• Previous Article
    Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints
  • DCDS-B Home
  • This Issue
  • Next Article
    A note on the existence and properties of evanescent solutions for nonlinear elliptic problems
October  2014, 19(8): 2641-2656. doi: 10.3934/dcdsb.2014.19.2641

On a nonlinear age-structured model of semelparous species

1. 

Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland

2. 

Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland

Received  November 2013 Revised  April 2014 Published  August 2014

We study a nonlinear age-structured model of a population such that individuals may give birth only at a given age. Properties of measure-valued periodic solutions of this system are investigated. We show that in some cases the age profile of the population tends to a Dirac measure, which means that the population asymptotically consists of individuals at the same age. This phenomenon is observed in nature in some insects populations.
Citation: Ryszard Rudnicki, Radosław Wieczorek. On a nonlinear age-structured model of semelparous species. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2641-2656. doi: 10.3934/dcdsb.2014.19.2641
References:
[1]

M. G. Bulmer, Periodical insects,, The American Naturalist, 111 (1977), 1099.  doi: 10.1086/283240.  Google Scholar

[2]

J. A. Carrillo, R. M. Colombo, P. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws,, J. Differential Equations, 252 (2012), 3245.  doi: 10.1016/j.jde.2011.11.003.  Google Scholar

[3]

P. Cull and A. Vogt, The periodic limit for the Leslie model,, Math. Biosci., 21 (1974), 39.  doi: 10.1016/0025-5564(74)90103-5.  Google Scholar

[4]

J. M. Cushing, Three stage semelparous Leslie models,, J. Math. Biol., 59 (2009), 75.   Google Scholar

[5]

O. Diekmann, N. Davydova and S. van Gils, On a boom and bust year class cycle,, J. Difference Equ. Appl., 11 (2005), 327.  doi: 10.1080/10236190412331335409.  Google Scholar

[6]

R. Dilão and A. Lakmeche, On the weak solutions of the McKendrick equation: Existence of demography cycles,, Math. Model. Nat. Phenom., 1 (2006), 1.  doi: 10.1051/mmnp:2006001.  Google Scholar

[7]

J. Z. Farkas, Stability conditions for the non-linear McKendrick equations,, Appl. Math. Comput., 156 (2004), 771.  doi: 10.1016/j.amc.2003.06.019.  Google Scholar

[8]

J. E. Franke and A.-A. Yakubu, Globally attracting attenuant versus resonant cycles in periodic compensatory Leslie models,, Math. Biosci., 204 (2006), 1.  doi: 10.1016/j.mbs.2006.08.016.  Google Scholar

[9]

G. Gripenberg, S. O. Londen and O. Staffans, Volterra Integral and Functional Equations,, Cambridge University Press, (1990).  doi: 10.1017/CBO9780511662805.  Google Scholar

[10]

M. E. Gurtin and R. C. MacCamy, Non-linear age-dependent populations dynamics,, Arch. Rational Mech. Anal., 54 (1974), 281.   Google Scholar

[11]

P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces,, J. Hyperbolic Differ. Equ., 7 (2010), 733.  doi: 10.1142/S021989161000227X.  Google Scholar

[12]

P. E. Jabin, Large time concentration for solutions to kinetic equations with energy dissipation,, Comm. Partial Differential Equations, 25 (2000), 541.  doi: 10.1080/03605300008821523.  Google Scholar

[13]

P. Michel, A singular asymptotic behavior of a transport equation,, C. R. Math. Acad. Sci. Paris, 346 (2008), 155.  doi: 10.1016/j.crma.2007.12.010.  Google Scholar

[14]

B. Perthame, Transport Equations in Biology,, Frontiers in Mathematics, (2007).   Google Scholar

[15]

S. I. Rubinow, A maturity time representation for cell populations,, Biophy. J., 8 (1968), 1055.  doi: 10.1016/S0006-3495(68)86539-7.  Google Scholar

[16]

R. Rudnicki and R. Wieczorek, Asymptotic analysis of a semelparous species model,, Fund. Inform., 103 (2010), 219.   Google Scholar

[17]

S. K. Tumuluri, Steady state analysis of a nonlinear renewal equation,, Math. Comput. Modelling, 53 (2011), 1420.  doi: 10.1016/j.mcm.2010.02.050.  Google Scholar

show all references

References:
[1]

M. G. Bulmer, Periodical insects,, The American Naturalist, 111 (1977), 1099.  doi: 10.1086/283240.  Google Scholar

[2]

J. A. Carrillo, R. M. Colombo, P. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws,, J. Differential Equations, 252 (2012), 3245.  doi: 10.1016/j.jde.2011.11.003.  Google Scholar

[3]

P. Cull and A. Vogt, The periodic limit for the Leslie model,, Math. Biosci., 21 (1974), 39.  doi: 10.1016/0025-5564(74)90103-5.  Google Scholar

[4]

J. M. Cushing, Three stage semelparous Leslie models,, J. Math. Biol., 59 (2009), 75.   Google Scholar

[5]

O. Diekmann, N. Davydova and S. van Gils, On a boom and bust year class cycle,, J. Difference Equ. Appl., 11 (2005), 327.  doi: 10.1080/10236190412331335409.  Google Scholar

[6]

R. Dilão and A. Lakmeche, On the weak solutions of the McKendrick equation: Existence of demography cycles,, Math. Model. Nat. Phenom., 1 (2006), 1.  doi: 10.1051/mmnp:2006001.  Google Scholar

[7]

J. Z. Farkas, Stability conditions for the non-linear McKendrick equations,, Appl. Math. Comput., 156 (2004), 771.  doi: 10.1016/j.amc.2003.06.019.  Google Scholar

