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November  2020, 13(11): 3029-3045. doi: 10.3934/dcdss.2020117

Dynamical stabilization and traveling waves in integrodifference equations

 1 Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5, Canada 2 Department of Mathematics and Statistics, Department of Biology, University of Ottawa, Ottawa, ON K1N 6N5, Canada

* Corresponding author: flutsche@uottawa.ca

Received  December 2018 Revised  April 2019 Published  November 2020 Early access  October 2019

Fund Project: VL and FL are supported by respective Discovery Grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada (RGPIN-2016-04318 and RGPIN-2016-04795). FL is grateful for a Discovery Accelerator Supplement award from NSERC (RGPAS 492878-2016). FL thanks the participants of the workshop "Integrodifference equations in spatial ecology: 30 years and counting" (16w5121) at the Banff International Research Station for their feedback on an oral presentation of this material

Integrodifference equations are discrete-time analogues of reaction-diffusion equations and can be used to model the spatial spread and invasion of non-native species. They support solutions in the form of traveling waves, and the speed of these waves gives important insights about the speed of biological invasions. Typically, a traveling wave leaves in its wake a stable state of the system. Dynamical stabilization is the phenomenon that an unstable state arises in the wake of such a wave and appears stable for potentially long periods of time, before it is replaced with a stable state via another transition wave. While dynamical stabilization has been studied in systems of reaction-diffusion equations, we here present the first such study for integrodifference equations. We use linear stability analysis of traveling-wave profiles to determine necessary conditions for the emergence of dynamical stabilization and relate it to the theory of stacked fronts. We find that the phenomenon is the norm rather than the exception when the non-spatial dynamics exhibit a stable two-cycle.

Citation: Adèle Bourgeois, Victor LeBlanc, Frithjof Lutscher. Dynamical stabilization and traveling waves in integrodifference equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3029-3045. doi: 10.3934/dcdss.2020117
References:
 [1] A. Bourgeois, Spreading Speeds and Travelling Waves in Intergrodifference Equations with Overcompensatory Dynamics, Master's Thesis, University of Ottawa, 2016, URL http://hdl.handle.net/10393/34578. Google Scholar [2] A. Bourgeois, V. LeBlanc and F. Lutscher, Spreading phenomena in integrodifference equations with non-monotone growth functions, SIAM Journal of Applied Mathematics, 78 (2018), 2950-2972.  doi: 10.1137/17M1126102.  Google Scholar [3] R. M. Coutinho, W. A. C. Godoy and R. A. Kraenkel, Integrodifference model for blowfly invasion, Theoretical Ecology, 5 (2012), 363-371.  doi: 10.1007/s12080-012-0157-1.  Google Scholar [4] A. S. Dagbovie and J. A. Sherratt, Absolute stability and dynamical stabilisation in predator-prey systems, Journal of Mathematical Biology, 68 (2014), 1403-1421.  doi: 10.1007/s00285-013-0672-8.  Google Scholar [5] G. de Vries, T. Hillen, M. Lewis, J. Müller and B. Schönfisch, A Course in Mathematical Biology: Quantitative Modeling with Mathematical and Computational Methods, Mathematical Modeling and Computation, 12. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. doi: 10.1137/1.9780898718256.  Google Scholar [6] R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, Vol. 20. Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar [7] A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, Journal of Differential Equations, 260 (2016), 8316-8357.  doi: 10.1016/j.jde.2016.02.023.  Google Scholar [8] W. F. Fagan and J. G. Bishop, Trophic interactions during primary succession: Herbivores slow a plant reinvasion at mount st. helens, The American Naturalist, 155 (2000), 238-251.  doi: 10.1086/303320.  Google Scholar [9] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Archive for Rational Mechanics and Analysis, 65 (1977), 335-361.  doi: 10.1007/BF00250432.  Google Scholar [10] A. Gharouni, M. A. Barbeau, A. Locke, L. Wang and J. Watmough, Sensitivity of invasion speed to dispersal and demography: An application of spreading speed theory to the green crab invasion on the northwest atlantic coast, Marine Ecology Progress Series, 541 (2015), 135-150.  