A Free Boundary Problem for the Predator–Prey Model with Double Free Boundaries

Abstract

In this paper we investigate a free boundary problem for the classical Lotka–Volterra type predator–prey model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species and are described by Stefan-like condition. We prove a spreading–vanishing dichotomy for this model, namely the predator species either successfully spreads to infinity as $t\rightarrow \infty$ at both fronts and survives in the new environment, or it spreads within a bounded area and dies out in the long run while the prey species stabilizes at a positive equilibrium state. The long time behavior of solution and criteria for spreading and vanishing are also obtained.

Appendix

1.1 Global Estimate of the Solution w to (4.3)

Proposition 7.1

Let (u, v, g, h) be any solution of (1.3) and assume $h_\infty -g_\infty <\infty$. If w(t, y) is the solution of (4.3), then there exists a constant $K_0$ such that

$$\begin{aligned} \Vert w\Vert _{C^{\frac{1+\alpha }{2},1+\alpha }([1,\infty )\times [-1,1])}<K_0. \end{aligned}$$

(7.1)

Proof

We are inspired by [1, Theorem A2]. For convenience, we denote $\varphi _n(t)=\varphi (t+n), \psi _n=\psi (t+n,y), z_n=z(t+n,y), w_n=w(t+n,y)$. Let $w(t+n,y)=a^n(t,y)+b^n(t,y)$, where $a^n$ and $b^n$ are solutions of

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} a^n_t=\varphi _n(t)a^n_{yy}, \ \ &{}t>0,-1<y<1,\\ a^n(t,-1)=a^n(t,1)=0,\ \ &{}t>0,\\ a^n(0,y)=w(n,y),&{}-1\le y\le 1 \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} b^n_t=\varphi _n(t)b^n_{yy}+\psi _nb^n_y+\psi _na^n_y+w_n(1-w_n+az_n), \ \ &{}t>0,-1<y<1,\\ b^n(t,-1)=b^n(t,1)=0, &{}t>0,\\ b^n(0,y)=0,&{}-1\le y\le 1, \end{array}\right. \end{aligned}$$

respectively. Let $0<\lambda _1\le \lambda _2\le \cdots$ be the eigenvalues of the problem

$$\begin{aligned} -\phi _{yy}=\lambda \phi ,\ -1<y<1;\ \ \phi (-1)=\phi (1)=0, \end{aligned}$$

and let $\phi _1,\phi _2,\cdots$ be the corresponding set of orthonormal eigenfunctions. We may express $a^n$ as

$$\begin{aligned} a^n(t,y)=\sum \limits _{k\ge 1}\exp \left( -\lambda _k\int _0^t\varphi _n(s)ds\right) w^n_k\phi _k, \end{aligned}$$

where $w^n_k=(\phi _k,w(n,\cdot ))_{L^2}$. In view of $0<u\le M_1$ (cf. Lemma 2.1), it follows that

$$\begin{aligned} \left\| \displaystyle \frac{\partial ^{2j}a^n(T)}{\partial y^{2j}}\right\| _{L^2(-1,1)}^2= & {} \sum \limits _{k\ge 1}\lambda _k^{2j}\exp \displaystyle \left( -2\lambda _k\int _0^T\varphi _n(s)ds\right) (w^n_k)^2\\\le & {} \sup \limits _{\ell \ge 0}\left\{ \ell ^{2j}\exp \left( \frac{-8T\ell }{(h_\infty -g_\infty )^2}\right) \right\} \Vert w(n,x)\Vert _{L^2(-1,1)}^2\\\le & {} 2M_1^2\left( \displaystyle \frac{(h_\infty -g_\infty )^2j}{4T}\right) ^{2j}\mathrm{e}^{-2j}. \end{aligned}$$

Thanks to the $L^p$ estimates and Sobolev’s imbedding theorem, we therefore have that $\Vert a^n(t)\Vert _{C^2[-1,1]}\le K_1(1+t^{-j})$, where $K_1$ is independent of n provided that $j\ge 2$. From this last estimate and the differential equation satisfied by $a^n$, we obtain $\Vert a^n_t(t)\Vert _{C[-1,1]}\le K_1h_0^{-2}(1+t^{-j})$. Hence, there exists positive constant $K_2$ such that $\Vert a^n\Vert _{C^{\frac{1+\alpha }{2},1+\alpha }(E_1)}<K_2$, where $E_1=[{1\over 2},2]\times [-1,1]$ and $K_2$ depends only on $K_1$ and $E_1$.

