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June  2020, 25(6): 2331-2349. doi: 10.3934/dcdsb.2020037

An adaptative model for a multistage structured population under fluctuating environment

B. Ainseba 1, and  S. M. Bouguima 2,

1. 

Bordeaux University, IMB UMR 5251, Talence, France
2. 

Tlemcen University, Department of Mathematics, Algeria, labo:Systèmes Dynamiques et applications

Received  February 2019 Revised  June 2019 Published  June 2020 Early access  February 2020

Fund Project: This research was supported by the BIOCASTANEA project (Aquitaine Region and FEDER) and the Cluster SysNum

We consider a modified version of a mathematical model describing the dynamics of the European Grapevine Moth, studied by Ainseba, Picart and Thiery. The improvment consists in including adaptation at the larval stage. We establish well-posedness of the model under suitable hypothesis.

Keywords: European grapevine moth, physiologically structured model, hyperbolic transport, fixed point theorem.
Mathematics Subject Classification: 35L02, 35L45, 35L50, 92D25, 92D40, 93A30.
Citation: B. Ainseba, S. M. Bouguima. An adaptative model for a multistage structured population under fluctuating environment. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : 2331-2349. doi: 10.3934/dcdsb.2020037
References:
[1]

B. E. Ainseba and D. Picart, Parameter identification in multistage poppulation dynamics model, Nonlinear Anal. Real World Appl., 12 (2011), 3315-3328.  doi: 10.1016/j.nonrwa.2011.05.030.

[2]

B. E. Ainseba, D. Picart and D. Thiery, An innovative multistage, physiologically structured, population model to understand the European grapevine moth dynamics, J. Math. Anal. Appl., 382 (2011), 34-46.  doi: 10.1016/j.jmaa.2011.04.021.

[3]

B. E. Ainseba, S. M. Bouguima and S. Fekih, Biological consistency of an epidemic model with both vertical and horizontal transmissions, Nonlinear Anal. Real World Appl., 28 (2016), 192-207.  doi: 10.1016/j.nonrwa.2015.09.010.

[4]

H. Amann, Ordinary Differential Equations. An Introduction to Nonlinear Analysis, De Gruter Studies in Mathematics, 13, Walter de Gruyter & Co., Berlin, 1990. doi: 10.1515/9783110853698.

[5]

A. Calsina, S. Cuadrado, L. Desvillettes and G. Raoul, Asymptotics of steady states of a selection-mutation equation for small mutation rate, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1123-1146.  doi: 10.1017/S0308210510001629.

[6]

A. Calsina and J. M. Palmada, Steady states of selection-mutation model for an age structured population, J. Math. Anal. Appl., 400 (2013), 386-395.  doi: 10.1016/j.jmaa.2012.11.042.

[7]

A. Calsina and J. Saldaña, A model of physiologically structured population dynamics with nonlinear growth structured population dynamics with nonlinear individual growth rate, J. Math. Biol., 33 (1995), 335-364.  doi: 10.1007/BF00176377.

[8]

J. Cleveland and A. S. Ackleh, Evolutionary game theory on measure spaces: Well-posedness, Nonlinear Anal. Real World Appl., 14 (2013), 785-797.  doi: 10.1016/j.nonrwa.2012.08.002.

[9]

O. Diekmann, P. E. Jabin, S. Mischler and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach, Theor. Popul. Biol., 67 (2005), 257-271.  doi: 10.1016/j.tpb.2004.12.003.

[10]

M. Farkas, On the stability of stationary age distributions, Appl. Math. Comput., 131 (2002), 107-123.  doi: 10.1016/S0096-3003(01)00131-X.

[11]

C. P. Ferreira, S. P. Lyra, F. Azevedo, D. Greenhalgh and E. Massad, Modelling the impact of the long-term use of insecticide-treated bet nets on Anopheles mosquito biting time, Malaria J., 16 (2017). doi: 10.1186/s12936-017-2014-6.

[12]

M. E. Gurtin and R. C. MacCamy, Non-linear age-dependent population dynamics, Arch. Rational. Mech. Anal., 54 (1974), 281-300.  doi: 10.1007/BF00250793.

[13]

M. Iannelli and F. Milner, The Basic Approach to Age-Structured Population Dynamics. Models, Methods and Numerics, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, Dordrecht, 2017. doi: 10.1007/978-94-024-1146-1.

[14]

Y. Iwasa and S. A. Levin, The timing of life history events, J. Theor. Biol., 172 (1995), 33-42.  doi: 10.1006/jtbi.1995.0003.

[15]

N. Kato, A general model of size-dependent population dynamics with nonlinear growth rate, J. Math. Anal. Appl., 297 (2004), 234-256.  doi: 10.1016/j.jmaa.2004.05.004.