doi: 10.3354/meps11508.  Google Scholar [11] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar [12] L. N. Howard and N. Kopell, Slowly varying waves and shock structures in reaction-diffusion equations, Studies in Applied Mathematics, 56 (1976/77), 95-145.  doi: 10.1002/sapm197756295.  Google Scholar [13] S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM Journal on Mathematical Analysis, 40 (2008), 776-789.  doi: 10.1137/070703016.  Google Scholar [14] M. Iida, R. Lui and H. Ninomiya, Stacked fronts for cooperative systems with equal diffusion coefficients, SIAM Journal on Mathematical Analysis, 43 (2011), 1369-1389.  doi: 10.1137/100792846.  Google Scholar [15] N. Kopell and L. N. Howard, Plane wave solutions to reaction-diffusion equations, Studies in Applied Mathematics, 52 (1973), 291-328.  doi: 10.1002/sapm1973524291.  Google Scholar [16] M. Kot, Discrete-time traveling waves: Ecological examples, Journal of Mathematical Biology, 30 (1992), 413-436.  doi: 10.1007/BF00173295.  Google Scholar [17] 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.  Google Scholar [18] M. Kot and W. Schaffer, Discrete-time growth-dispersal models, Mathematical Biosciences, 80 (1986), 109-136.  doi: 10.1016/0025-5564(86)90069-6.  Google Scholar [19] M. A. Lewis, S. V. Petrovskii and J. R. Potts, The Mathematics Behind Biological Invasions, Interdisciplinary Applied Mathematics, 44. Springer, 2016. doi: 10.1007/978-3-319-32043-4.  Google Scholar [20] B. T. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, Journal of Mathematical Biology, 58 (2009), 323-338.  doi: 10.1007/s00285-008-0175-1.  Google Scholar [21] G. Lin, Traveling wave solutions for integro-difference systems, Journal of Differential Equations, 258 (2015), 2908-2940.  doi: 10.1016/j.jde.2014.12.030.  Google Scholar [22] H. Malchow and S. V. Petrovskii, Dynamical stabilization of an unstable equilibrium in chemical and biological systems. Mathematical modelling of nonlinear systems, Mathematical and Computer Modelling, 36 (2002), 307-319.  doi: 10.1016/S0895-7177(02)00127-9.  Google Scholar [23] H. Malchow, S. V. Petrovskii and E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulation, Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, FL, 2008.  Google Scholar [24] N. G. Marculis and R. Lui, Modelling the biological invasion of Carcinus maenas (the European green crab), Journal of Biological Dynamics, 10 (2016), 140-163.  doi: 10.1080/17513758.2015.1115563.  Google Scholar [25] R. M. May, Biological populations obeying difference equations: Stable points, stable cycles, and chaos, Journal of Theoretical Biology, 51 (1975), 511-524.  doi: 10.1016/0022-5193(75)90078-8.  Google Scholar [26] M. G. Neubert, M. Kot and M. A. Lewis, Dispersal and pattern formation in a discrete-time predator-prey model, Theoretical Population Biology, 48 (1995), 7-43.  doi: 10.1006/tpbi.1995.1020.  Google Scholar [27] M. R. Owen and M. A. Lewis, How predation can slow, stop or reverse a prey invasion, Bulletin of Mathematical Biology, 63 (2001), 665-684.  doi: 10.1006/bulm.2001.0239.  Google Scholar [28] S. X. Pan, Invasion speed of a predator-prey system, Applied Mathematics Letters, 74 (2017), 46-51.  doi: 10.1016/j.aml.2017.05.014.  Google Scholar [29] S. X. Pan and G. Lin, Propagation of second order integrodifference equations with local monotonicity, Nonlinear Analysis: Real World Applications, 12 (2011), 535-544.  doi: 10.1016/j.nonrwa.2010.06.038.  Google Scholar [30] S. V. Petrovskii and H. Malchow, A minimal model of pattern formation in a prey-predator system, Mathematical and Computer Modelling, 29 (1999), 49-63.  doi: 10.1016/S0895-7177(99)00070-9.  Google Scholar [31] S. V. Petrovskii and H. Malchow, Critical phenomena in plankton communities: KISS model revisited, Nonlinear Analysis: Real World Applications, 1 (2000), 37-51.  doi: 10.1016/S0362-546X(99)00392-2.  Google Scholar [32] J.-M. Roquejoffre, D. Terman and V. A. Volpert, Global stability of traveling fronts and convergence towards stacked families of waves in monotone parabolic systems, SIAM Journal on Mathematical Analysis, 27 (1996), 1261-1269.  doi: 10.1137/S0036141094267522.  