Abstract
In this paper, we study the speed selection mechanism for traveling wave solutions to a twospecies Lotka–Volterra competition model. After transforming the partial differential equations into a cooperative system, the speed selection mechanism (linear vs. nonlinear) is investigated for the new system. Hosono conjectured that there is a critical value \(r_c\) of the birth rate so that the speed selection mechanism changes only at this value. In the absence of diffusion for the second species, we obtain the speed selection mechanism and successfully prove a modified version of the Hosono’s conjecture. Estimation of the critical value is given and some new conditions for linear or nonlinear selection are established.
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References
 1.
Diekmann, O.: Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol. 6, 109–130 (1979)
 2.
Fei, N., Carr, J.: Existence of travelling waves with their minimal speed for a diffusing Lotka–Volterra system. Nonlinear Anal. 4, 504–524 (2003)
 3.
Guo, J., Liang, X.: The minimal speed of traveling fronts for the Lotka–Volterra competition system. J. Dyn. Differ. Equ. 2, 353–363 (2011)
 4.
Hosono, Y.: Singular perturbation analysis of traveling waves for diffusive Lotka–Volterra competing models. Numer. Appl. Math. 2, 687–692 (1989)
 5.
Hosono, Y.: Traveling waves for diffusive Lotka–Volterra competition model ii: a geometric approach. Forma 10, 235–257 (1995)
 6.
Hosono, Y.: The minimal speed of traveling fronts for diffusive Lotka–Volterra competition model. Bull. Math. Biol. 60, 435–448 (1998)
 7.
Huang, W.: Problem on minimum wave speed for Lotka–Volterra reaction–diffusion competition model. J. Dym. Differ. Equ. 22, 285–297 (2010)
 8.
Huang, W., Han, M.: Nonlinear determinacy of minimum wave speed for Lotka–Volterra competition model. J. Differ. Equ. 251, 1549–1561 (2011)
 9.
Kanon, Y.: Fisher wave fronts for the Lotka–Volterra competition model with diffusion. Nonlinear Anal. 28, 145–164 (1997)
 10.
Lewis, M.A., Li, B., Weinberger, H.F.: Spreading speed and linear determinacy for twospecies competition models. J. Math. Biol. 45, 219–233 (2002)
 11.
Li, B., Weinberger, H.F., Lewis, M.A.: Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosci. 196, 82–98 (2005)
 12.
Liang, X., Zhao, X.Q.: Asymptotic speed of spread and traveling waves for monotone semiflows with applications. Commun. Pure Appl. Math. 60, 1–40 (2007)
 13.
Lucia, M., Muratov, C.B., Novaga, M.: Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction–diffusion equations invading an unstable equilibrium. Commun. Pure Appl. Math. 57, 616–636 (2004)
 14.
Ma, S.: Traveling wavefronts for delayed reaction–diffusion systems via a fixed point theorem. J. Differ. Equ. 171, 294–314 (2001)
 15.
Murray, J.D.: Mathematical Biology: I and II. Springer, Heidelberg (1989)
 16.
Okubo, A., Maini, P.K., Williamson, M.H., Murray, J.D.: On the spatial spread of the grey squirrel in britain. Proc. R. Soc. Lond. Ser. B Biol. Sci. 238, 113–125 (1989)
 17.
Puckett, M.: Minimum wave speed and uniqueness of monotone traveling wave solutions. Ph.D. Thesis, The University of Alabama in Huntsville (2009)
 18.
Rothe, F.: Convergence to pushed fronts. J. Rocky Mt. J. Math. 11(4), 617–633 (1981)
 19.
Sabelnikov, V.A., Lipatnikov, A.N.: Speed selection for travelingwave solutions to the diffusion–reaction equation with cubic reaction term and burgers nonlinear convection. Phys. Rev. E 90, 033004 (2014)
 20.
Volpert, A.I., Volpert, V.A., Volpert, V.A.: Traveling wave solutions of parabolic systems. Translations of Mathematical Monographs, vol. 140. American Mathematical Society (1994)
 21.
Weinberger, H.: On sufficient conditions for a linearly determinate spreading speed. Discrete Contin. Dyn. Syst. Ser. B 17(6), 2267–2280 (2012)
 22.
Weinberger, H.F., Lewis, M.A., Li, B.: Analysis of linear determinacy for spread in cooperative models. J. Math. Biol. 45, 183–218 (2002)
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Chunhua Ou: This work is partially supported by the NSERC discovery Grant.
Appendix: Upper–Lower Solution Method
Appendix: Upper–Lower Solution Method
A useful method to prove the existence of monotone traveling wave solution is the upper–lower solution technique originated in Diekmann [1]. Here we illustrate the main idea. By transforming the system (2.1) to a system of integral equations, we can define a monotone iteration scheme in terms of the integral system. By construction an upper and a lower solutions to the system and using the iteration scheme, we can give the existence of traveling wave solutions.
Let \(\alpha \) be a sufficiently large positive number so that
and
are monotone in U and V, respectively. Equations in (2.1) are equivalent to
Define constants \(\lambda ^\pm _1\) as
By applying the variationofparameter method to the first equation in the system (6.1), and the first order differential equation theory to the second equation, the system can be written in the form
where
Definition 2
A pair of continuous functions (U(z), V(z)) is an upper (a lower) solution to the integral equations system (6.2) if
Lemma 6.1
A continuous function (U, V)(z) which is differentiable on \(\mathbb {R}\) except at finite number of points \(z_i, i=1,\ldots ,n\), and satisfies
for \(z \not =z_i\), and \(U'(z_i^)\ge U'(z_i^+)\), for all \(z_i\), is an upper solution to the integral equations system (6.2). The same result is true for the lower solution by reversing the inequalities.
Proof
We give the proof for the upper solution where the same argument can be applied for the lower solution. From
we have
Simple computations as that in [14, proof of Lemma 2.5] yield
Similarly \(T_2(U,V)\le V(z)\). This implies that (U, V)(z) is an upper solution to the system (6.2). \(\square \)
The existence of an upper and a lower solution to the system (6.2) will give the existence of the actual traveling wave solution. Indeed, for our problem, we assume that the following hypothesis is true.
Hypothesis 1
There exists a monotone nonincreasing upper solution \((\bar{U},\bar{V})(z)\) and a nonzero lower solution \((\underline{U},\underline{V})(z)\) to the system (6.2) with the following properties:

