Global dynamics of a generalist predator–prey model in open advective environments

Abstract

This paper deals with a system of reaction–diffusion–advection equations for a generalist predator–prey model in open advective environments, subject to an unidirectional flow. In contrast to the specialist predator–prey model, the dynamics of this system is more complex. It turns out that there exist some critical advection rates and predation rates, which classify the global dynamics of the generalist predator–prey system into three or four scenarios: (1) coexistence; (2) persistence of prey only; (3) persistence of predators only; and (4) extinction of both species. Moreover, the results reveal significant differences between the specialist predator–prey system and the generalist predator–prey system, including the evolution of the critical predation rates with respect to the ratio of the flow speeds; the take-over of the generalist predator; and the reduction in parameter range for the persistence of prey species alone. These findings may have important biological implications on the invasion of generalist predators in open advective environments.

Appendix

For completeness and the reader’s convenience, we provide the proof of Lemma 4.5 here via the comparison principle and uniform persistence theory although its proof is exactly similar to Theorems 1.1 and 1.2 of Nie et al. (2020).

Proof of Lemma 4.5

(i) Since the positive solution of (1.2) satisfies $N(x,t)>0$ and $P(x,t)>0$ for $x\in [0,1]$ and $t>0$ (see Lemma 4.4), we have

$$\begin{aligned} N_{t}\le d_{1}N_{xx}-q_1N_{x}+r_{1}N(1-\frac{N}{K_{1}}),\ x\in (0,1),\ t>0. \end{aligned}$$

Let ${\mathcal {N}}(x,t)$ be the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {N}}_{t}= d_{1}{\mathcal {N}}_{xx}-q_1{\mathcal {N}}_{x}+r_{1}{\mathcal {N}}(1-\frac{{\mathcal {N}}}{K_{1}}),&{} x\in (0,1),\ t>0,\\ d_{1}{\mathcal {N}}_{x}(0,t)-q_1{\mathcal {N}}(0,t)=0,\ {\mathcal {N}}_{x}(1,t)=0,&{} t>0,\\ {\mathcal {N}}(x,0)=N_{0}(x)\ge 0,\not \equiv 0, &{} x\in [0,1].\\ \end{array}\right. } \end{aligned}$$

(7.1)

The comparison principle for parabolic equations yields that $N(x, t)\le {\mathcal {N}}(x, t)$ for all $x\in [0,1],\ t>0$. In view of $q_1\ge q_{1}^{*}$, by Lemma 2.1, we conclude that ${\mathcal {N}}(x,t)\rightarrow 0,\ x\in [0,1]$ as $t\rightarrow +\infty$. Thus $\lim \limits _{t\rightarrow +\infty }N(x,t)=0$ uniformly in [0, 1]. Hence for any $\epsilon >0$, there exists $T_{\epsilon }>0$ such that $N(x,t)\le \epsilon$ for all $x\in [0,1],\ t\ge T_{\epsilon }$. Furthermore,

$$\begin{aligned} P_{t}\le d_{2}P_{xx}-q_2P_{x}+(r_{2}+ea\epsilon )P\ \text{ for }\ x\in (0,1),\ t\ge T_{\epsilon }. \end{aligned}$$

Let ${\mathcal {P}}(x, t)$ be the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {P}}_{t}=d_{2}{\mathcal {P}}_{xx}-q_2{\mathcal {P}}_{x}+(r_{2}+ea\epsilon ){\mathcal {P}},&{} x\in (0,1),\ t\ge T_{\epsilon },\\ d_{2}{\mathcal {P}}_{x}(0,t)-q_2{\mathcal {P}}(0,t)=0,\ {\mathcal {P}}_{x}(1,t)=0,&{} t\ge T_{\epsilon },\\ {\mathcal {P}}(x,T_{\epsilon })=P(x,T_{\epsilon }),&{} x\in [0,1].\\ \end{array}\right. } \end{aligned}$$

The comparison principle implies $P(x,t)\le {\mathcal {P}}(x,t)$ for all $x\in [0,1]$, $t\ge T_{\epsilon }$. Since $\mu _{1}(d_{2},q_2,r_{2})<0$ when $q_2>q_{2}^{*}$, we conclude that there exists $\epsilon >0$ sufficiently small such that $\mu _{1}(d_{2},q_2,r_{2}+ea\epsilon )<0$ for $q_2>q_{2}^{*}$. By the method of variable separation we get $\lim \limits _{t\rightarrow +\infty }{\mathcal {P}}(x,t)=0$, $x\in [0,1]$, which implies that $\lim \limits _{t\rightarrow +\infty }P(x,t)=0$ for all $x\in [0,1]$. Thus, the solution (N(x, t), P(x, t)) of system (1.2) converges to (0, 0) uniformly for $x\in [0,1]$ as $t\rightarrow +\infty .$

