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On oscillations to a 2D age-dependent predation equations characterizing Beddington-DeAngelis type schemes
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 |
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.
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. Caswell, M. 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. Doherty, A. S. Glen, D. G. Nimmo, E. 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. Gallardo, M. Clavero, M. 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. Hastings, K. Cuddington, K. F. Davies, C. J. Dugaw, S. Elmendorf, A. Freestone, S. Harrison, M. Holland, J. 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. Kot, M. 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. Lewis, B. 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. Li, H. 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. Lutscher, R. 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. Mesk, T. Mahdjoub, S. Gourbière, J. 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. Neubert, M. 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. Powell, I. 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. Roques, F. Hamel, J. Fayard, B. 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. Stevens, G. 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. Strayer, V. T. Eviner, J. 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 Bosch, J. 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. Weinberger, M. 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. Caswell, M. 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. Doherty, A. S. Glen, D. G. Nimmo, E. 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. Gallardo, M. Clavero, M. 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. Hastings, K. Cuddington, K. F. Davies, C. J. Dugaw, S. Elmendorf, A. Freestone, S. Harrison, M. Holland, J. 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. Kot, M. 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. Lewis, B. 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. Li, H. 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. Lutscher, R. 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. Mesk, T. Mahdjoub, S. Gourbière, J. 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. Neubert, M. 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. Powell, I. 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. Roques, F. Hamel, J. Fayard, B. 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. Stevens, G. 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. Strayer, V. T. Eviner, J. 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 Bosch, J. 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. Weinberger, M. 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. |
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