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doi: 10.3934/dcdsb.2021210
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On an upper bound for the spreading speed

1. 

University of Tlemcen, Department of Ecology and Environment, Algeria, Laboratoire d'Analyse Non linéaire et Mathématiques Appliquées

2. 

University of Tlemcen, Department of Mathematics, Algeria, Laboratoire d'Analyse Non linéaire et Mathématiques Appliquées

* Corresponding author: Ali Moussaoui

Received  March 2021 Revised  June 2021 Early access August 2021

Fund Project: This research was supported by DGRSDT, Algeria

In this paper, we use the exponential transform to give a unified formal upper bound for the asymptotic rate of spread of a population propagating in a one dimensional habitat. We show through examples how this upper bound can be obtained directly for discrete and continuous time models. This upper bound has the form $ \min_{s>0} \ln (\rho(s))/s $ and coincides with the speeds of several models found in the literature.

Citation: Mohammed Mesk, Ali Moussaoui. On an upper bound for the spreading speed. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021210
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Springer, (1975), 5–49.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

R. D. Benguria and M. C. Depassier, Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation, Comm. Math. Phys., 175 (1996), 221-227.  doi: 10.1007/BF02101631.  Google Scholar

[4]

M. Bramson, Convergence of Solutions of the Kolmogorov Equation to Travelling Waves, Mem. Amer. Math. Soc., 1983. doi: 10.1090/memo/0285.  Google Scholar

[5]

H. Caswell, Matrix Population Models, 2$^nd$ Edition, Sinauer Associates Inc. Sunderland, USA, 2000. Google Scholar

[6]

H. CaswellM. G. Neubert and C. M. Hunter, Demography and dispersal: Invasion speeds and sensitivity analysis in periodic and stochastic environments, Theor. Ecol., 4 (2011), 407-421.  doi: 10.1007/s12080-010-0091-z.  Google Scholar

[7]

T. S. DohertyA. S. GlenD. G. NimmoE. G. Ritchie and C. R. Dickman, Invasive predators and global biodiversity loss, Proc. Natl. Acad. Sci. USA, 113 (2016), 11261-11265.  doi: 10.1073/pnas.1602480113.  Google Scholar

[8]

D. Finkelshtein and P. Tkachov, Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line, Appl. Anal., 98 (2019), 756-780.  doi: 10.1080/00036811.2017.1400537.  Google Scholar

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R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen., 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[10]

B. GallardoM. ClaveroM. I. Sánchez and M. Vilà, Global ecological impacts of invasive species in aquatic ecosystems, Glob. Chang. Biol., 22 (2016), 151-163.  doi: 10.1111/gcb.13004.  Google Scholar

[11]

F. R. Gantmacher and J. L. Brenner, Applications of the Theory of Matrices, Courier Corporation, 2005. Google Scholar

[12]

K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.  doi: 10.1007/BF00277154.  Google Scholar

[13]

A. HastingsK. CuddingtonK. F. DaviesC. J. DugawS. ElmendorfA. FreestoneS. HarrisonM. HollandJ. Lambrinos and U. Malvadkar, The spatial spread of invasions: New developments in theory and evidence, Ecol. Lett., 8 (2005), 91-101.  doi: 10.1111/j.1461-0248.2004.00687.x.  Google Scholar

[14]

C. S. Kolar and D. M. Lodge, Progress in invasion biology: Predicting invaders, Trends Ecol. Evol., 16 (2001), 199-204.  doi: 10.1016/S0169-5347(01)02101-2.  Google Scholar

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A. N. Kolmogorov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. Moskow, Ser. Internat., Sec. A, 1 (1937), 1-25.   Google Scholar

[16]

M. KotM. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042.  doi: 10.2307/2265698.  Google Scholar

[17]

M. Kot and M. G. Neubert, Saddle-point approximations, integrodifference equations, and invasions, Bull. Math. Biol., 70 (2008), 1790-1826.  doi: 10.1007/s11538-008-9325-2.  Google Scholar

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M. Kot and W. M. Schaffer, Discrete-time growth-dispersal models, Math. Biosci., 80 (1986), 109-136.  doi: 10.1016/0025-5564(86)90069-6.  Google Scholar