Google Scholar [33] J. A. Sherratt, Invasion generates periodic traveling waves (wavetrains) in predator-prey models with nonlocal dispersal, SIAM Journal on Applied Mathematics, 76 (2016), 291-313.  doi: 10.1137/15M1027991.  Google Scholar [34] J. A. Sherratt, A. S. Dagbovie and F. M. Hilker, A mathematical biologist's guide to convective and absolute stability, Bulletin of Mathematical Biology, 76 (2014), 1-26.  doi: 10.1007/s11538-013-9911-9.  Google Scholar [35] H. F. Weinberger, Asymptotic behavior of a model in population genetics, Nonlinear Partial Differential Equations and Applications, Lecture Notes in Math., Springer, Berlin, 648 (1978), 47-96.   Google Scholar [36] H. F. Weinberger, Long-time behavior of a class of biological models, SIAM Journal on Mathematical Analysis, 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar [37] H. Weinberger and X.-Q. Zhao, An extension of the formula for spreading speeds, Mathematical Biosciences and Engineering, 7 (2010), 187-194.  doi: 10.3934/mbe.2010.7.187.  Google Scholar [38] T. S. Yi and X. F. Zou, Asymptotic behavior, spreading speeds and traveling waves of nonmonotone dynamical systems, SIAM Journal on Mathematical Analysis, 47 (2015), 3005-3034.  doi: 10.1137/14095412X.  Google Scholar [39] Z.-X. Yu and R. Yuan, Properties of traveling waves for integrodifference equation with nonmonotone growth functions, Zeitschrift fuer Angewandte Mathematik und Physik, 63 (2012), 249-259.  doi: 10.1007/s00033-011-0170-z.  Google Scholar

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References:
 [1] A. Bourgeois, Spreading Speeds and Travelling Waves in Intergrodifference Equations with Overcompensatory Dynamics, Master's Thesis, University of Ottawa, 2016, URL http://hdl.handle.net/10393/34578. Google Scholar [2] A. Bourgeois, V. LeBlanc and F. Lutscher, Spreading phenomena in integrodifference equations with non-monotone growth functions, SIAM Journal of Applied Mathematics, 78 (2018), 2950-2972.  doi: 10.1137/17M1126102.  Google Scholar [3] R. M. Coutinho, W. A. C. Godoy and R. A. Kraenkel, Integrodifference model for blowfly invasion, Theoretical Ecology, 5 (2012), 363-371.  doi: 10.1007/s12080-012-0157-1.  Google Scholar [4] A. S. Dagbovie and J. A. Sherratt, Absolute stability and dynamical stabilisation in predator-prey systems, Journal of Mathematical Biology, 68 (2014), 1403-1421.  doi: 10.1007/s00285-013-0672-8.  Google Scholar [5] G. de Vries, T. Hillen, M. Lewis, J. Müller and B. Schönfisch, A Course in Mathematical Biology: Quantitative Modeling with Mathematical and Computational Methods, Mathematical Modeling and Computation, 12. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. doi: 10.1137/1.9780898718256.  Google Scholar [6] R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, Vol. 20. Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar [7] A. Ducrot, Spatial propagation for a two component reaction-diffusion system arising in population dynamics, Journal of Differential Equations, 260 (2016), 8316-8357.  doi: 10.1016/j.jde.2016.02.023.  Google Scholar [8] W. F. Fagan and J. G. Bishop, Trophic interactions during primary succession: Herbivores slow a plant reinvasion at mount st. helens, The American Naturalist, 155 (2000), 238-251.  doi: 10.1086/303320.  Google Scholar [9] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Archive for Rational Mechanics and Analysis, 65 (1977), 335-361.  doi: 10.1007/BF00250432.  Google Scholar [10] A. Gharouni, M. A. Barbeau, A. Locke, L. Wang and J. Watmough, Sensitivity of invasion speed to dispersal and demography: An application of spreading speed theory to the green crab invasion on the northwest atlantic coast, Marine Ecology Progress Series, 541 (2015), 135-150.  doi: 10.3354/meps11508.  Google Scholar [11] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar [12] L. N. Howard and N. Kopell, Slowly varying waves and shock structures in reaction-diffusion equations, Studies in Applied Mathematics, 56 (1976/77), 95-145.  doi: 10.1002/sapm197756295.  Google Scholar [13] S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM Journal on Mathematical Analysis, 40 (2008), 776-789.  doi: 10.1137/070703016.  Google Scholar [14] M. Iida, R. Lui and H. Ninomiya, Stacked fronts for cooperative systems with equal diffusion coefficients, SIAM Journal on Mathematical Analysis, 43 (2011), 1369-1389.  