• Previous Article
    Transitions between metastable long-run consumption behaviors in a stochastic peer-driven consumer network
  • DCDS-B Home
  • This Issue
  • Next Article
    Dynamical analysis of a banking duopoly model with capital regulation and asymmetric costs
November  2021, 26(11): 5827-5848. doi: 10.3934/dcdsb.2021178

Population dynamics and economic development

1. 

Department of Economics, University of Foggia, Largo Papa Giovanni Paolo Ⅱ, 1, I–71121 Foggia (FG), Italy

2. 

Department of Law, University of Pisa, Via Collegio Ricci, 10, I–56126 Pisa (PI), Italy

3. 

Department of Law, University of Naples 'Federico Ⅱ', Via Mezzocannone 16, I–80134 Naples (NA), Italy

4. 

Department of Finance, Faculty of Economics, Technical University of Ostrava, Ostrava, Czech Republic

* Corresponding author: Luca Gori

Received  October 2020 Revised  May 2021 Published  November 2021 Early access  July 2021

This research develops a continuous-time optimal growth model that accounts for population dynamics resembling the historical pattern of the demographic transition. The Ramsey model then becomes able to generate multiple determinate or indeterminate stationary equilibria and explain the process of the transition from a state with high fertility and low income per capita to a state with low fertility and high income per capita. The article also investigates the emergence of damped or persistent cyclical dynamics.

Citation: Andrea Caravaggio, Luca Gori, Mauro Sodini. Population dynamics and economic development. Discrete and Continuous Dynamical Systems - B, 2021, 26 (11) : 5827-5848. doi: 10.3934/dcdsb.2021178
References:
[1]

E. AccinelliJ. G Brida and S. London, Crecimiento económico y trampas de pobreza: ¿cuáles el papel del capital humano?, Inv. Econ., 66 (2007), 97-118. 

[2]

C. Azariadis and A. Drazen, Threshold externalities in economic development, Q. J. Econ., 105 (1990), 501-526. 

[3]

R. J. Barro and X. Sala-i-Martin, Economic Growth, 2$^{nd}$ edition. Cambridge (MA) US: MIT Press, 2003.

[4]

G. S. Becker, An Economic Analysis of Fertility, Demographic and economic change in developing countries. National Bureau Committee for Economic Research. Princeton (NJ) US: Princeton University Press, 1960.

[5]

J. Bhattacharya and S. Chakraborty, Contraception and the demographic transition, Econ. J., 127 (2017), 2263-2301. 

[6]

K. Blackburn and G. P. Cipriani, A model of longevity, fertility and growth, J. Econ. Dynam. Control, 26 (2002), 187-204. 

[7]

D. E. Bloom, D. Canning and J. Sevilla, The Demographic Dividend. A New Perspective on the Economic Consequences of Population Change, Santa Monica (CA) US: RAND Corporation, 2003.

[8]

D. E. BloomD. CanningG. Fink and J. E. Finlay, Fertility, female labor force participation, and the demographic dividend, J. Econ. Growth, 14 (2009), 79-101. 

[9]

J. G. Brida and E. Accinelli, The Ramsey model with logistic population growth, Econ. Bull., 3 (2007), 1-8. 

[10]

A. Bucci and L. Guerrini, Transitional dynamics in the Solow-Swan growth model with AK technology and logistic population change, BE J. Macroecon., 9 (2009).

[11]

D. Cass, Optimum growth in an aggregative model of capital accumulation, Rev. Econ. Stud., 32, 233–240.

[12]

E. Canton and L. Meijdam, Altruism and the macroeconomic effects of demographic changes, J. Popul. Econ., 10 (1965), 317-334. 

[13]

P. Diamond, National debt in a neoclassical growth model, Am. Econ. Rev., 55 (1965), 1126-1150. 

[14]

L. Fanti and L. Gori, Fertility-related pensions and cyclical instability, J. Popul. Econ., 26 (2013), 1209-1232. 

[15]

L. Fanti and P. Manfredi, The Solow's model with endogenous population: A neoclassical growth cycle model, J. Econ. Dev., 28 (2003), 103-115. 