(ii) Recall that $N(x, t)\le {\mathcal {N}}(x, t)$ for all $x\in [0,1],\ t>0$, where ${\mathcal {N}}(x, t)$ is the solution of (7.1). Observe that the existence of $\theta _1$ means $q_1<q_{1}^{*}$. It follows from Lemma 2.1 that $\lim \limits _{t\rightarrow +\infty }{\mathcal {N}}(x,t)=\theta _{1}$ uniformly for $x\in [0,1]$. This implies that

$$\begin{aligned} \limsup \limits _{t\rightarrow +\infty }N(x,t)\le \theta _{1}\ \text{ uniformly } \text{ for }\ x\in [0,1]. \end{aligned}$$

(7.2)

Then for any $\epsilon >0$, there exists $T_{1}>0$ such that $N(x,t)<\theta _{1}+\epsilon$ for all $x\in [0,1],\ t\ge T_{1}$. Let $\mathbf{P} (x,t)$ satisfy the following equation,

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathbf {P}_{t}=d_{2}\mathbf {P}_{xx}-q_2\mathbf {P}_{x}+r_{2}\mathbf {P}(1-\frac{\mathbf {P}}{K_{2}})+ea\mathbf {P}(\theta _{1}+\epsilon ),&{} x\in (0,1),\ t\ge T_{1},\\ d_{2}\mathbf {P}_{x}(0,t)-q_2\mathbf {P}(0,t)=0,\ \mathbf {P}_{x}(1,t)=0,&{} t\ge T_{1},\\ \mathbf {P}(x,T_{1})=P(x,T_{1}),&{} x\in [0,1].\\ \end{array}\right. } \end{aligned}$$

Easily we know that $P(x, t)\le \mathbf{P} (x,t)$ for all $x\in [0,1],\ t\ge T_{1}$ by using the comparison principle. Since $\mu _{1}(d_{2},q_2, r_{2}+ea\theta _1(q_1))<0$, there exists $\epsilon >0$ small enough such that $\mu _{1}(d_{2},q_2, r_{2}+ea(\theta _1(q_1)+\epsilon ))<0$. Similar arguments as in Lemma 2.1 yield that $\lim \limits _{t\rightarrow +\infty }{} \mathbf{P} (x,t)=0,\ x\in [0,1]$, thus $\lim \limits _{t\rightarrow +\infty }P(x,t)=0$ uniformly for $x\in [0,1]$. Therefore, for any $\epsilon >0$, there exists $T_{2}>T_{1}$ such that $P(x,t)\le \epsilon$ for all $x\in [0,1],\ t\ge T_{2}$, which leads to

$$\begin{aligned} N_{t}\ge d_{1}N_{xx}-q_1N_{x}+r_{1}N(1-\frac{N}{K_{1}})-aN\epsilon ,\ x\in (0,1),\ t\ge T_{2}. \end{aligned}$$

Let $\mathbf{N} (x,t)$ be the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathbf {N}_{t}= d_{1}\mathbf {N}_{xx}-q_1\mathbf {N}_{x}+r_{1}\mathbf {N}(1-\frac{\mathbf {N}}{K_{1}})-a\mathbf {N}\epsilon ,&{} x\in (0,1),\ t\ge T_{2},\\ d_{1}\mathbf {N}_{x}(0,t)-q_1\mathbf {N}(0,t)=0,\ \mathbf {N}_{x}(1,t)=0,&{} t\ge T_{2},\\ \mathbf {N}(x,T_{2})=N(x,T_{2}), &{} x\in [0,1].\\ \end{array}\right. } \end{aligned}$$

(7.3)