[16]

N. Kato and H. Torikata, Local existence for a general model of size-dependent population dynamics, Abstr. Appl. Anal., 2 (1997), 207-226.  doi: 10.1155/S1085337597000353.

[17]

P. Koeller, Basin-scale coherence in phenology of shrimps and phytoplankton in the North Atlantic Ocean, Science, 324 (2009), 791-793.  doi: 10.1126/science.1170987.

[18]

A. Laurila, J. Kujasalo and E. Ranta, Predator-induced changes in life history in two anuran tadpoles: Effects of predator diet, Oikos, 83 (1998), 307-317.  doi: 10.2307/3546842.

[19]

D. Ludwig and L. Rowe, Life-history strategies for energy gain and predator avoidance under time constraints, Am. Nat., 135 (1990), 686-707.  doi: 10.1086/285069.

[20]

P. Marcati, On the global stability of the logistic age-dependent population growth, J. Math. Biol., 15 (1982), 215-226.  doi: 10.1007/BF00275074.

[21]

A. G. M'Kendrick, Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 44 (1925), 98-130.  doi: 10.1017/S0013091500034428.

[22]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lectures Notes in Biomathematics, 68, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.

[23]

J. V. Moore, Biotic control of stream fluxes: Spawning salmon drive nutrient and matter export, Ecology, 88 (2007), 1278-1291.  doi: 10.1890/06-0782.

[24]

D. K. Skelly and L. K. Freidenburg, Effect of beaver on the thermal biology of an amphibian, Ecol. Lett., 3 (2000), 483-486.  doi: 10.1111/j.1461-0248.2000.00186.x.

[25]

F. R. Sharp and A. J. Lotka, A problem in age-distribution, in Mathematical Demography, Biomathematics, 6, Springer, Berlin, 97–100. doi: 10.1007/978-3-642-81046-6_13.

[26]

L. B. Slobodkin, Populations dynamics in Daphnia obtusa Kurz, Ecol. Monog., 24 (1954), 69-89.  doi: 10.2307/1943511.

[27]

J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918.  doi: 10.2307/1934533.

[28]

H. Stibor, Predator induced life-history shifts in a freshwater cladoceran, Oecolgia, 92 (1992), 162-165.  doi: 10.1007/BF00317358.

[29]

D. Thiéry and J. Moreau, Relative performance of European grapevine moth (Lobesia botrana) on grapes and other hosts, Oecologia, 143 (2005), 548-557.  doi: 10.1007/s00442-005-0022-7.

[30]

D. Thiéry, K. Monceau and J. Moreau, Different emergence phenology of European grapevine moth (Lobesia botrana, Lepidoptera: Tortricidae) on six varieties of grapes, Bull. Etymological Res., 104 (2014), 277-287. 

[31]

G. F. Webb, Theory of Nonlinear Age-Dependent Populations Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, Inc., New York, 1985.

show all references

References:
[1]

B. E. Ainseba and D. Picart, Parameter identification in multistage poppulation dynamics model, Nonlinear Anal. Real World Appl., 12 (2011), 3315-3328.  doi: 10.1016/j.nonrwa.2011.05.030.

[2]

B. E. Ainseba, D. Picart and D. Thiery, An innovative multistage, physiologically structured, population model to understand the European grapevine moth dynamics, J. Math. Anal. Appl., 382 (2011), 34-46.  doi: 10.1016/j.jmaa.2011.04.021.

[3]

B. E. Ainseba, S. M. Bouguima and S. Fekih, Biological consistency of an epidemic model with both vertical and horizontal transmissions, Nonlinear Anal. Real World Appl., 28 (2016), 192-207.  doi: 10.1016/j.nonrwa.2015.09.010.

[4]

H. Amann, Ordinary Differential Equations. An Introduction to Nonlinear Analysis, De Gruter Studies in Mathematics, 13, Walter de Gruyter & Co., Berlin, 1990. doi: 10.1515/9783110853698.

[5]

A. Calsina, S. Cuadrado, L. Desvillettes and G. Raoul, Asymptotics of steady states of a selection-mutation equation for small mutation rate, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 1123-1146.  doi: 10.1017/S0308210510001629.

[6]

A. Calsina and J. M. Palmada, Steady states of selection-mutation model for an age structured population, J. Math. Anal. Appl., 400 (2013), 386-395.  doi: 10.1016/j.jmaa.2012.11.042.

[7]

A. Calsina and J. Saldaña, A model of physiologically structured population dynamics with nonlinear growth structured population dynamics with nonlinear individual growth rate, J. Math. Biol., 33 (1995), 335-364.  doi: 10.1007/BF00176377.