[16] R. W. Fogel, The Escape from Hunger and Premature Death, Cambridge University Press, New York (NY) US, 2004. 
[17]

H. G. Gaddy, A decade of TFR declines suggests no relationship between development and sub-replacement fertility rebounds, Demogr. Res., 44 (2021), 125-142. 

[18]

D. Gollin, Getting income shares right, J. Polit. Econ., 110 (2002), 458-474. 

[19] O. Galor, Unified Growth Theory, Princeton University Press, Princeton (NJ) US, 2011. 
[20]

O. Galor and D. N. Weil, The gender gap, fertility, and growth, Am. Econ. Rev., 86 (1996), 374-387. 

[21]

O. Galor and D. N. Weil, Population, technology, and growth: From Malthusian stagnation to the demographic transition and beyondm, Am. Econ. Rev., 90 (2000), 806-828. 

[22]

L. Gori and M. Sodini, Endogenous labour supply, endogenous lifetime and economic development, Struct. Change Econ. Dynam., 52 (2020), 238-259. 

[23]

L. Gori and M. Sodini, A contribution to the theory of fertility and economic development, Macroecon. Dynam., 25 (2021), 753-775. 

[24]

L. Gori, P. Manfredi and M. Sodini, A Parsimonious Model of Longevity, Fertility, HIV Transmission and Development, Macroecon. Dynam., Forthcoming, 2019. doi: 10.1017/S1365100519000609.

[25]

L. GoriE. LupiP. Manfredi and M. Sodini, A contribution to the theory of economic development and the demographic transition: Fertility reversal under the HIV epidemic, J. Demogr. Econ., 86 (2020), 125-155. 

[26]

L. Guerrini, The Solow–Swan model with a bounded population growth rate, J. Math. Econ., 42 (2006), 14-21.  doi: 10.1016/j.jmateco.2005.05.001.

[27]

C. I. Jones, The Shape of Production Functions And the Direction of Technical Change, NBER Working, 2004.

[28]

S. Kalemli-Ozcan, Does the mortality decline promote economic growth?, J. Econ. Growth, 7 (2002), 411-439. 

[29]

S. Kalemli-Ozcan, AIDS, "reversal" of the demographic transition and economic development: Evidence from Africa, J. Popul. Econ., 25 (2012), 871-897. 

[30]

S. Kalemli-Ozcan and B. Turan, HIV and fertility revisited, J. Dev. Econ., 96 (2011), 61-65. 

[31]

T. C. Koopmans, On the Concept of Optimal Economic Growth, Koopmans TC, ed. The Econometric Approach to Development Planning, Amsterdam: North-Holland, 1965.

[32]

A. B. Krueger, Measuring labor's share, Am. Econ. Rev., 89 (1999), 45-51. 

[33]

P. Krugman, History versus expectations, Q. J. Econ., 106 (1991), 651-667. 

[34]

N. P. Lagerlöf, The Galor–Weil model revisited: A quantitative exercise, Rev. Econ. Dynam., 9 (2006), 116–142.

[35]

H. Leibenstein, Economic Backwardness and Economic Growth, New York (NY) US: Wiley, 1957.

[36]

P. LorentzenJ. McMillan and R. Wacziarg, Death and development, J. Econ. Growth., 13 (2008), 81-124. 

[37]

M. Livi-Bacci, A Concise History of World Population, 6$^{th}$ edition, Malden (MA) US: Wiley-Blackwell, 2017.

[38]

K. Matsuyama, Increasing returns, industrialization, and indeterminacy of equilibrium, Q. J. Econ., 106 (1991), 617-650. 

[39]

S. Marsiglio and D. L. Torre, Population dynamics and utilitarian criteria in the Lucas–Uzawa model, Econ. Model., 29 (2012), 1197-1204. 

[40]

T. Palivos, Endogenous fertility, multiple growth paths, and economic convergence, J. Econ. Dynam. Control, 19 (1995), 1489-1510. 

[41]

K. Prettner and H. Strulik, It's a sin. Contraceptive use, religious beliefs, and long-run economic development, Rev. Dev. Econ., 21 (2017), 543-566. 

[42]

F. P. Ramsey, A mathematical theory of saving, Econ. J., 38 (1928), 543-559. 

[43]

M. Roser, H. Ritchie and E. Ortiz-Ospina, World Population Growth, WorldInData.org., 2 (2013), Retrieved from: https://ourworldindata.org/world-population-growth.