The comparison principle implies $N(x,t)\ge \mathbf{N} (x,t)$ for all $x\in [0,1],\ t\ge T_{2}$. Noting that $\mu _{1}(d_{1},q_1,r_{1})>0$ based on $q_1<q_{1}^{*}$, we can choose $\epsilon >0$ sufficiently small such that $\mu _{1}(d_{1},q_1,r_{1}-a\epsilon )>0$. Similar arguments as in Lemma 2.1 yield that $\lim \limits _{t\rightarrow +\infty }N(x,t)=\mathbf{N} _{\epsilon }^{*}$ uniformly for $x\in [0,1]$, where $\mathbf{N} _{\epsilon }^{*}$ is the unique positive steady-state solution of (7.3). Just as Lemma 4.2, we can obtain that $0<\mathbf{N} _{\epsilon }^{*}<K_{1}-\frac{aK_{1}\epsilon }{r_{1}}$. Integrating the steady-state system of (7.3) over (0, x), easily we have both $(\mathbf{N} _{\epsilon }^{*})_{x}$ and $(\mathbf{N} _{\epsilon }^{*})_{xx}$ are uniformly bounded in [0, 1]. By $L^{p}$ estimates and Sobolev embedding theorem, we can deduce that $\mathbf{N} _{\epsilon }^{*}\rightarrow \theta _{1}$ as $\epsilon \rightarrow 0$. That is

$$\begin{aligned} \liminf \limits _{t\rightarrow +\infty }N(x,t)\ge \theta _{1}\ \text{ uniformly } \text{ for }\ x\in [0,1]. \end{aligned}$$

(7.4)

It follows from (7.2) and (7.4) that (ii) holds.

(iii) Easily we have

$$\begin{aligned} P_{t}\ge d_{2}P_{xx}-q_2P_{x}+r_{2}P\left( 1-\frac{P}{K_{2}}\right) ,\ x\in (0,1),\ t>0 \end{aligned}$$

since the positive solution of (1.2) satisfies $N(x,t)>0$ and $P(x,t)>0$ for $x\in [0,1]$ and $t>0$. Let ${\hat{P}}(x,t)$ be the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} {\hat{P}}_{t}= d_{2}{\hat{P}}_{xx}-q_2{\hat{P}}_{x}+r_{2}{\hat{P}}(1-\frac{{\hat{P}}}{K_{2}}),&{} x\in (0,1),\ t>0,\\ d_{2}{\hat{P}}_{x}(0,t)-q_2{\hat{P}}(0,t)=0,\ {\hat{P}}_{x}(1,t)=0,&{} t>0,\\ {\hat{P}}(x,0)=P_{0}(x)\ge 0,\not \equiv 0, &{} x\in [0,1].\\ \end{array}\right. } \end{aligned}$$

The comparison principle for parabolic equations yields that $P(x, t)\ge {\hat{P}}(x, t)$ for all $x\in [0,1],\ t>0$. Observe that the existence of $\theta _2$ means $q_2<q_{2}^{*}$. It follows from Lemma 2.1 that $\lim \limits _{t\rightarrow +\infty }{\hat{P}}(x,t)=\theta _{2}$ uniformly for $x\in [0,1]$. This implies that

$$\begin{aligned} \liminf \limits _{t\rightarrow +\infty }P(x,t)\ge \theta _{2}\ \text{ uniformly } \text{ for }\ x\in [0,1]. \end{aligned}$$

(7.5)

Then for any $\epsilon >0$, there exists $T_{3}>0$ such that $P(x,t)>\theta _{2}-\epsilon$ for all $x\in [0,1],\ t\ge T_{3}$. Let ${\hat{N}}(x,t)$ satisfy the following equation,

$$\begin{aligned} {\left\{ \begin{array}{ll} {\hat{N}}_{t}=d_{1}{\hat{N}}_{xx}-q_1{\hat{N}}_{x}+r_{1}{\hat{N}}(1-\frac{{\hat{N}}}{K_{1}})-a{\hat{N}}(\theta _{2}-\epsilon ),&{} x\in (0,1),\ t\ge T_{3},\\ d_{1}{\hat{N}}_{x}(0,t)-q_1{\hat{N}}(0,t)=0,\ {\hat{N}}_{x}(1,t)=0,&{} t\ge T_{3},\\ {\hat{N}}(x,T_{3})=N(x,T_{3}),&{} x\in [0,1].\\ \end{array}\right. } \end{aligned}$$

It is not hard to know that $N(x,t)\le {\hat{N}}(x,t)$ for all $x\in [0,1],\ t\ge T_{3}$ by using the comparison principle. Since $\mu _{1}(d_1,q_1, r_1-a\theta _2)<0$, we can choose $\epsilon >0$ small enough such that $\mu _{1}(d_{1},q, r_{1}-a(\theta _{2}-\epsilon ))<0$. Similar arguments as in Lemma 2.1 yield that $\lim \limits _{t\rightarrow +\infty }{\hat{N}}(x,t)=0,\ x\in [0,1]$, thus $\lim \limits _{t\rightarrow +\infty }N(x,t)=0$ uniformly for $x\in [0,1]$. Therefore, for any $\epsilon >0$, there exists $T_{4}>T_{3}$ such that $N(x,t)<\epsilon$ for all $x\in [0,1],\ t\ge T_{4}$, which leads to

$$\begin{aligned} P_{t}\le d_{2}P_{xx}-q_2P_{x}+r_{2}P(1-\frac{P}{K_{2}})+eaP\epsilon ,\ x\in (0,1),\ t\ge T_{4}. \end{aligned}$$