Next we estimate $b^n$. It is obvious that the function $c^n=\mathrm{e}^{-\frac{1}{t}}b^n$ satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} c^n_t=\varphi _n(t)c^n_{yy}+\psi _nc^n_y+f_n(t,x), \ \ &{}t>0,-1<y<1,\\ b^n(t,-1)=b^n(t,1)=0, &{}t>0,\\ b^n(0,y)=0,&{}-1\le y\le 1, \end{array}\right. \end{aligned}$$

where $f_n(0,x)=0$ and

$$\begin{aligned} f_n(t,x)=t^{-2}\mathrm{e}^{-\frac{1}{t}}w_n+(\psi _n+t^{-2}) \mathrm{e}^{-\frac{1}{t}}a^n_y+\mathrm{e}^{-\frac{1}{t}}w_n(1-w_n+az_n),\ t>0. \end{aligned}$$

Note that $\lim \limits _{t\rightarrow 0^+}\displaystyle t^{-j}\mathrm{e}^{-\frac{1}{t}}=0$ for any $j>0$, we have $f_n(t,x)$ is continuous in $E=[0,3]\times [-1,1]$ and $\Vert f_n\Vert _{C(E)}\le K_3$ where $K_3$ is dependent on $K_1$ and independent of n. By using of [14, Theorem 4,p191], we can obtain that $\Vert c^n\Vert _{C^{\frac{1+\alpha }{2},1+\alpha }(E_1)}<\tilde{K}_3$ where $\tilde{K}_3$ depends on $K_3$. It follows that $\Vert b^n\Vert _{C^{\frac{1+\alpha }{2},1+\alpha }(E_1)}<K_4$. We therefore have that

$$\begin{aligned} \Vert w\Vert _{C^{\frac{1+\alpha }{2},1+\alpha }(E_n)}\le \Vert a^n\Vert _{C^{\frac{1+\alpha }{2},1+\alpha }(E_1)}+\Vert b^n\Vert _{C^{\frac{1+\alpha }{2},1+\alpha }(E_1)} <K_2+K_4=K_0, \end{aligned}$$

where $E_n=[n+\frac{1}{2},n+2]\times [-1,1]$. It easily to get (7.1) since the intervals $E_n$ overlap and $K_0$ is independent of n. $\square$

1.2 Estimates of Solutions to Parabolic Partial Differential Inequalities

Let $d,\beta$ and $\theta$ be fixed positive constants. In order to investigate the long time behavior of the solution (u, v) to (1.3), we should prove the following proposition.

Proposition 8.1

Let k be a non-negative constant. For any given $\varepsilon , L>0$, there exist $T_\varepsilon >0$ and $l_\varepsilon >\max \big \{L,\frac{\pi }{2}\sqrt{d/\beta }\big \}$, such that when the continuous and non-negative function z(t, x) satisfies

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} z_t-dz_{xx}\ge \, (\le )\, z(\beta -\theta z), \ \ &{}t>0, \ \ -l_\varepsilon <x<l_\varepsilon ,\\ z(0,x)>0, &{}-l_\varepsilon <x<l_\varepsilon , \end{array}\right. \end{aligned}$$

and for $t>0$, $z(t,\pm l_\varepsilon )\ge \, (\le )\, k$ if $k>0$, while $z(t,\pm l_\varepsilon )\ge \, (=)\, 0$ if $k=0$, then we have

$$\begin{aligned} z(t,x)>\beta /\theta -\varepsilon \ \ \big (z(t,x)<\beta /\theta +\varepsilon \big ), \ \ \ \forall \ t\ge T_\varepsilon , \ \ x\in [-L,L]. \end{aligned}$$

This implies

$$\begin{aligned} \liminf _{t\rightarrow \infty }z(t,x)\ge \beta /\theta -\varepsilon \ \ \left( \limsup _{t\rightarrow \infty }z(t,x)<\beta /\theta +\varepsilon \right) \ \ \ \hbox {uniformly on }\, [-L,L]. \end{aligned}$$

Proof

Let $l>\frac{\pi }{2}\sqrt{d/\beta }$ be a parameter, and $z_l(x)$ the unique positive solution of

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -dz_{xx}=z(\beta -\theta z),\ \ -l<x<l,\\ z(\pm l)=k. \end{array}\right. \end{aligned}$$

(8.1)

(refer to the proof of Lemma 2.3 in [12]). We claim that

$$\begin{aligned} \lim _{l\rightarrow \infty }z_l(x)=\beta /\theta \ \ \hbox { uniformly in any compact subset of} \, \ {\mathbb {R}}. \end{aligned}$$

(8.2)

For the case $k>\beta /\theta$. By the maximum principle we see that $\beta /\theta \le z_l(x)\le k$ for all $x\in [-l,l]$. Note that $z_l(x)\le k$, by the comparison principle we have that $z_l(x)$ is decreasing in l. Therefore, the limit $\lim _{l\rightarrow \infty }z_l(x)=z(x)$ exists, and $z(x)\ge \beta /\theta$ and z(x) satisfies

$$\begin{aligned} -dz_{xx}=z(\beta -\theta z),\ \ x\in {\mathbb {R}}. \end{aligned}$$

By Theorem 1.2 of [12], $z(x)\equiv \beta /\theta$. Using the interior estimate we assert that $\lim _{l\rightarrow \infty }z_l(x)=z(x)$ uniformly in any compact subset of ${\mathbb {R}}$. Hence (8.2) holds.