[44]

I. Seidl and C. A. Tisdell, Carrying capacity reconsidered: From Malthus' population theory to cultural carrying capacity, Ecol. Econ., 31 (1999), 395-408. 

[45]

A. Sen, The concept of development, Chenery, H. and T. N. Srinivasan, eds., Handbook of Development Economics, North-Holland, Amsterdam, 1 (1988), 9–26.

[46]

L. Spataro and L. Fanti, The optimal level of debt in an OLG model with endogenous fertility, Ger. Econ. Rev., 12 (2011), 351-369.  doi: 10.1111/j.1468-0475.2011.00534.x.

[47]

E. Spolaore and R. Wacziarg, How deep are the roots of economic development?, Journal of Economic Literature, 51 (2013), 325-369. 

[48]

R. M. Solow, A contribution to the theory of economic growth, Q. J. Econ., 70 (1956), 65-94. 

[49]

K. Tabata, Inverted U-shaped fertility dynamics, the poverty trap and growth, Econ. Lett., 81 (2003), 241-248.  doi: 10.1016/S0165-1765(03)00188-5.

[50]

F. Wirl, Stability and limit cycles in one-dimensional dynamic optimisations of competitive agents with a market externality, J. Evol. Econ., 7 (1997), 73-89. 

[51]

A. Yakita, Human capital accumulation, fertility and economic development, J. Econ., 99 (2010), 97-116. 

show all references

References:
[1]

E. AccinelliJ. G Brida and S. London, Crecimiento económico y trampas de pobreza: ¿cuáles el papel del capital humano?, Inv. Econ., 66 (2007), 97-118. 

[2]

C. Azariadis and A. Drazen, Threshold externalities in economic development, Q. J. Econ., 105 (1990), 501-526. 

[3]

R. J. Barro and X. Sala-i-Martin, Economic Growth, 2$^{nd}$ edition. Cambridge (MA) US: MIT Press, 2003.

[4]

G. S. Becker, An Economic Analysis of Fertility, Demographic and economic change in developing countries. National Bureau Committee for Economic Research. Princeton (NJ) US: Princeton University Press, 1960.

[5]

J. Bhattacharya and S. Chakraborty, Contraception and the demographic transition, Econ. J., 127 (2017), 2263-2301. 

[6]

K. Blackburn and G. P. Cipriani, A model of longevity, fertility and growth, J. Econ. Dynam. Control, 26 (2002), 187-204. 

[7]

D. E. Bloom, D. Canning and J. Sevilla, The Demographic Dividend. A New Perspective on the Economic Consequences of Population Change, Santa Monica (CA) US: RAND Corporation, 2003.

[8]

D. E. BloomD. CanningG. Fink and J. E. Finlay, Fertility, female labor force participation, and the demographic dividend, J. Econ. Growth, 14 (2009), 79-101. 

[9]

J. G. Brida and E. Accinelli, The Ramsey model with logistic population growth, Econ. Bull., 3 (2007), 1-8. 

[10]

A. Bucci and L. Guerrini, Transitional dynamics in the Solow-Swan growth model with AK technology and logistic population change, BE J. Macroecon., 9 (2009).

[11]

D. Cass, Optimum growth in an aggregative model of capital accumulation, Rev. Econ. Stud., 32, 233–240.

[12]

E. Canton and L. Meijdam, Altruism and the macroeconomic effects of demographic changes, J. Popul. Econ., 10 (1965), 317-334. 

[13]

P. Diamond, National debt in a neoclassical growth model, Am. Econ. Rev., 55 (1965), 1126-1150. 

[14]

L. Fanti and L. Gori, Fertility-related pensions and cyclical instability, J. Popul. Econ., 26 (2013), 1209-1232. 

[15]

L. Fanti and P. Manfredi, The Solow's model with endogenous population: A neoclassical growth cycle model, J. Econ. Dev., 28 (2003), 103-115. 

[16] R. W. Fogel, The Escape from Hunger and Premature Death, Cambridge University Press, New York (NY) US, 2004. 
[17]

H. G. Gaddy, A decade of TFR declines suggests no relationship between development and sub-replacement fertility rebounds, Demogr. Res., 44 (2021), 125-142. 

