Skip to main content

On a Conjecture Raised by Yuzo Hosono


In this paper, we study the speed selection mechanism for traveling wave solutions to a two-species 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.

This is a preview of subscription content, access via your institution.

Fig. 1


  1. 1.

    Diekmann, O.: Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol. 6, 109–130 (1979)

    MathSciNet  Article  MATH  Google Scholar 

  2. 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)

    MathSciNet  MATH  Google Scholar 

  3. 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)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Hosono, Y.: Singular perturbation analysis of traveling waves for diffusive Lotka–Volterra competing models. Numer. Appl. Math. 2, 687–692 (1989)

    Google Scholar 

  5. 5.

    Hosono, Y.: Traveling waves for diffusive Lotka–Volterra competition model ii: a geometric approach. Forma 10, 235–257 (1995)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Hosono, Y.: The minimal speed of traveling fronts for diffusive Lotka–Volterra competition model. Bull. Math. Biol. 60, 435–448 (1998)

    Article  MATH  Google Scholar 

  7. 7.

    Huang, W.: Problem on minimum wave speed for Lotka–Volterra reaction–diffusion competition model. J. Dym. Differ. Equ. 22, 285–297 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Huang, W., Han, M.: Non-linear determinacy of minimum wave speed for Lotka–Volterra competition model. J. Differ. Equ. 251, 1549–1561 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Kan-on, Y.: Fisher wave fronts for the Lotka–Volterra competition model with diffusion. Nonlinear Anal. 28, 145–164 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Lewis, M.A., Li, B., Weinberger, H.F.: Spreading speed and linear determinacy for two-species competition models. J. Math. Biol. 45, 219–233 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Li, B., Weinberger, H.F., Lewis, M.A.: Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosci. 196, 82–98 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  12. 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)

    MathSciNet  Article  MATH  Google Scholar 

  13. 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)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Ma, S.: Traveling wavefronts for delayed reaction–diffusion systems via a fixed point theorem. J. Differ. Equ. 171, 294–314 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Murray, J.D.: Mathematical Biology: I and II. Springer, Heidelberg (1989)

    Book  Google Scholar 

  16. 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)

    Article  Google Scholar 

  17. 17.

    Puckett, M.: Minimum wave speed and uniqueness of monotone traveling wave solutions. Ph.D. Thesis, The University of Alabama in Huntsville (2009)

  18. 18.

    Rothe, F.: Convergence to pushed fronts. J. Rocky Mt. J. Math. 11(4), 617–633 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Sabelnikov, V.A., Lipatnikov, A.N.: Speed selection for traveling-wave solutions to the diffusion–reaction equation with cubic reaction term and burgers nonlinear convection. Phys. Rev. E 90, 033004 (2014)

    Article  Google Scholar 

  20. 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. 21.

    Weinberger, H.: On sufficient conditions for a linearly determinate spreading speed. Discrete Contin. Dyn. Syst. Ser. B 17(6), 2267–2280 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  22. 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)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Chunhua Ou.

Additional information

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

$$\begin{aligned} \alpha U+U(1-a_1-U+V):=F(U,V) \end{aligned}$$


$$\begin{aligned} \alpha V +r(1-V)(a_2U-V):=G(U,V) \end{aligned}$$

are monotone in U and V, respectively. Equations in (2.1) are equivalent to

$$\begin{aligned} \left\{ \begin{aligned} U''+cU'-\alpha U&=-F(U,V),\\ cV'-\alpha V&=-G(U,V).\\ \end{aligned} \right. \end{aligned}$$

Define constants \(\lambda ^\pm _1\) as

$$\begin{aligned} \lambda ^-_1=\frac{-c-\sqrt{c^2+4\alpha }}{2}<0 \ \ \text {and }\ \ \lambda ^+_1=\frac{-c+\sqrt{c^2+4\alpha }}{2}>0. \end{aligned}$$

By applying the variation-of-parameter 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

$$\begin{aligned} \left\{ \begin{array}{c} U(z)=T_1(U,V)(z),\\ V(z)=T_2(U,V)(z), \end{array}\right. \end{aligned}$$


$$\begin{aligned} T_1(U,V)(z)&=\frac{1}{\lambda ^+_1-\lambda ^-_1}\left\{ \int _{-\infty }^{z} e^{\lambda ^-_1(z-s)}F(U,V)(s)ds+\int _{z}^{\infty } e^{\lambda ^+_1(z-s)}F(U,V)(s)ds\right\} ,\\ T_2(U,V)(z)&=\frac{1}{c}\int _{z}^{\infty } e^{\frac{\alpha }{c}(z-s)}G(U,V)(s) ds. \end{aligned}$$

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

$$\begin{aligned} \left\{ \begin{array}{c} U(z)\ge (\le ) \ T_1(U,V)(z),\\ V(z)\ge (\le ) \ T_2(U,V)(z). \end{array}\right. \end{aligned}$$

Lemma 6.1

A continuous function (UV)(z) which is differentiable on \(\mathbb {R}\) except at finite number of points \(z_i, i=1,\ldots ,n\), and satisfies

$$\begin{aligned} \left\{ \begin{array}{l} U''+ cU'+U(1-a_1-U+a_1V) \le \ 0,\\ cV'+ r(1-V)(a_2U-V) \le \ 0 \end{array}\right. \end{aligned}$$

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.


We give the proof for the upper solution where the same argument can be applied for the lower solution. From

$$\begin{aligned}&U''+cU'-\alpha U +F(U,V)\le 0\\&cV'-\alpha V + G(U,V) \le 0, \end{aligned}$$

we have

$$\begin{aligned} T_1(U,V)(z)&=\frac{1}{\lambda ^+_1-\lambda ^-_1} \left\{ \int _{-\infty }^{z} e^{\lambda ^-_1(z-s)}F(U,V)(s)ds +\int _{z}^{\infty } e^{\lambda ^+_1(z-s)}F(U,V)(s)ds\right\} \\&\le \frac{-1}{\lambda ^+_1-\lambda ^-_1} \left\{ \int _{-\infty }^{z} e^{\lambda ^-_1(z-s)}(U''+cU'-\alpha U)(s)ds\right. \\&\quad \left. +\int _{z}^{\infty } e^{\lambda ^+_1(z-s)}(U''+cU'-\alpha U)(s)ds\right\} . \end{aligned}$$

Simple computations as that in [14, proof of Lemma 2.5] yield

$$\begin{aligned} T_1(U,V)(z)\le U(z). \end{aligned}$$

Similarly \(T_2(U,V)\le V(z)\). This implies that (UV)(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 non-increasing upper solution \((\bar{U},\bar{V})(z)\) and a non-zero lower solution \((\underline{U},\underline{V})(z)\) to the system (6.2) with the following properties:

  1. (1)

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

  2. (2)

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

  3. (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

$$\begin{aligned} \left\{ \begin{aligned}&(U_0,V_0) =(\bar{U},\bar{V}),\\&U_{n+1} = T_1(U_n,V_n),&n=0,1,2,\ldots ,\\&V_{n+1}= T_2(U_n,V_n),&n=0,1,2,\ldots , \end{aligned}\right. \end{aligned}$$

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 non-increasing function (UV)(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 \(z\in \mathbb {R}\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alhasanat, A., Ou, C. On a Conjecture Raised by Yuzo Hosono. J Dyn Diff Equat 31, 287–304 (2019).

Download citation


  • Lotka–Volterra
  • Traveling waves
  • Speed selection

Mathematics Subject Classification

  • 35K40
  • 35K57
  • 92D25