Let ${\hat{P}}_{\epsilon }(x,t)$ be the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} ({\hat{P}}_{\epsilon })_{t}= d_{2}({\hat{P}}_{\epsilon })_{xx}-q_2({\hat{P}}_{\epsilon })_{x}+r_{2}{\hat{P}}_{\epsilon }(1-\frac{{\hat{P}}_{\epsilon }}{K_{2}})+ea\epsilon {\hat{P}}_{\epsilon },&{} x\in (0,1),\ t\ge T_{4},\\ d_{2}({\hat{P}}_{\epsilon })_{x}(0,t)-q_2{\hat{P}}_{\epsilon }(0,t)=0,\ ({\hat{P}}_{\epsilon })_{x}(1,t)=0,&{} t\ge T_{4},\\ {\hat{P}}_{\epsilon }(x,0)=P_{0}(x)\ge 0,\not \equiv 0, &{} x\in [0,1].\\ \end{array}\right. }\nonumber \\ \end{aligned}$$

(7.6)

The comparison principle implies $P(x,t)\le {\hat{P}}_{\epsilon }(x,t)$ for all $x\in [0,1],\ t\ge T_{4}$. Noting that $\mu _{1}(d_{2},q_2,r_{2})>0$ when $0\le q_2<q_{2}^{*}$, obviously we have that $\mu _{1}(d_{2},q_2,r_{2}+ea\epsilon )>0$. By similar arguments as in Lemma 2.1, we deduce that $\lim \limits _{t\rightarrow +\infty }P(x,t)={\hat{P}}_{\epsilon }^{*}$ uniformly for $x\in [0,1]$, where ${\hat{P}}_{\epsilon }^{*}$ is the steady-state solution of (7.6). Similar to Lemma 4.2, we get $0<{\hat{P}}_{\epsilon }^{*}<K_{2}+\frac{eaK_{2}\epsilon }{r_{2}}$. Integrating the steady-state system of (7.6) over (0, x), easily we have both $({\hat{P}}_{\epsilon }^{*})_{x}$ and $({\hat{P}}_{\epsilon }^{*})_{xx}$ are uniformly bounded in [0, 1]. By $L^{p}$ estimates and Sobolev embedding theorem, we can deduce that ${\hat{P}}_{\epsilon }^{*}\rightarrow \theta _{2}$ as $\epsilon \rightarrow 0$. That is

$$\begin{aligned} \limsup \limits _{t\rightarrow +\infty }P(x,t)\le \theta _{2}\ \text{ uniformly } \text{ for }\ x\in [0,1]. \end{aligned}$$

(7.7)

It follows from (7.5) and (7.7) that (iii) holds.

(iv) To prove the uniform persistence of system (1.2), let $\Theta (t)$ be the solution semiflow generated by system (1.2) on the state space ${\mathbb {P}}$, where

$$\begin{aligned} {\mathbb {P}}=\{(N,P)\in C[0,1]\times C[0,1]: N\ge 0,\ P\ge 0,\ x\in [0,1]\}. \end{aligned}$$

Define

$$\begin{aligned} {\mathbb {P}}_{0}=\{(N,P)\in {\mathbb {P}}: N(x)\not \equiv 0\ \text{ and }\ P(x)\not \equiv 0\} \end{aligned}$$

and $\partial {\mathbb {P}}_{0}={\mathbb {P}}\setminus {\mathbb {P}}_{0}$. Let

$$\begin{aligned} M_{\partial }=\{(N_{0},P_{0})\in \partial {\mathbb {P}}_{0}: \Theta (t)(N_{0},P_{0})\in \partial {\mathbb {P}}_{0},\ \forall t\ge 0\} \end{aligned}$$

and $\omega ((N_{0},P_{0}))$ be the omega limit set of the forward orbit $\gamma ^{+}((N_{0},P_{0}))=\{\Theta (t)(N_{0},P_{0}): t\ge 0\}$. By the strong maximum principle of the parabolic equation, we conclude that ${\mathbb {P}}_{0}$ is open in ${\mathbb {P}}$ and forward invariant under the dynamics generated by system (1.2), and $\partial {\mathbb {P}}_{0}$ contains steady state points (0, 0), $(\theta _1,0)$ and $(0,\theta _{2}).$