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K.-S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov, J. Differ. Equ., 59 (1985), 44-70.  doi: 10.1016/0022-0396(85)90137-8.  Google Scholar

[20]

M.-R. Leung and M. Kot, Models for the spread of white pine blister rust, J. Theor. Biol., 382 (2015), 328-336.  doi: 10.1016/j.jtbi.2015.07.018.  Google Scholar

[21]

M. A. LewisB. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.  doi: 10.1007/s002850200144.  Google Scholar

[22]

B. LiH. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[23]

B. R. Liu and M. Kot, Accelerating invasions and the asymptotics of fat-tailed dispersal, J. Theor. Biol., 471 (2019), 22-41.  doi: 10.1016/j.jtbi.2019.03.016.  Google Scholar

[24]

J. A. Lubina and S. A. Levin, The spread of a reinvading species: Range expansion in the california sea otter, Am. Nat., 131 (1988), 526-543.  doi: 10.1086/284804.  Google Scholar

[25]

R. Lui, Biological growth and spread modeled by systems of recursions. i. mathematical theory, Math. Biosci., 93 (1989), 269-295.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar

[26]

F. Lutscher, Integrodifference Equations in Spatial Ecology, Springer, 2019. doi: 10.1007/978-3-030-29294-2.  Google Scholar

[27]

F. LutscherR. M. Nisbet and E. Pachepsky, Population persistence in the face of advection, Theor. Ecol., 3 (2010), 271-284.  doi: 10.1007/s12080-009-0068-y.  Google Scholar

[28]

J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.  doi: 10.1016/S0025-5564(03)00041-5.  Google Scholar

[29]

M. MeskT. MahdjoubS. GourbièreJ. E. Rabinovich and F. Menu, Invasion speeds of triatoma dimidiata, vector of chagas disease: An application of orthogonal polynomials method, J. Theor. Biol., 395 (2016), 126-143.  doi: 10.1016/j.jtbi.2016.01.017.  Google Scholar

[30]

D. Mollison, Spatial contact models for ecological and epidemic spread, J. R. Stat. Soc. B (Methodological), 39 (1977), 283-313.  doi: 10.1111/j.2517-6161.1977.tb01627.x.  Google Scholar

[31]

D. Mollison, Dependence of epidemic and population velocities on basic parameters, Math. Biosci., 107 (1991), 255-287.  doi: 10.1016/0025-5564(91)90009-8.  Google Scholar

[32]

A. Moussaoui and V. Volpert, Speed of wave propagation for a nonlocal reaction-diffusion equation, Appl. Anal., 99 (2020), 2307-2321.  doi: 10.1080/00036811.2018.1559303.  Google Scholar

[33]

M. G. Neubert and H. Caswell, Demography and dispersal: Calculation and sensitivity analysis of invasion speed for structured populations, Ecology, 81 (2000), 1613-1628.  doi: 10.1890/0012-9658(2000)081[1613:DADCAS]2.0.CO;2.  Google Scholar

[34]

M. G. NeubertM. Kot and M. A. Lewis, Invasion speeds in fluctuating environments, Proc. R. Soc. B, 267 (2000), 1603-1610.  doi: 10.1098/rspb.2000.1185.  Google Scholar

[35]

S. V. Petrovskii and B.-L. Li, Exactly Solvable Models of Biological Invasion, CRC Press, 2005. doi: 10.1201/9781420034967.  Google Scholar

[36]

J. A. PowellI. Slapničar and W. van der Werf, Epidemic spread of a lesion-forming plant pathogen-analysis of a mechanistic model with infinite age structure, Linear Algebra Appl., 398 (2005), 117-140.  doi: 10.1016/j.laa.2004.10.020.  Google Scholar

[37]

J. Radcliffe and L. Rass, Saddle point approximations in n-type epidemics and contact birth processes, Rocky Mountain J. Math., 14 (1984), 599-617.  doi: 10.1216/RMJ-1984-14-3-599.  Google Scholar

[38]

J. Radcliffe and L. Rass, Reducible epidemics: Choosing your saddle, Rocky Mountain J. Math., 23 (1993), 725-752.  doi: 10.1216/rmjm/1181072587.  Google Scholar