doi: 10.1137/100792846.  Google Scholar [15] N. Kopell and L. N. Howard, Plane wave solutions to reaction-diffusion equations, Studies in Applied Mathematics, 52 (1973), 291-328.  doi: 10.1002/sapm1973524291.  Google Scholar [16] M. Kot, Discrete-time traveling waves: Ecological examples, Journal of Mathematical Biology, 30 (1992), 413-436.  doi: 10.1007/BF00173295.  Google Scholar [17] 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.  Google Scholar [18] M. Kot and W. Schaffer, Discrete-time growth-dispersal models, Mathematical Biosciences, 80 (1986), 109-136.  doi: 10.1016/0025-5564(86)90069-6.  Google Scholar [19] M. A. Lewis, S. V. Petrovskii and J. R. Potts, The Mathematics Behind Biological Invasions, Interdisciplinary Applied Mathematics, 44. Springer, 2016. doi: 10.1007/978-3-319-32043-4.  Google Scholar [20] B. T. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, Journal of Mathematical Biology, 58 (2009), 323-338.  doi: 10.1007/s00285-008-0175-1.  Google Scholar [21] G. Lin, Traveling wave solutions for integro-difference systems, Journal of Differential Equations, 258 (2015), 2908-2940.  doi: 10.1016/j.jde.2014.12.030.  Google Scholar [22] H. Malchow and S. V. Petrovskii, Dynamical stabilization of an unstable equilibrium in chemical and biological systems. Mathematical modelling of nonlinear systems, Mathematical and Computer Modelling, 36 (2002), 307-319.  doi: 10.1016/S0895-7177(02)00127-9.  Google Scholar [23] H. Malchow, S. V. Petrovskii and E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulation, Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, FL, 2008.  Google Scholar [24] N. G. Marculis and R. Lui, Modelling the biological invasion of Carcinus maenas (the European green crab), Journal of Biological Dynamics, 10 (2016), 140-163.  doi: 10.1080/17513758.2015.1115563.  Google Scholar [25] R. M. May, Biological populations obeying difference equations: Stable points, stable cycles, and chaos, Journal of Theoretical Biology, 51 (1975), 511-524.  doi: 10.1016/0022-5193(75)90078-8.  Google Scholar [26] M. G. Neubert, M. Kot and M. A. Lewis, Dispersal and pattern formation in a discrete-time predator-prey model, Theoretical Population Biology, 48 (1995), 7-43.  doi: 10.1006/tpbi.1995.1020.  Google Scholar [27] M. R. Owen and M. A. Lewis, How predation can slow, stop or reverse a prey invasion, Bulletin of Mathematical Biology, 63 (2001), 665-684.  doi: 10.1006/bulm.2001.0239.  Google Scholar [28] S. X. Pan, Invasion speed of a predator-prey system, Applied Mathematics Letters, 74 (2017), 46-51.  doi: 10.1016/j.aml.2017.05.014.  Google Scholar [29] S. X. Pan and G. Lin, Propagation of second order integrodifference equations with local monotonicity, Nonlinear Analysis: Real World Applications, 12 (2011), 535-544.  doi: 10.1016/j.nonrwa.2010.06.038.  Google Scholar [30] S. V. Petrovskii and H. Malchow, A minimal model of pattern formation in a prey-predator system, Mathematical and Computer Modelling, 29 (1999), 49-63.  doi: 10.1016/S0895-7177(99)00070-9.  Google Scholar [31] S. V. Petrovskii and H. Malchow, Critical phenomena in plankton communities: KISS model revisited, Nonlinear Analysis: Real World Applications, 1 (2000), 37-51.  doi: 10.1016/S0362-546X(99)00392-2.  Google Scholar [32] J.-M. Roquejoffre, D. Terman and V. A. Volpert, Global stability of traveling fronts and convergence towards stacked families of waves in monotone parabolic systems, SIAM Journal on Mathematical Analysis, 27 (1996), 1261-1269.  doi: 10.1137/S0036141094267522.  Google Scholar [33] J. A. Sherratt, Invasion generates periodic traveling waves (wavetrains) in predator-prey models with nonlocal dispersal, SIAM Journal on Applied Mathematics, 76 (2016), 291-313.  doi: 10.1137/15M1027991.  Google Scholar [34] J. A. Sherratt, A. S. Dagbovie and F. M. Hilker, A mathematical biologist's guide to convective and absolute stability, Bulletin of Mathematical Biology, 76 (2014), 1-26.  doi: 10.1007/s11538-013-9911-9.  Google Scholar [35] H. F. Weinberger, Asymptotic behavior of a model in population genetics, Nonlinear Partial Differential Equations and Applications, Lecture Notes in Math., Springer, Berlin, 648 (1978), 47-96.   