We first claim that

$$\begin{aligned} \cup _{(N_{0},P_{0})\in M_{\partial }}\omega ((N_{0},P_{0}))\subset \{(0,0)\}\cup \{(\theta _{1},0)\}\cup \{(0,\theta _{2})\}. \end{aligned}$$

Indeed, for any given $(N_{0},P_{0})\in M_{\partial }$, we have $\Theta (t)(N_{0},P_{0})\in \partial {\mathbb {P}}_{0},\ \forall t\ge 0$. That is, $N(x,t,(N_{0},P_{0}))\equiv 0\ \text{ or }\ P(x,t,(N_{0},P_{0}))\equiv 0$ for each $x\in [0,1],\ t\ge 0$. Clearly, in the case where $N(x,t,(N_{0},P_{0}))\equiv 0$ for all $x\in [0,1],\ t\ge 0$, $P(x,t,(N_{0},P_{0}))$ satisfies the single species system (2.2). It follows from Lemma 2.1 that either $\lim \limits _{t\rightarrow +\infty }P(x,t)=0$, or $\lim \limits _{t\rightarrow +\infty }P(x,t)=\theta _{2}$, $x\in [0,1]$. In the case where $N(x,\tau _0,(N_{0},P_{0}))\not \equiv 0$ for $x\in [0,1]$ and some $\tau _0>0$, we have $N(x,t,(N_{0},P_{0}))>0$ for all $x\in [0,1],\ t>\tau _0$ by strong maximum principle, which implies that $P(x,t,(N_{0},P_{0}))\equiv 0$ for all $x\in [0,1],\ t>\tau _0$. Thus $N(x,t,(N_{0},P_{0}))$ is the solution of (2.1). By Lemma 2.1 we have that either $\lim \limits _{t\rightarrow +\infty }N(x,t)=0$, or $\lim \limits _{t\rightarrow +\infty }N(x,t)=\theta _1,\ x\in [0,1]$. Hence, $\cup _{(N_{0},P_{0})\in M_{\partial }}\omega ((N_{0},P_{0}))\subset \{(0,0)\}\cup \{(\theta _1,0)\}\cup \{(0,\theta _2)\}.$

We next claim that (0, 0), $(\theta _{1},0)$ and $(0,\theta _{2})$ are uniform weak repellers in the sense that

$$\begin{aligned} \limsup \limits _{t\rightarrow +\infty }\Vert \Theta (t)(N_{0},P_{0})-(0,0)\Vert \ge \delta _{1}\ \text{ for } \text{ all }\ (N_{0},P_{0})\in {\mathbb {P}}_{0}, \end{aligned}$$

(7.8)

$$\begin{aligned} \limsup \limits _{t\rightarrow +\infty }\Vert \Theta (t)(N_{0},P_{0})-(\theta _1,0)\Vert \ge \delta _{2}\ \text{ for } \text{ all }\ (N_{0},P_{0})\in {\mathbb {P}}_{0}, \end{aligned}$$

(7.9)

and

$$\begin{aligned} \limsup \limits _{t\rightarrow +\infty }\Vert \Theta (t)(N_{0},P_{0})-(0,\theta _{2})\Vert \ge \delta _{3}\ \text{ for } \text{ all }\ (N_{0},P_{0})\in {\mathbb {P}}_{0}. \end{aligned}$$

(7.10)

In fact, (7.8), (7.9) and (7.10) are equivalent to the linear instability of (0, 0), $(\theta _{1},0)$ and $(0,\theta _{2})$ respectively, which is guaranteed by the conditions $\mu _{1}(d_{1},q_1,r_{1}-a\theta _{2})>0$ and $\mu _{1}(d_{2},q_2,r_{2}+ea\theta _{1})>0$ (see Lemmas 5.1–5.3). For the detailed proof, please see Theorem 4.3 of Nie et al. (2020).