[39]

J. Radcliffe and L. Rass, Discrete time spatial models arising in genetics, evolutionary game theory, and branching processes, Math. Biosci., 140 (1997), 101-129.  doi: 10.1016/S0025-5564(97)00154-5.  Google Scholar

[40]

L. RoquesF. HamelJ. FayardB. Fady and E. K. Klein, Recolonisation by diffusion can generate increasing rates of spread, Theor. Pop. Biol., 77 (2010), 205-212.  doi: 10.1016/j.tpb.2010.02.002.  Google Scholar

[41]

S. J. Schreiber and M. E. Ryan, Invasion speeds for structured populations in fluctuating environments, Theor. Ecol., 4 (2011), 423-434.  doi: 10.1007/s12080-010-0098-5.  Google Scholar

[42]

N. Shigesada, K. Kawasaki et al., Invasion and the range expansion of species: Effects of long-distance dispersal, In: Dispersal Ecology (eds. J. Bullock, R. Kenward & R. Hails), (2002) 350–373. Google Scholar

[43]

A. StevensG. Papanicolaou and S. Heinze, Variational principles for propagation speeds in inhomogeneous media, SIAM J. Appl. Math., 62 (2001), 129-148.  doi: 10.1137/S0036139999361148.  Google Scholar

[44]

D. L. StrayerV. T. EvinerJ. M. Jeschke and M. L. Pace, Understanding the long-term effects of species invasions, Trends Ecol. Evol., 21 (2006), 645-651.  doi: 10.1016/j.tree.2006.07.007.  Google Scholar

[45]

A. E. Taylor, L'hospital's rule, Am. Math. Mon., 59 (1952), 20-24.  doi: 10.1080/00029890.1952.11988058.  Google Scholar

[46]

F. Van den BoschJ. A. J. Metz and O. Diekmann, The velocity of spatial population expansion, J. Math. Biol., 28 (1990), 529-565.  doi: 10.1007/BF00164162.  Google Scholar

[47]

V. Volpert, Elliptic Partial Differential Equations: Volume 2: Reaction-Diffusion Equations, vol. 104, Springer, 2014. doi: 10.1007/978-3-0348-0813-2.  Google Scholar

[48]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar

[49]

H. F. WeinbergerM. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218.  doi: 10.1007/s002850200145.  Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Springer, (1975), 5–49.  Google Scholar

[2]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[3]

R. D. Benguria and M. C. Depassier, Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation, Comm. Math. Phys., 175 (1996), 221-227.  doi: 10.1007/BF02101631.  Google Scholar

[4]

M. Bramson, Convergence of Solutions of the Kolmogorov Equation to Travelling Waves, Mem. Amer. Math. Soc., 1983. doi: 10.1090/memo/0285.  Google Scholar

[5]

H. Caswell, Matrix Population Models, 2$^nd$ Edition, Sinauer Associates Inc. Sunderland, USA, 2000. Google Scholar

[6]

H. CaswellM. G. Neubert and C. M. Hunter, Demography and dispersal: Invasion speeds and sensitivity analysis in periodic and stochastic environments, Theor. Ecol., 4 (2011), 407-421.  doi: 10.1007/s12080-010-0091-z.  Google Scholar

[7]

T. S. DohertyA. S. GlenD. G. NimmoE. G. Ritchie and C. R. Dickman, Invasive predators and global biodiversity loss, Proc. Natl. Acad. Sci. USA, 113 (2016), 11261-11265.  doi: 10.1073/pnas.1602480113.  Google Scholar

[8]

D. Finkelshtein and P. Tkachov, Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line, Appl. Anal., 98 (2019), 756-780.  doi: 10.1080/00036811.2017.1400537.  Google Scholar

[9]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen., 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x.  Google Scholar

[10]

B. GallardoM. ClaveroM. I. Sánchez and M. Vilà, Global ecological impacts of invasive species in aquatic ecosystems, Glob. Chang. Biol., 22 (2016), 151-163.  doi: 10.1111/gcb.13004.  Google Scholar

[11]

F. R. Gantmacher and J. L. Brenner, Applications of the Theory of Matrices, Courier Corporation, 2005. Google Scholar

[12]

