On a Conjecture Raised by Yuzo Hosono

Abstract

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.

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

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

and

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

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

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

(6.1)

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

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

(6.2)

where

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

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 (U, V)(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.

Proof

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 (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 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) $(\underline{U},\underline{V})(z)\le (\bar{U},\bar{V})(z),$ 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

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

(6.3)

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 (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 $z\in \mathbb {R}$.