Google Scholar [36] H. F. Weinberger, Long-time behavior of a class of biological models, SIAM Journal on Mathematical Analysis, 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar [37] H. Weinberger and X.-Q. Zhao, An extension of the formula for spreading speeds, Mathematical Biosciences and Engineering, 7 (2010), 187-194.  doi: 10.3934/mbe.2010.7.187.  Google Scholar [38] T. S. Yi and X. F. Zou, Asymptotic behavior, spreading speeds and traveling waves of nonmonotone dynamical systems, SIAM Journal on Mathematical Analysis, 47 (2015), 3005-3034.  doi: 10.1137/14095412X.  Google Scholar [39] Z.-X. Yu and R. Yuan, Properties of traveling waves for integrodifference equation with nonmonotone growth functions, Zeitschrift fuer Angewandte Mathematik und Physik, 63 (2012), 249-259.  doi: 10.1007/s00033-011-0170-z.  Google Scholar
Solution of the integrodifference equation (left) and its 'phase plane' (right), where $F$ is the Ricker function with $r = 0.8$ (solid) and $r = 1.03$ (dashed) and $K$ is the Laplace kernel with $a = 15$
Numerical solution of the IDE in 1, plotted for even (top panel) and odd (bottom panel) generations every 10 time steps. The solid lines correspond to the Ricker growth function 2 with $r = 2.2$ and the dashed line to the logistic growth function 3 with $r = 2.44.$ The dispersal kernel is the Laplace kernel 4 with $a = 15.$ The initial condition was the function $N_0(x) = n_-\chi_{x\leq 10}.$
Plot of the implicit functions defined by equations 15 (thin blue lines) and 16 (thicker red line) with $c = c^*$ for the Ricker function with $r = 1.0327$ (left) and $r = 2.526$ (right). Note that there are no negative real roots as we chose $r>r^*.$ Only the upper half plane is plotted; the lower half plane is symmetric
Solution of the integrodifference equation (left) and its phase plane (right), where $F$ is the Ricker function with $r = 1.8$ and $K$ is the Laplace kernel with $a = 15$
. The solid curve corresponds to the Ricker function, the dashed curve to the logistic function, both with parameter $r = 2.2.$ $K$ is the Laplace kernel with $a = 15.$">Figure 5.  Phase plane of the solution in Figure 1. The solid curve corresponds to the Ricker function, the dashed curve to the logistic function, both with parameter $r = 2.2.$ $K$ is the Laplace kernel with $a = 15.$
Solution of the integrodifference equation (left) and its phase plane (right), where $F$ is the Ricker function with $r = 2.525$ and $K$ is the Laplace kernel with $a = 15$
Solution of the integrodifference equation (left) and its phase plane (right), where $F$ is the logistic function with $r = 2.5$ and $K$ is the Laplace kernel with $a = 15$
Solution of the IDE, with Ricker function and Laplace kernel ($a = 15$). The growth parameter is $r = 2.6,$ so that the two-cycle $n_\pm$ is unstable for the Ricker dynamics and a stable four-cycle exist (denoted by $n^-_-, n^-_+, n^+_-, n^+_+$). Initial conditions are $N_0 = n^+_+\chi_{[x\geq 10]}.$
Dynamic behavior of the map $N \mapsto F(N)$ with $F$ as in 2 or 3. The abbreviation 'g.a.s.' stands for globally asymptotically stable within all non-stationary, non-negative solutions
 Dynamic behavior Ricker function 2 Logistic function 3 $N^*=1$ g.a.s. $0< r< 1$ $0< r< 1$ monotone approach $N^*=1$ g.a.s. $1< r< 2$ $1< r< 2$ oscillatory approach $N^*=1$ unstable $2< r< 2.526$ $2  Dynamic behavior Ricker function 2 Logistic function 3$ N^*=1 $g.a.s.$ 0< r< 1  0< r< 1 $monotone approach$ N^*=1 $g.a.s.$ 1< r< 2  1< r< 2 $oscillatory approach$ N^*=1 $unstable$ 2< r< 2.526  2
Shape of the traveling profile emerging from $N^* = 0$ in IDE 1 as a function of parameter $r$ for the Ricker and the logistic function and with Laplace dispersal kernel. When the kernel has compact support, monotone traveling waves may not exist even if they do with a Laplace kernel [29]
 Shape of the traveling profile Ricker function Logistic function Monotone on $[0,1]$ $0< r< 1.0327$ $0< r< 1.0686$ Damped oscillations at $N=1$ $1.0327< r< 2.5072$ $1.0686< r< 2.570$ Wavetrain around $N = 1$ $2.5072< r< 2.692$ NA
 Shape of the traveling profile Ricker function Logistic function Monotone on $[0,1]$ $0< r< 1.0327$ $0< r< 1.0686$ Damped oscillations at $N=1$ $1.0327< r< 2.5072$ $1.0686< r< 2.570$ Wavetrain around $N = 1$ $2.5072< r< 2.692$ NA
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