[8]

J. Cleveland and A. S. Ackleh, Evolutionary game theory on measure spaces: Well-posedness, Nonlinear Anal. Real World Appl., 14 (2013), 785-797.  doi: 10.1016/j.nonrwa.2012.08.002.

[9]

O. Diekmann, P. E. Jabin, S. Mischler and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach, Theor. Popul. Biol., 67 (2005), 257-271.  doi: 10.1016/j.tpb.2004.12.003.

[10]

M. Farkas, On the stability of stationary age distributions, Appl. Math. Comput., 131 (2002), 107-123.  doi: 10.1016/S0096-3003(01)00131-X.

[11]

C. P. Ferreira, S. P. Lyra, F. Azevedo, D. Greenhalgh and E. Massad, Modelling the impact of the long-term use of insecticide-treated bet nets on Anopheles mosquito biting time, Malaria J., 16 (2017). doi: 10.1186/s12936-017-2014-6.

[12]

M. E. Gurtin and R. C. MacCamy, Non-linear age-dependent population dynamics, Arch. Rational. Mech. Anal., 54 (1974), 281-300.  doi: 10.1007/BF00250793.

[13]

M. Iannelli and F. Milner, The Basic Approach to Age-Structured Population Dynamics. Models, Methods and Numerics, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, Dordrecht, 2017. doi: 10.1007/978-94-024-1146-1.

[14]

Y. Iwasa and S. A. Levin, The timing of life history events, J. Theor. Biol., 172 (1995), 33-42.  doi: 10.1006/jtbi.1995.0003.

[15]

N. Kato, A general model of size-dependent population dynamics with nonlinear growth rate, J. Math. Anal. Appl., 297 (2004), 234-256.  doi: 10.1016/j.jmaa.2004.05.004.

[16]

N. Kato and H. Torikata, Local existence for a general model of size-dependent population dynamics, Abstr. Appl. Anal., 2 (1997), 207-226.  doi: 10.1155/S1085337597000353.

[17]

P. Koeller, Basin-scale coherence in phenology of shrimps and phytoplankton in the North Atlantic Ocean, Science, 324 (2009), 791-793.  doi: 10.1126/science.1170987.

[18]

A. Laurila, J. Kujasalo and E. Ranta, Predator-induced changes in life history in two anuran tadpoles: Effects of predator diet, Oikos, 83 (1998), 307-317.  doi: 10.2307/3546842.

[19]

D. Ludwig and L. Rowe, Life-history strategies for energy gain and predator avoidance under time constraints, Am. Nat., 135 (1990), 686-707.  doi: 10.1086/285069.

[20]

P. Marcati, On the global stability of the logistic age-dependent population growth, J. Math. Biol., 15 (1982), 215-226.  doi: 10.1007/BF00275074.

[21]

A. G. M'Kendrick, Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 44 (1925), 98-130.  doi: 10.1017/S0013091500034428.

[22]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lectures Notes in Biomathematics, 68, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.

[23]

J. V. Moore, Biotic control of stream fluxes: Spawning salmon drive nutrient and matter export, Ecology, 88 (2007), 1278-1291.  doi: 10.1890/06-0782.

[24]

D. K. Skelly and L. K. Freidenburg, Effect of beaver on the thermal biology of an amphibian, Ecol. Lett., 3 (2000), 483-486.  doi: 10.1111/j.1461-0248.2000.00186.x.

[25]

F. R. Sharp and A. J. Lotka, A problem in age-distribution, in Mathematical Demography, Biomathematics, 6, Springer, Berlin, 97–100. doi: 10.1007/978-3-642-81046-6_13.

[26]

L. B. Slobodkin, Populations dynamics in Daphnia obtusa Kurz, Ecol. Monog., 24 (1954), 69-89.  doi: 10.2307/1943511.

[27]

J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918.  doi: 10.2307/1934533.

[28]

H. Stibor, Predator induced life-history shifts in a freshwater cladoceran, Oecolgia, 92 (1992), 162-165.  doi: 10.1007/BF00317358.

[29]

D. Thiéry and J. Moreau, Relative performance of European grapevine moth (Lobesia botrana) on grapes and other hosts, Oecologia, 143 (2005), 548-557.  doi: 10.1007/s00442-005-0022-7.

[30]

D. Thiéry, K. Monceau and J. Moreau, Different emergence phenology of European grapevine moth (Lobesia botrana, Lepidoptera: Tortricidae) on six varieties of grapes, Bull. Etymological Res., 104 (2014), 277-287. 

[31]

G. F. Webb, Theory of Nonlinear Age-Dependent Populations Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, Inc., New York, 1985.

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