Asymptotic Stability of an Abstract Delay Functional-Differential Equation

We study the exponential asymptotic stability of an abstract functional-differential equation with a mixed type of deviating arguments, namely: c which might represent the gestation period of the population and r(u(t)), a suitably defined function. The equation is reduced to its equivalent integral form and solved via Laplace transform method. An interesting connection of our study is with generalizations of populations with potentially complex (chaotic) dynamics

Age-dependent diffusive Lotka-Volterra type systems

In this paper it is shown that the sub-supersolution method works for age-dependent diffusive nonlinear systems with non-local initial conditions. As application, we prove the existence and uniqueness of positive solution for a kind of Lotka-Volterra systems, as well as the blow-up in finite time in a particular case.Ministerio de Ciencia y Tecnologí

A Stochastic Model for Population and Well-Being Dynamics

This article presents a stochastic dynamic model to study the demographic evolution per sexes and the corresponding well-being of a general human population. The main model variables are population per sexes and well-being. The considered well-being variable is the Gender-Related Development Index (GDI), a United Nations index. The model's objectives are to improve future well-being and to reach a stable population in a country. The application case consists of adapting, validating, and using the model for Spain in the 2000–2006 period. Some instance strategies have been tested in different scenarios for the 2006–2015 period to meet these objectives by calculating the reliability of the results. The optimal strategy is “government invests more in education and maintains the present health investment tendency.”Sanz, MT.; Micó Ruiz, JC.; Caselles, A.; Soler Fernández, D. (2014). A Stochastic Model for Population and Well-Being Dynamics. Journal of Mathematical Sociology. 38(2):75-94. doi:10.1080/0022250X.2011.629064S7594382Alho , J. M. & Spencer , B. D. ( 2005 ).Statistical demography and forecasting(pp. 166 – 193 ). Berlin , Germany : Springer .Almeder , C. ( 2004 ). Solution methods for age-structured optimal control models with feedback . In I. Lirkov , S. Margenov , J. Wasniewski , & P. Yalamov (Eds.),Large-scale scientific computing(Lecture Notes in Computer Science, Vol. 2907 , pp. 197–203). Berlin , Germany : Springer .Anand , S. & Sen , A. ( 1994 ).Human development index: Methodology and measurement(Human Development Report Office Occasional Paper 12). New York , NY : Human Development Report Office .Park, E. J., Iannelli, M., Kim, M. Y., & Anita, S. (1998). Optimal Harvesting for Periodic Age-Dependent Population Dynamics. SIAM Journal on Applied Mathematics, 58(5), 1648-1666. doi:10.1137/s0036139996301180Bacaër, N., Abdurahman, X., & Ye, J. (2006). Modeling the HIV/AIDS Epidemic Among Injecting Drug Users and Sex Workers in Kunming, China. Bulletin of Mathematical Biology, 68(3), 525-550. doi:10.1007/s11538-005-9051-yBarbu, V., Iannelli, M., & Martcheva, M. (2001). On the Controllability of the Lotka–McKendrick Model of Population Dynamics. Journal of Mathematical Analysis and Applications, 253(1), 142-165. doi:10.1006/jmaa.2000.7075CASELLES, A. (1992). STRUCTURE AND BEHAVIOR IN GENERAL SYSTEMS THEORY. Cybernetics and Systems, 23(6), 549-560. doi:10.1080/01969729208927481CASELLES, A. (1993). SYSTEMS DECOMPOSITION AND COUPLING. Cybernetics and Systems, 24(4), 305-323. doi:10.1080/01969729308961712CASELLES, A. (1994). IMPROVEMENTS IN THE SYSTEMS-BASED MODELS GENERATOR SIGEM. Cybernetics and Systems, 25(1), 81-103. doi:10.1080/01969729408902317Caselles , A. , Micó , J. C. , Soler , D. & Sanz , M. T. ( 2008 ). Population growth and social well-being: A dynamic model approach. In Associação Portuguesa de Complexidade Sistémica (Ed.),Proceedings of the 7th Congress of the UES (Systems Science European Union). Lisbon, Portugal: Associação Portuguesa de Complexidade Sistémica. Retrieved from http://www.afscet.asso.fr/resSystemica/Lisboa08/entete08.htm .Caswell, H., & Weeks, D. E. (1986). Two-Sex Models: Chaos, Extinction, and Other Dynamic Consequences of Sex. The American Naturalist, 128(5), 707-735. doi:10.1086/284598Chowdhury, M., & Allen, E. J. (2001). A