Walther, J. Wu, Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback, Fields Institute Monographs, vol. 11, AMS, Providence, RI, 1999. · Zbl 1004.34002
[35] Kuang, Y.; Smith, H.L., Periodic solutions of differential delay equations with threshold-type delays, (), 153-176 · Zbl 0762.34044
[36] Kuang, Y.; Smith, H.L., Slowly oscillating periodic solutions of autonomous state-dependent delay differential equations, Nonlinear anal., 19, 855-872, (1992) · Zbl 0774.34054
[37] Lions, J.L., Quelques Méthodes de Résolution des problèmes aux limites non linéaires, (1969), Dunod Paris · Zbl 0189.40603
[38] Mahaffy, J.; Belair, J.; Mackey, M., Hematopoietic model with moving boundary condition and state-dependent delay: applications in erythropoiesis, J. theoret. biol., 190, 135-146, (1998)
[39] Mallet-Paret, J.; Nussbaum, R.D., Boundary layer phenomena for differential-delay equations with state-dependent time lags I, Arch. rational mech. anal., 120, 99-146, (1992) · Zbl 0763.34056
[40] Mallet-Paret, J.; Nussbaum, R.D., Boundary layer phenomena for differential-delay equations with state-dependent time lags II, J. reine angew. math., 477, 129-197, (1996) · Zbl 0854.34072
[41] Mallet-Paret, J.; Nussbaum, R.D.; Paraskevopoulos, P., Periodic solutions for functional-differential with multiple state-dependent time lags, Topol. methods nonlinear anal., 3, 1, 101-162, (1994) · Zbl 0808.34080
[42] A.V. Rezounenko, On singular limit dynamics for a class of retarded nonlinear partial differential equations, Mat. Fiz. Anal. Geom.4 (1/2) (1997) C 193-211. · Zbl 0904.35095
[43] Rezounenko, A.V., A short introduction to the theory of ordinary delay differential equations, lecture notes, (2004), Kharkov University Press Kharkov
[44] Smith, H.; Thieme, H., Strongly order preserving semiflows generated by functional differential equations, J. differential equations, 93, 332-363, (1991) · Zbl 0735.34065
[45] So, J.W.-H.; Wu, J.; Yang, Y., Numerical steady state and Hopf bifurcation analysis on the diffusive Nicholson’s blowflies equation, Appl. math. comput., 111, 1, 33-51, (2000) · Zbl 1028.65138
[46] So, J.W.-H.; Wu, J.; Zou, X., A reaction diffusion model for a single species with age structure. I. travelling wavefronts on unbounded domains, Proc. roy. soc. London A, 457, 1841-1853, (2001) · Zbl 0999.92029
[47] So, J.W.-H.; Yang, Y., Dirichlet problem for the diffusive Nicholson’s blowflies equation, J. differential equations, 150, 2, 317-348, (1998) · Zbl 0923.35195
[48] Temam, R., Infinite dimensional dynamical systems in mechanics and physics, (1988), Springer Berlin, Heidelberg, New York · Zbl 0662.35001
[49] Travis, C.C.; Webb, G.F., Existence and stability for partial functional differential equations, Trans. AMS, 200, 395-418, (1974) · Zbl 0299.35085
[50] Unnikrishnan, K.P.; Hopfield, J.J.; Tank, D.W., Connected-digit speaker dependent speech recognition using a neural network with time-delayed connections, IEEE trans. signal process., 39, 698-713, (1991)
[51] Walther, H.-O., Stable periodic motion of a system with state dependent delay, Differential integral equations, 15, 923-944, (2002) · Zbl 1034.34085
[52] Walther, H.-O., The solution manifold and \(C^1\)-smoothness for differential equations with state-dependent delay, J. differential equations, 195, 1, 46-65, (2003) · Zbl 1045.34048
[53] Wu, J., Theory and applications of partial functional differential equations, (1996), Springer New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.