For the case $k\le \beta /\theta$. Choose $k_0>\beta /\theta$ and let $z_l^0$ be the unique positive solution of (8.1) with $k=k_0$. By the comparison principle we have $w_l(x)\le z_l(x)\le z_l^0(x)$ in $[-l,l]$, where $w_l(x)$ is the unique positive solution of

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -dw_{xx}=w(\beta -\theta w),\ \ -l<x<l,\\ w(\pm l)=0. \end{array}\right. \end{aligned}$$

with $l>\frac{\pi }{2}\sqrt{d/\beta }$. By Lemma 2.2 of [12], $\lim _{l\rightarrow \infty }w_l(x)=\beta /\theta$ uniformly in any compact subset of ${\mathbb {R}}$. Take into account the result we have proved in the above, it is deduced that (8.2) holds.

In view of (8.2), for any given $L>0$ and $\varepsilon >0$, there is $l_\varepsilon >\max \big \{L,\frac{\pi }{2}\sqrt{d/\beta }\big \}$, which also depends on $d,\beta ,\theta$ and k, such that

$$\begin{aligned} \beta /\theta -\varepsilon /2<z_l(x)<\beta /\theta +\varepsilon /2, \ \ \ \forall \ l\ge l_\varepsilon , \ x\in [-L,L]. \end{aligned}$$

(8.3)

Let $z_0(x)\in C([-l_\varepsilon ,l_\varepsilon ])$ be a positive function and $z_\varepsilon (t,x)$ be the solution of

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} z_{t}-dz_{xx}=z(\beta -\theta z), \ \ &{}t>0, \ \ -l_\varepsilon <x<l_\varepsilon ,\\ z(t,\pm l_\varepsilon )=k,&{}t\ge 0,\\ z(0,x)=z_0(x),&{}-l_\varepsilon \le x\le l_\varepsilon . \end{array}\right. \end{aligned}$$

Recall $l_\varepsilon >\frac{\pi }{2}\sqrt{d/\beta }$, we shall illustrate that

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty } z_\varepsilon (t,x)=z_{l_\varepsilon }(x)\ \ \hbox { uniformly in the compact subset of} \ \ (-l_\varepsilon ,l_\varepsilon ). \end{aligned}$$

(8.4)

In fact, take a positive constant q and let $\phi _q(t,x)$ be the unique solution of

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \phi _t-d \phi _{xx}=\phi (\beta -\theta \phi ), \ \ &{}t>0, \ \ -l_\varepsilon <x<l_\varepsilon ,\\ \phi (t,\pm l_\varepsilon )=k,&{}t\ge 0,\\ \phi (0,x)=q,&{}-l_\varepsilon \le x\le l_\varepsilon . \end{array}\right. \end{aligned}$$

Let $M\gg 1$ and $0<m\ll 1$. Then M and m are the ordered upper and lower solutions of (8.1) with $l=l_\varepsilon$. Therefore, $\phi _M(t,x)$ is monotone decreasing and $\phi _m(t,x)$ is monotone increasing in t. So, the limits $\lim _{t\rightarrow \infty }\phi _M(t,x)=\phi _M(x)$ and $\lim _{t\rightarrow \infty }\phi _m(t,x)=\phi _m(x)$ exist, and they are all positive solution of (8.1) with $l=l_\varepsilon$. Hence, $\phi _M(x)=\phi _m(x)=z_{l_\varepsilon }(x)$. Meanwhile, the comparison principle yields $\phi _m(t,x)\le z_\varepsilon (t,x)\le \phi _M(t,x)$. Consequently, $\lim _{t\rightarrow \infty } z_\varepsilon (t,x)=z_{l_\varepsilon }(x)$. By use of the interior estimate, it can be shown that this limit converges uniformly in the compact subset of $(-l_\varepsilon ,l_\varepsilon )$.

Thanks to (8.3) and (8.4), there is $T_\varepsilon \gg 1$ such that

$$\begin{aligned} \beta /\theta -\varepsilon <z_\varepsilon (t,x)<\beta /\theta +\varepsilon , \ \ \ \forall \ t\ge T_\varepsilon , \ \ x\in [-L,L]. \end{aligned}$$

By sue of this fact and the comparison principle, the proof is immediately completed. $\square$