[8]

J. E. Franke and A.-A. Yakubu, Globally attracting attenuant versus resonant cycles in periodic compensatory Leslie models,, Math. Biosci., 204 (2006), 1.  doi: 10.1016/j.mbs.2006.08.016.  Google Scholar

[9]

G. Gripenberg, S. O. Londen and O. Staffans, Volterra Integral and Functional Equations,, Cambridge University Press, (1990).  doi: 10.1017/CBO9780511662805.  Google Scholar

[10]

M. E. Gurtin and R. C. MacCamy, Non-linear age-dependent populations dynamics,, Arch. Rational Mech. Anal., 54 (1974), 281.   Google Scholar

[11]

P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces,, J. Hyperbolic Differ. Equ., 7 (2010), 733.  doi: 10.1142/S021989161000227X.  Google Scholar

[12]

P. E. Jabin, Large time concentration for solutions to kinetic equations with energy dissipation,, Comm. Partial Differential Equations, 25 (2000), 541.  doi: 10.1080/03605300008821523.  Google Scholar

[13]

P. Michel, A singular asymptotic behavior of a transport equation,, C. R. Math. Acad. Sci. Paris, 346 (2008), 155.  doi: 10.1016/j.crma.2007.12.010.  Google Scholar

[14]

B. Perthame, Transport Equations in Biology,, Frontiers in Mathematics, (2007).   Google Scholar

[15]

S. I. Rubinow, A maturity time representation for cell populations,, Biophy. J., 8 (1968), 1055.  doi: 10.1016/S0006-3495(68)86539-7.  Google Scholar

[16]

R. Rudnicki and R. Wieczorek, Asymptotic analysis of a semelparous species model,, Fund. Inform., 103 (2010), 219.   Google Scholar

[17]

S. K. Tumuluri, Steady state analysis of a nonlinear renewal equation,, Math. Comput. Modelling, 53 (2011), 1420.  doi: 10.1016/j.mcm.2010.02.050.  Google Scholar

[1]

Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032

[2]

Cleopatra Christoforou, Myrto Galanopoulou, Athanasios E. Tzavaras. Measure-valued solutions for the equations of polyconvex adiabatic thermoelasticity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6175-6206. doi: 10.3934/dcds.2019269

[3]

Azmy S. Ackleh, Vinodh K. Chellamuthu, Kazufumi Ito. Finite difference approximations for measure-valued solutions of a hierarchically size-structured population model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 233-258. doi: 10.3934/mbe.2015.12.233

[4]

Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020041

[5]

C. Connell McCluskey. Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes. Mathematical Biosciences & Engineering, 2012, 9 (4) : 819-841. doi: 10.3934/mbe.2012.9.819

[6]

Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control & Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017

[7]

John Cleveland. Basic stage structure measure valued evolutionary game model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 291-310. doi: 10.3934/mbe.2015.12.291

[8]

Leonardi Filippo. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 941-961. doi: 10.3934/dcdss.2018056

[9]

Simona Fornaro, Stefano Lisini, Giuseppe Savaré, Giuseppe Toscani. Measure valued solutions of sub-linear diffusion equations with a drift term. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1675-1707. doi: 10.3934/dcds.2012.32.1675

[10]

Odo Diekmann, Yi Wang, Ping Yan. Carrying simplices in discrete competitive systems and age-structured semelparous populations. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 37-52. doi: 10.3934/dcds.2008.20.37

[11]

Claudio Qureshi, Daniel Panario, Rodrigo Martins. Cycle structure of iterating Redei functions. Advances in Mathematics of Communications, 2017, 11 (2) : 397-407. doi: 10.3934/amc.2017034

[12]

Verena Bögelein, Frank Duzaar, Ugo Gianazza. Very weak solutions of singular porous medium equations with measure data. Communications on Pure & Applied Analysis, 2015, 14 (1) : 23-49. doi: 10.3934/cpaa.2015.14.23

[13]

Hiroshi Morishita, Eiji Yanagida, Shoji Yotsutani. Structure of positive radial solutions including singular solutions to Matukuma's equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 871-888. doi: 10.3934/cpaa.2005.4.871

[14]

Young-Pil Choi, Seung-Yeal Ha, Seok-Bae Yun. Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia. Networks & Heterogeneous Media, 2013, 8 (4) : 943-968. doi: 10.3934/nhm.2013.8.943

[15]

Todd Young. Asymptotic measures and distributions of Birkhoff averages with respect to Lebesgue measure. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 359-378. doi: 10.3934/dcds.2003.9.359

[16]

Ke Lin, Chunlai Mu. Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2233-2260. doi: 10.3934/dcdsb.2017094

[17]

Hai-Yang Jin, Tian Xiang. Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes sytstem with competitive kinetics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1919-1942. doi: 10.3934/dcdsb.2018249

[18]

Mario E. Chávez-Gordillo, Bernardo San Martín, Jaime Vera. Persistent singular attractors arising from singular cycle under symmetric conditions. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 671-685. doi: 10.3934/dcds.2011.30.671

[19]

Jacek Banasiak, Eddy Kimba Phongi, MirosŁaw Lachowicz. A singularly perturbed SIS model with age structure. Mathematical Biosciences & Engineering, 2013, 10 (3) : 499-521. doi: 10.3934/mbe.2013.10.499

[20]

Amin Sakzad, Mohammad-Reza Sadeghi, Daniel Panario. Cycle structure of permutation functions over finite fields and their applications. Advances in Mathematics of Communications, 2012, 6 (3) : 347-361. doi: 10.3934/amc.2012.6.347

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]