stochastic continuous-time age-structured population model. Nonlinear Analysis: Theory, Methods & Applications, 47(3), 1477-1488. doi:10.1016/s0362-546x(01)00283-8Clemons, C. B., Hariharan, S. I., & Quinn, D. D. (2001). Amplitude Equations for Time-Dependent Solutions of the McKendrick Equations. SIAM Journal on Applied Mathematics, 62(2), 684-705. doi:10.1137/s003613990037813xDjidjeli, K., Price, W. G., Temarel, P., & Twizell, E. H. (1998). Partially implicit schemes for the numerical solutions of some non-linear differential equations. Applied Mathematics and Computation, 96(2-3), 177-207. doi:10.1016/s0096-3003(97)10133-3Farkas, J. Z. (2004). Stability conditions for the non-linear McKendrick equations. Applied Mathematics and Computation, 156(3), 771-777. doi:10.1016/j.amc.2003.06.019Feichtinger, G., Tragler, G., & Veliov, V. M. (2003). Optimality conditions for age-structured control systems. Journal of Mathematical Analysis and Applications, 288(1), 47-68. doi:10.1016/j.jmaa.2003.07.001Guo, B.-Z., & Sun, B. (2005). Numerical solution to the optimal birth feedback control of a population dynamics: viscosity solution approach. Optimal Control Applications and Methods, 26(5), 229-254. doi:10.1002/oca.759Hagerty, M. R., & Land, K. C. (2007). Constructing Summary Indices of Quality of Life. Sociological Methods & Research, 35(4), 455-496. doi:10.1177/0049124106292354Inaba, H. (2001). Kermack and McKendrick revisited: The variable susceptibility model for infectious diseases. Japan Journal of Industrial and Applied Mathematics, 18(2), 273-292. doi:10.1007/bf03168575KIM, M.-Y. (2006). DISCONTINUOUS GALERKIN METHODS FOR THE LOTKA–MCKENDRICK EQUATION WITH FINITE LIFE-SPAN. Mathematical Models and Methods in Applied Sciences, 16(02), 161-176. doi:10.1142/s0218202506001108Land, K. C., Yang, Y., & Zeng, Y. (s. f.). Mathematical Demography. Handbook of Population, 659-717. doi:10.1007/0-387-23106-4_23Letellier, C., Elaydi, S., Aguirre, L. A., & Alaoui, A. (2004). Difference equations versus differential equations, a possible equivalence for the Rössler system? Physica D: Nonlinear Phenomena, 195(1-2), 29-49. doi:10.1016/j.physd.2004.02.007Marchetti, C., Meyer, P. S., & Ausubel, J. H. (1996). Human population dynamics revisited with the logistic model: How much can be modeled and predicted? Technological Forecasting and Social Change, 52(1), 1-30. doi:10.1016/0040-1625(96)00001-7MICÓ, J. C., SOLER, D., & CASELLES, A. (2006). Age-Structured Human Population Dynamics. The Journal of Mathematical Sociology, 30(1), 1-31. doi:10.1080/00222500500323143MICÓ, J. C., CASELLES, A., SOLER, D., SANZ, T., & MARTÍNEZ, E. (2008). A Side-by-Side Single Sex Age-Structured Human Population Dynamic Model: Exact Solution and Model Validation. The Journal of Mathematical Sociology, 32(4), 285-321. doi:10.1080/00222500802352758MISCHLER, S., PERTHAME, B., & RYZHIK, L. (2002). STABILITY IN A NONLINEAR POPULATION MATURATION MODEL. Mathematical Models and Methods in Applied Sciences, 12(12), 1751-1772. doi:10.1142/s021820250200232xMurphy, L. F., & Smith, S. J. (1991). Maximum sustainable yield of a nonlinear population model with continuous age structure. Mathematical Biosciences, 104(2), 259-270. doi:10.1016/0025-5564(91)90064-pNoymer, A. (2001). The transmission and persistence of ‘urban legends’: Sociological application of age‐structured epidemic models. The Journal of Mathematical Sociology, 25(3), 299-323. doi:10.1080/0022250x.2001.9990256Patten, S. B. (1999). Epidemics of violence. Medical Hypotheses, 53(3), 217-220. doi:10.1054/mehy.1998.0748Pollak, R. A. (1986). A Reformulation of the Two-Sex Problem. Demography, 23(2), 247. doi:10.2307/2061619Pollak, R. A. (1990). Two-Sex Demographic Models. Journal of Political Economy, 98(2), 399-420. doi:10.1086/261683Schoen, R. (1988). Modeling Multigroup Populations. The Springer Series on Demographic Methods and Population Analysis. doi:10.1007/978-1-4899-2055-3Segarra, J., Jeger, M. J., & van den Bosch, F. (2001). Epidemic Dynamics and Patterns of Plant Diseases. Phytopathology, 91(10), 1001-1010. doi:10.1094/phyto.2001.91.10.1001Takada, T., & Caswell, H. (1997). Optimal Size at Maturity in Size-Structured Populations. Journal of Theoretical Biology, 187(1), 81-93. doi:10.1006/jtbi.1997.042