K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol., 2 (1975), 251-263.  doi: 10.1007/BF00277154.  Google Scholar

[13]

A. HastingsK. CuddingtonK. F. DaviesC. J. DugawS. ElmendorfA. FreestoneS. HarrisonM. HollandJ. Lambrinos and U. Malvadkar, The spatial spread of invasions: New developments in theory and evidence, Ecol. Lett., 8 (2005), 91-101.  doi: 10.1111/j.1461-0248.2004.00687.x.  Google Scholar

[14]

C. S. Kolar and D. M. Lodge, Progress in invasion biology: Predicting invaders, Trends Ecol. Evol., 16 (2001), 199-204.  doi: 10.1016/S0169-5347(01)02101-2.  Google Scholar

[15]

A. N. Kolmogorov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. Moskow, Ser. Internat., Sec. A, 1 (1937), 1-25.   Google Scholar

[16]

M. KotM. A. Lewis and P. van den Driessche, Dispersal data and the spread of invading organisms, Ecology, 77 (1996), 2027-2042.  doi: 10.2307/2265698.  Google Scholar

[17]

M. Kot and M. G. Neubert, Saddle-point approximations, integrodifference equations, and invasions, Bull. Math. Biol., 70 (2008), 1790-1826.  doi: 10.1007/s11538-008-9325-2.  Google Scholar

[18]

M. Kot and W. M. Schaffer, Discrete-time growth-dispersal models, Math. Biosci., 80 (1986), 109-136.  doi: 10.1016/0025-5564(86)90069-6.  Google Scholar

[19]

K.-S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovsky, and Piscounov, J. Differ. Equ., 59 (1985), 44-70.  doi: 10.1016/0022-0396(85)90137-8.  Google Scholar

[20]

M.-R. Leung and M. Kot, Models for the spread of white pine blister rust, J. Theor. Biol., 382 (2015), 328-336.  doi: 10.1016/j.jtbi.2015.07.018.  Google Scholar

[21]

M. A. LewisB. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233.  doi: 10.1007/s002850200144.  Google Scholar

[22]

B. LiH. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008.  Google Scholar

[23]

B. R. Liu and M. Kot, Accelerating invasions and the asymptotics of fat-tailed dispersal, J. Theor. Biol., 471 (2019), 22-41.  doi: 10.1016/j.jtbi.2019.03.016.  Google Scholar

[24]

J. A. Lubina and S. A. Levin, The spread of a reinvading species: Range expansion in the california sea otter, Am. Nat., 131 (1988), 526-543.  doi: 10.1086/284804.  Google Scholar

[25]

R. Lui, Biological growth and spread modeled by systems of recursions. i. mathematical theory, Math. Biosci., 93 (1989), 269-295.  doi: 10.1016/0025-5564(89)90026-6.  Google Scholar

[26]

F. Lutscher, Integrodifference Equations in Spatial Ecology, Springer, 2019. doi: 10.1007/978-3-030-29294-2.  Google Scholar

[27]

F. LutscherR. M. Nisbet and E. Pachepsky, Population persistence in the face of advection, Theor. Ecol., 3 (2010), 271-284.  doi: 10.1007/s12080-009-0068-y.  Google Scholar

[28]

J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Math. Biosci., 184 (2003), 201-222.  doi: 10.1016/S0025-5564(03)00041-5.  Google Scholar

[29]

M. MeskT. MahdjoubS. GourbièreJ. E. Rabinovich and F. Menu, Invasion speeds of triatoma dimidiata, vector of chagas disease: An application of orthogonal polynomials method, J. Theor. Biol., 395 (2016), 126-143.  doi: 10.1016/j.jtbi.2016.01.017.  Google Scholar

[30]

D. Mollison, Spatial contact models for ecological and epidemic spread, J. R. Stat. Soc. B (Methodological), 39 (1977), 283-313.  doi: 10.1111/j.2517-6161.1977.tb01627.x.  Google Scholar

[31]

D. Mollison, Dependence of epidemic and population velocities on basic parameters, Math. Biosci., 107 (1991), 255-287.  doi: 10.1016/0025-5564(91)90009-8.  Google Scholar

[32]