[18]

D. Gollin, Getting income shares right, J. Polit. Econ., 110 (2002), 458-474. 

[19] O. Galor, Unified Growth Theory, Princeton University Press, Princeton (NJ) US, 2011. 
[20]

O. Galor and D. N. Weil, The gender gap, fertility, and growth, Am. Econ. Rev., 86 (1996), 374-387. 

[21]

O. Galor and D. N. Weil, Population, technology, and growth: From Malthusian stagnation to the demographic transition and beyondm, Am. Econ. Rev., 90 (2000), 806-828. 

[22]

L. Gori and M. Sodini, Endogenous labour supply, endogenous lifetime and economic development, Struct. Change Econ. Dynam., 52 (2020), 238-259. 

[23]

L. Gori and M. Sodini, A contribution to the theory of fertility and economic development, Macroecon. Dynam., 25 (2021), 753-775. 

[24]

L. Gori, P. Manfredi and M. Sodini, A Parsimonious Model of Longevity, Fertility, HIV Transmission and Development, Macroecon. Dynam., Forthcoming, 2019. doi: 10.1017/S1365100519000609.

[25]

L. GoriE. LupiP. Manfredi and M. Sodini, A contribution to the theory of economic development and the demographic transition: Fertility reversal under the HIV epidemic, J. Demogr. Econ., 86 (2020), 125-155. 

[26]

L. Guerrini, The Solow–Swan model with a bounded population growth rate, J. Math. Econ., 42 (2006), 14-21.  doi: 10.1016/j.jmateco.2005.05.001.

[27]

C. I. Jones, The Shape of Production Functions And the Direction of Technical Change, NBER Working, 2004.

[28]

S. Kalemli-Ozcan, Does the mortality decline promote economic growth?, J. Econ. Growth, 7 (2002), 411-439. 

[29]

S. Kalemli-Ozcan, AIDS, "reversal" of the demographic transition and economic development: Evidence from Africa, J. Popul. Econ., 25 (2012), 871-897. 

[30]

S. Kalemli-Ozcan and B. Turan, HIV and fertility revisited, J. Dev. Econ., 96 (2011), 61-65. 

[31]

T. C. Koopmans, On the Concept of Optimal Economic Growth, Koopmans TC, ed. The Econometric Approach to Development Planning, Amsterdam: North-Holland, 1965.

[32]

A. B. Krueger, Measuring labor's share, Am. Econ. Rev., 89 (1999), 45-51. 

[33]

P. Krugman, History versus expectations, Q. J. Econ., 106 (1991), 651-667. 

[34]

N. P. Lagerlöf, The Galor–Weil model revisited: A quantitative exercise, Rev. Econ. Dynam., 9 (2006), 116–142.

[35]

H. Leibenstein, Economic Backwardness and Economic Growth, New York (NY) US: Wiley, 1957.

[36]

P. LorentzenJ. McMillan and R. Wacziarg, Death and development, J. Econ. Growth., 13 (2008), 81-124. 

[37]

M. Livi-Bacci, A Concise History of World Population, 6$^{th}$ edition, Malden (MA) US: Wiley-Blackwell, 2017.

[38]

K. Matsuyama, Increasing returns, industrialization, and indeterminacy of equilibrium, Q. J. Econ., 106 (1991), 617-650. 

[39]

S. Marsiglio and D. L. Torre, Population dynamics and utilitarian criteria in the Lucas–Uzawa model, Econ. Model., 29 (2012), 1197-1204. 

[40]

T. Palivos, Endogenous fertility, multiple growth paths, and economic convergence, J. Econ. Dynam. Control, 19 (1995), 1489-1510. 

[41]

K. Prettner and H. Strulik, It's a sin. Contraceptive use, religious beliefs, and long-run economic development, Rev. Dev. Econ., 21 (2017), 543-566. 

[42]

F. P. Ramsey, A mathematical theory of saving, Econ. J., 38 (1928), 543-559. 

[43]

M. Roser, H. Ritchie and E. Ortiz-Ospina, World Population Growth, WorldInData.org., 2 (2013), Retrieved from: https://ourworldindata.org/world-population-growth.

