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 and 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.

[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.

[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.

[4]

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

[5]

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

[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.

[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.

[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.

[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.

[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.

[11]

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

[12]

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

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[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.

[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.

[38]

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

[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.

[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.

[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.

[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.

[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.

[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.

[45]

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

[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.

[47]

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

[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.

[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.

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.

[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.

[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.

[4]

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

[5]

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

[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.

[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.

[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.

[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.

[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.

[11]

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

[12]

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

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[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.

[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.

[38]

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

[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.

[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.

[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.

[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.

[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.

[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.

[45]

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

[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.

[47]

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

[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.

[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.

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