A. Moussaoui and V. Volpert, Speed of wave propagation for a nonlocal reaction-diffusion equation, Appl. Anal., 99 (2020), 2307-2321.  doi: 10.1080/00036811.2018.1559303.  Google Scholar

[33]

M. G. Neubert and H. Caswell, Demography and dispersal: Calculation and sensitivity analysis of invasion speed for structured populations, Ecology, 81 (2000), 1613-1628.  doi: 10.1890/0012-9658(2000)081[1613:DADCAS]2.0.CO;2.  Google Scholar

[34]

M. G. NeubertM. Kot and M. A. Lewis, Invasion speeds in fluctuating environments, Proc. R. Soc. B, 267 (2000), 1603-1610.  doi: 10.1098/rspb.2000.1185.  Google Scholar

[35]

S. V. Petrovskii and B.-L. Li, Exactly Solvable Models of Biological Invasion, CRC Press, 2005. doi: 10.1201/9781420034967.  Google Scholar

[36]

J. A. PowellI. Slapničar and W. van der Werf, Epidemic spread of a lesion-forming plant pathogen-analysis of a mechanistic model with infinite age structure, Linear Algebra Appl., 398 (2005), 117-140.  doi: 10.1016/j.laa.2004.10.020.  Google Scholar

[37]

J. Radcliffe and L. Rass, Saddle point approximations in n-type epidemics and contact birth processes, Rocky Mountain J. Math., 14 (1984), 599-617.  doi: 10.1216/RMJ-1984-14-3-599.  Google Scholar

[38]

J. Radcliffe and L. Rass, Reducible epidemics: Choosing your saddle, Rocky Mountain J. Math., 23 (1993), 725-752.  doi: 10.1216/rmjm/1181072587.  Google Scholar

[39]

J. Radcliffe and L. Rass, Discrete time spatial models arising in genetics, evolutionary game theory, and branching processes, Math. Biosci., 140 (1997), 101-129.  doi: 10.1016/S0025-5564(97)00154-5.  Google Scholar

[40]

L. RoquesF. HamelJ. FayardB. Fady and E. K. Klein, Recolonisation by diffusion can generate increasing rates of spread, Theor. Pop. Biol., 77 (2010), 205-212.  doi: 10.1016/j.tpb.2010.02.002.  Google Scholar

[41]

S. J. Schreiber and M. E. Ryan, Invasion speeds for structured populations in fluctuating environments, Theor. Ecol., 4 (2011), 423-434.  doi: 10.1007/s12080-010-0098-5.  Google Scholar

[42]

N. Shigesada, K. Kawasaki et al., Invasion and the range expansion of species: Effects of long-distance dispersal, In: Dispersal Ecology (eds. J. Bullock, R. Kenward & R. Hails), (2002) 350–373. Google Scholar

[43]

A. StevensG. Papanicolaou and S. Heinze, Variational principles for propagation speeds in inhomogeneous media, SIAM J. Appl. Math., 62 (2001), 129-148.  doi: 10.1137/S0036139999361148.  Google Scholar

[44]

D. L. StrayerV. T. EvinerJ. M. Jeschke and M. L. Pace, Understanding the long-term effects of species invasions, Trends Ecol. Evol., 21 (2006), 645-651.  doi: 10.1016/j.tree.2006.07.007.  Google Scholar

[45]

A. E. Taylor, L'hospital's rule, Am. Math. Mon., 59 (1952), 20-24.  doi: 10.1080/00029890.1952.11988058.  Google Scholar

[46]

F. Van den BoschJ. A. J. Metz and O. Diekmann, The velocity of spatial population expansion, J. Math. Biol., 28 (1990), 529-565.  doi: 10.1007/BF00164162.  Google Scholar

[47]

V. Volpert, Elliptic Partial Differential Equations: Volume 2: Reaction-Diffusion Equations, vol. 104, Springer, 2014. doi: 10.1007/978-3-0348-0813-2.  Google Scholar

[48]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar

[49]

H. F. WeinbergerM. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218.  doi: 10.1007/s002850200145.  Google Scholar

Figure 1.  Locations $ X_t(a) $ and $ \tilde{X}_t(a) $ for level $ a $
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