(1)
\((\underline{U},\underline{V})(z)\le (\bar{U},\bar{V})(z),\) for all \(z\in \mathbb {R}\),

(2)
\((\bar{U},\bar{V})(+\infty )=e_0 , \ \ \ \ (\bar{U},\bar{V})(\infty )=(\bar{k}_1,\bar{k}_2),\)

(3)
\((\underline{U},\underline{V})(+\infty )=e_0 , \ \ \ (\underline{U},\underline{V})(\infty )=(\underline{k}_1,\underline{k}_2),\)
for \(e_0\le (\underline{k}_1,\underline{k}_2)\le e_1\) and \((\bar{k}_1,\bar{k}_2)\ge e_1=(1,1)\) so that no equilibrium solution to (2.1) exists in the set \(\{(U,V) e_1<(U,V)\le (\bar{k}_1,\bar{k}_2)\}\). \(\square \)
From the integral system, we define an iteration scheme as
and arrive at the following result by the upper–lower solution method, see e.g. [1].
Theorem 6.2
If Hypothesis 1 holds, then the iteration (6.3) converges to a nonincreasing function (U, V)(z), which is a solution to the system (2.1) with \((U,V)(\infty )=e_1\) and \((U,V)(\infty )=e_0\). Moreover, \((\underline{U},\underline{V})(z)\le (U,V)(z)\le (\bar{U},\bar{V})(z)\) for all \(z\in \mathbb {R}\).
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Alhasanat, A., Ou, C. On a Conjecture Raised by Yuzo Hosono. J Dyn Diff Equat 31, 287–304 (2019). https://doi.org/10.1007/s1088401896515
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Keywords
 Lotka–Volterra
 Traveling waves
 Speed selection
Mathematics Subject Classification
 35K40
 35K57
 92D25