[44]

I. Seidl and C. A. Tisdell, Carrying capacity reconsidered: From Malthus' population theory to cultural carrying capacity, Ecol. Econ., 31 (1999), 395-408. 

[45]

A. Sen, The concept of development, Chenery, H. and T. N. Srinivasan, eds., Handbook of Development Economics, North-Holland, Amsterdam, 1 (1988), 9–26.

[46]

L. Spataro and L. Fanti, The optimal level of debt in an OLG model with endogenous fertility, Ger. Econ. Rev., 12 (2011), 351-369.  doi: 10.1111/j.1468-0475.2011.00534.x.

[47]

E. Spolaore and R. Wacziarg, How deep are the roots of economic development?, Journal of Economic Literature, 51 (2013), 325-369. 

[48]

R. M. Solow, A contribution to the theory of economic growth, Q. J. Econ., 70 (1956), 65-94. 

[49]

K. Tabata, Inverted U-shaped fertility dynamics, the poverty trap and growth, Econ. Lett., 81 (2003), 241-248.  doi: 10.1016/S0165-1765(03)00188-5.

[50]

F. Wirl, Stability and limit cycles in one-dimensional dynamic optimisations of competitive agents with a market externality, J. Evol. Econ., 7 (1997), 73-89. 

[51]

A. Yakita, Human capital accumulation, fertility and economic development, J. Econ., 99 (2010), 97-116. 

Figure 1.  A portion of the stable manifold on which the trajectories converging to the steady state $ P^{\ast } $ occur. Parameter set: $ \alpha = 0.18 $, $ \beta = 0.84 $, $ \delta = 0.13 $, $ \varepsilon = 0.4 $, $ \rho = 0.3 $, $ \sigma = 1.25 $, $ a = 40.5 $, $ b = 0.5 $. Stationary state equilibrium: $ P^{\ast }: = (k^{\ast },c^{\ast },L^{\ast }) = (0.257,0.749,82.315) $
Figure 2.  Existence of multiple equilibria depending on $ \beta $
Figure 3.  Global indeterminacy. Starting from the same initial conditions $ k_{0} = 0.455 $ and $ L_{0} = 104.843 $, an infinity of initial choices on $ c_{0} $ lead to the indeterminate stationary state equilibrium $ P_{2}^{\ast } $, while there exists a unique choice ($ c_{0}^{1} = 0.833 $) leading to the saddle $ P_{1}^{\ast } $. Parameter set: $ \alpha = 0.18 $, $ \beta = 0.84 $, $ \delta = 0.13 $, $ \varepsilon = 0.4 $, $ \rho = 0.3 $, $ \sigma = 1.25 $, $ a = 40.5 $, $ b = 0.5 $ and $ x = 0.7 $. Stationary state equilibria: $ P_{1}^{\ast }: = (k_{1}^{\ast },c_{1}^{\ast },L_{1}^{\ast }) = (0.594,0.833,80.677) $ and $ P_{2}^{\ast }: = (k_{2}^{\ast },c_{2}^{\ast },L_{2}^{\ast }) = (4.372,2.925,79.676) $
Figure 4.  Local indeterminacy scenario in terms of capital (Panel A) and population size (Panel B)
Figure 5.  Three-dimensional phase portrait in the space $ (k,c,L) $ showing: (i) the trajectory converging to $ P_{1}^{\ast } $ starting from the initial condition $ (k_{0},c_{0}^{1},L_{0}) = (0.346,0.565,3.178) $, and (ii) a trajectory converging to a limit cycle $ \Gamma $ around $ P_{2}^{\ast } $ starting from the initial condition $ (k_{0},c_{0}^{2},L_{0}) = (0.346,0.452,3.178) $. Economic and demographic variables permanently fluctuate around $ P_{2}^{\ast } $. Parameter set: $ \alpha = 0.54 $, $ \beta = 1 $, $ \gamma = 1.5672 $, $ \delta = 0.7 $, $ \varepsilon = 0.4 $, $ \rho = 0.3 $, $ \sigma = 7.04 $, $ a = 0.37 $, $ b = 0.2 $ and $ x = 1.1 $. Stationary state equilibria: $ P_{1}^{\ast }: = (k_{1}^{\ast },c_{1}^{\ast },L_{1}^{\ast }) = (0.221,0.288,3.305) $ and $ P_{2}^{\ast }: = (k_{2}^{\ast },c_{2}^{\ast },L_{2}^{\ast }) = (0.673,0.336,3.03) $
[1]