Now we define a continuous function ${\mathcal {D}}:{\mathbb {P}}\rightarrow [0,\infty )$ by

$$\begin{aligned} {\mathcal {D}}((N,P))=\min \limits _{x\in [0,1]}\{\min N(x),\ \min P(x)\}\ \text{ for } \text{ any }\ (N,P)\in {\mathbb {P}}. \end{aligned}$$

It follows from the standard comparison principle that ${\mathcal {D}}^{-1}(0,\infty )\subseteq {\mathbb {P}}_{0}$ and ${\mathcal {D}}$ satisfies that if ${\mathcal {D}}((N,P))>0\ \text{ or }\ (N,P)\in {\mathbb {P}}_{0}\ \text{ with }\ {\mathcal {D}}((N,P))=0$, then ${\mathcal {D}}(\Theta (t)(N,P))>0,\ \forall t>0$. That is, ${\mathcal {D}}$ is a generalized distance function for the semiflow $\Theta (t):{\mathbb {P}}\rightarrow {\mathbb {P}}$ (see Smith and Zhao 2001). It follows from Lemma 4.4 that $\Theta (t)$ is point dissipative on ${\mathbb {P}}$. Obviously, $\Theta (t):{\mathbb {P}}\rightarrow {\mathbb {P}}$ is compact for any $t>0$. By Theorem 2.6 of Magal and Zhao (2005), $\Theta (t):{\mathbb {P}}\rightarrow {\mathbb {P}},\ t\ge 0$ admits a global compact attractor. It follows from $\cup _{\Psi \in M_{\partial }}\omega (\Psi )\subset \{(0,0)\}\cup \{(\theta _{1},0)\}\cup \{(0,\theta _{2})\}$ that any forward orbit of $\Theta (t)$ in $M_{\partial }$ converges to (0, 0), $(\theta _{1},0)$ or $(0,\theta _{2})$. Recalling that (0, 0), $(\theta _{1},0)$ and $(0,\theta _{2})$ are uniform weak repellers (see (7.8) – (7.10)), we conclude that $\{(0,0)\},\ \{(\theta _{1},0)\}\text{ and }\ \{(0,\theta _{2})\}$ are isolated invariant sets in ${\mathbb {P}}$, and

$$\begin{aligned}&W^{S}\{(0,0)\}\cap {\mathcal {D}}^{-1}(0,\infty )=\emptyset ,\ W^{S}\{(\theta _{1},0)\}\cap {\mathcal {D}}^{-1}(0,\infty )\\&\quad =\emptyset \ \text{ and } \ W^{S}\{(0,\theta _{2})\}\cap {\mathcal {D}}^{-1}(0,\infty )=\emptyset . \end{aligned}$$

Here $W^{S}\{(0,0)\}$, $W^{S}\{(\theta _{1},0)\}$ and $W^{S}\{(0,\theta _{2})\}$ are the stable sets of (0, 0), $(\theta _{1},0)$ and $(0,\theta _{2})$, respectively (see Hale and Waltman 1989; Smith and Zhao 2001). Furthermore, no subsets of $\{(0,0)\}\cup \{(\theta _{1},0)\}\cup \{(0,\theta _{2})\}$ form a cycle in $\partial {\mathbb {P}}_0$. By Theorem 3 of Smith and Zhao (2001), there exists $\eta >0$ such that for any $(N_{0},P_{0})\in {\mathbb {P}}_{0},$

$$\begin{aligned} \min \limits _{(N_{0},P_{0})\in \omega ((N,P))}{\mathcal {D}}((N_{0},P_{0}))>\eta . \end{aligned}$$

This implies that for any $(N,P)\in {\mathbb {P}}_{0},\ \liminf \limits _{t\rightarrow +\infty }N(x,t)\ge \eta \ \text{ and }\ \liminf \limits _{t\rightarrow +\infty }P(x,t)\ge \eta ,\ x\in [0,1].$

It follows from Theorem 3.7 and Remark 3.10 of Magal and Zhao (2005) that $\Theta (t):{\mathbb {P}}_0\rightarrow {\mathbb {P}}_0$ admits a global attractor $A_0$. Then by Theorem 4.7 of Magal and Zhao (2005), we conclude that $\Theta (t)$ admits at least one steady-state solution $({\bar{N}}(\cdot ),{\bar{P}}(\cdot ))\in {\mathbb {P}}_0$. Furthermore, we deduce that ${\bar{N}}(\cdot ), {\bar{P}}(\cdot )>0$ by the strong maximum principle (see Protter and Weinberger 1984). Thus, system (1.2) admits at least one positive steady state solution $({\bar{N}}(\cdot ),{\bar{P}}(\cdot ))$. The uniqueness of positive steady state to system (1.2) follows from similar arguments as in Step 3 of Theorem 3.1 of Nie et al. (2020), see also the proof of Lemma 3.3 and Theorem 3.4 of Nie et al. (2015). $\square$