Gennadi M. Henkin, Victor M. Polterovich. A difference-differential analogue of the Burgers equations and some models of economic development. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 697-728. doi: 10.3934/dcds.1999.5.697

[2]

M. Dolfin, D. Knopoff, L. Leonida, D. Maimone Ansaldo Patti. Escaping the trap of 'blocking': A kinetic model linking economic development and political competition. Kinetic and Related Models, 2017, 10 (2) : 423-443. doi: 10.3934/krm.2017016

[3]

Adam Gregosiewicz. Asymptotics of the Lebowitz-Rubinow-Rotenberg model of population development. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2443-2472. doi: 10.3934/dcdsb.2018260

[4]

Angelo Antoci, Marcello Galeotti, Mauro Sodini. Environmental degradation and indeterminacy of equilibrium selection. Discrete and Continuous Dynamical Systems - B, 2021, 26 (11) : 5755-5767. doi: 10.3934/dcdsb.2021179

[5]

Alexander J. Zaslavski. The turnpike property of discrete-time control problems arising in economic dynamics. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 861-880. doi: 10.3934/dcdsb.2005.5.861

[6]

Martha G. Alatriste-Contreras, Juan Gabriel Brida, Martin Puchet Anyul. Structural change and economic dynamics: Rethinking from the complexity approach. Journal of Dynamics and Games, 2019, 6 (2) : 87-106. doi: 10.3934/jdg.2019007

[7]

Wei Feng, Xin Lu, Richard John Donovan Jr.. Population dynamics in a model for territory acquisition. Conference Publications, 2001, 2001 (Special) : 156-165. doi: 10.3934/proc.2001.2001.156

[8]

Luca Gerardo-Giorda, Pierre Magal, Shigui Ruan, Ousmane Seydi, Glenn Webb. Preface: Population dynamics in epidemiology and ecology. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : i-ii. doi: 10.3934/dcdsb.2020125

[9]

Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks and Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53

[10]

Masahiro Yamaguchi, Yasuhiro Takeuchi, Wanbiao Ma. Population dynamics of sea bass and young sea bass. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 833-840. doi: 10.3934/dcdsb.2004.4.833

[11]

MirosŁaw Lachowicz, Tatiana Ryabukha. Equilibrium solutions for microscopic stochastic systems in population dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 777-786. doi: 10.3934/mbe.2013.10.777

[12]

Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633

[13]

Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643

[14]

Dianmo Li, Zengxiang Gao, Zufei Ma, Baoyu Xie, Zhengjun Wang. Two general models for the simulation of insect population dynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 623-628. doi: 10.3934/dcdsb.2004.4.623

[15]

Hui Wan, Huaiping Zhu. A new model with delay for mosquito population dynamics. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1395-1410. doi: 10.3934/mbe.2014.11.1395

[16]

Henri Berestycki, Jean-Michel Roquejoffre, Luca Rossi. The periodic patch model for population dynamics with fractional diffusion. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 1-13. doi: 10.3934/dcdss.2011.4.1

[17]

B. E. Ainseba, W. E. Fitzgibbon, M. Langlais, J. J. Morgan. An application of homogenization techniques to population dynamics models. Communications on Pure and Applied Analysis, 2002, 1 (1) : 19-33. doi: 10.3934/cpaa.2002.1.19

[18]

Chris Cosner, Andrew L. Nevai. Spatial population dynamics in a producer-scrounger model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1591-1607. doi: 10.3934/dcdsb.2015.20.1591

[19]

Alain Miranville, Mazen Saad, Raafat Talhouk. Preface: Workshop in fluid mechanics and population dynamics. Discrete and Continuous Dynamical Systems - S, 2014, 7 (2) : i-i. doi: 10.3934/dcdss.2014.7.2i

[20]

Dianmo Li, Zhen Zhang, Zufei Ma, Baoyu Xie, Rui Wang. Allee effect and a catastrophe model of population dynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 629-634. doi: 10.3934/dcdsb.2004.4.629

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (187)
  • HTML views (156)
  • Cited by (0)

Other articles
by authors

[Back to Top]