2. Main results

We will state the main results in this paper. For the definition and properties of the set of all bounded and uniformly continuous functions $\text{BUC}$ , see Section 3, as well as $\text{BUC}^1$ .

We will explain the strategy of the proof of Theorem 1, briefly. Using the heat semigroups, (P) is written as the forms of integral equations:

(1) \begin{align} u (t) = e^{t \Delta } u_0 + \int _0^t e^{(t-s) \Delta } \left [ \left ( 1 - u \right ) u - \frac{\gamma u v}{u + h} \right ] (s) \, ds, \end{align}

(2) \begin{align} v (t) = e^{d t \Delta } v_0 + \int _0^t e^{d (t-s) \Delta } \left [ \frac{\mu u v}{u + h} + \alpha w - (m + v) v \right ] (s) \, ds, \end{align}

(3) \begin{align} w (t) = e^{-\rho t} w_0 + \int _0^t e^{-\rho (t-s)} \left [ \frac{\nu u v}{u + h} + \theta v \right ] (s) \, ds. \end{align}

Once we obtain the existence of solutions to (1)–(3), the uniqueness and the regularity of solutions follow from these forms. However, it is not easy to get the existence of solutions, at least directly. Because, the nonnegativity of solutions or its approximation seems to be not ensured, in general. In fact, the following standard iteration scheme is often employed:

\begin{equation*} \bar u_{\ell +1} (t) \,:\!=\, e^{t \Delta } u_0 + \int _0^t e^{(t-s) \Delta } \left [ \left ( 1 - \bar u_\ell \right ) \bar u_\ell - \frac {\gamma \bar u_\ell \bar v_\ell }{\bar u_\ell + h} \right ] (s) \, ds. \end{equation*}

See, for example, the book by Smoller [Reference Smoller7]. With this approximation, it is not clear how to show $\bar u_\ell \gt - h$ for $\ell \geq 2$ , unfortunately. Thus, we have to look for the another approximation or integral forms for proving the existence of nonnegative solutions.

There are many mathematical articles to prove the existence of partial differential equation (PDE) with the Holling type $\rm I\!I$ nonlinear terms. As far as the authors know, the exiting techniques are as follows.

• Under the a priori assumption of the positivity of solutions or approximation, the existence of solutions is easily proved. However, it is not clear how to get the positivity.

• Apply the fixed point theorem of the mapping in the set of positive functions $K_+ \,:\!=\, \{ u, v, w \gt 0 \}$ . However, we have to verify its domain and range in $K_+$ .

• Consider the modified system taking the absolute value to $u$ or $\bar u_\ell$ in the denominator. To do so, we obtain weak solutions. However, it is not clear how to show that the weak solutions satisfy (P) in the classical sense, that is, $u \in C ((0, T);\, C^2)$ .

• By standard arguments, the corresponding ordinary differential equation (ODE) admit time-local solutions. So, the solution to (P) is estimated from below by that to ODE. However, the comparison principle can be applicable, after showing the existence of classical solutions.

Each approach in above has a flaw. Hence, we employ the same sprits in [Reference Kondo, Novrianti and Tsuge6]. To overcome difficulties, we will construct the solutions as the limits of the following successive approximations of abstract forms:

\begin{align*} \partial _t u_{\ell +1} & = \Delta u_{\ell +1} - \left ( u_\ell + \frac{\gamma v_\ell }{u_\ell + h} \right ) u_{\ell +1} + u_\ell, \quad u_{\ell +1}|_{t=0} = u_0,\\ \partial _t v_{\ell +1} & = d \Delta v_{\ell +1} - \left ( m + v_\ell \right ) v_{\ell +1} + \frac{\mu u_\ell v_\ell }{u_\ell + h} + \alpha w_\ell, \quad v_{\ell +1}|_{t=0} = v_0,\\ \partial _t w_{\ell +1} & = -\rho w_{\ell +1} + \frac{\nu u_\ell v_\ell }{u_\ell + h} + \theta v_\ell, \quad w_{\ell +1}|_{t=0} = w_0 \end{align*}

for $\ell \in{\mathbb N}$ . Our idea is to involve the coefficients of negative terms into the generators. We can rewrite them as

(4) \begin{align} u_{\ell + 1} (t) & = U_\ell (t, 0) u_0 + \int _0^t U_\ell (t, s) \big [ u_\ell \big ] (s) \, ds, \end{align}

(5) \begin{align} v_{\ell + 1} (t) & = V_\ell (t, 0) v_0 + \int _0^t V_\ell (t, s) \left [ \frac{\mu u_\ell v_\ell }{u_\ell + h} + \alpha w_\ell \right ] (s) \, ds, \end{align}

(6) \begin{align} w_{\ell + 1} (t) & = e^{-\rho t} w_0 + \int _0^t e^{-\rho (t-s)} \left [ \frac{\nu u_\ell v_\ell }{u_\ell + h} + \theta v_\ell \right ] (s) \, ds \end{align}

for $\ell \in{\mathbb N}$ . Here, we have used the time-evolution operators $ \{ U_\ell (t, s) \}$ and $ \{ V_\ell (t, s) \}$ associated with

\begin{equation*} A_\ell \,:\!=\, \Delta - u_\ell - \gamma v_\ell/ (u_\ell + h) \quad \text {and} \quad B_\ell \,:\!=\, d \Delta - m - v_\ell \end{equation*}

for regarding $u_\ell$ , $v_\ell$ and $w_\ell$ as given nonnegative functions, respectively, starting at

(7) \begin{equation} u_1 (t) \,:\!=\, e^{t \Delta } u_0, \quad v_1 (t) \,:\!=\, e^{t (d \Delta - m)} v_0 \quad \textrm{and} \quad w_1 (t) \,:\!=\, e^{- \rho t} w_0. \end{equation}

The precise definition and estimates of time-evolution operators are given in Section 3. These approximations enable us to show the nonnegativities of $u_\ell$ , $v_\ell$ and $w_\ell$ for all $\ell \in{\mathbb N}$ , as well as its limit $u$ , $v$ and $w$ . We will derive the estimates $\| u_\ell, v_\ell, w_\ell \|_\infty$ by (4)–(6), inductively, in the fixed point arguments. Besides, for estimates $\| \partial _i u_\ell, \partial _i v_\ell, \partial _i w_\ell \|_\infty$ , we apply the heat semigroup representation of solutions. Once we derive uniform bounds of $u_\ell$ , $v_\ell$ , $w_\ell$ , $\partial _i u_\ell$ , $\partial _i v_\ell$ and $\partial _i w_\ell$ , we can easily see that the limit $(u, v, w)$ becomes a classical solution to (P).

On the other hand, it is rather standard to extend the obtained solutions time-globally, deriving a priori estimates of solutions. The key idea is to apply the maximum principle to the classical solutions. We can also investigate asymptotic behaviours of solutions, more precisely. Via analysis of solutions to the system of corresponding ODE, we obtain invariant regions.

Before stating results, we provide two stationary solutions. It is easy to verify that $(1, \overline v, \overline w)$ is a stationary solution to (P), where

\begin{align*} \overline v & \,:\!=\, \mu/ (1+h) + \alpha (\nu + \theta + \theta h)/ (\rho + \rho h) - m,\\ \overline w & \,:\!=\, (\nu + \theta + \theta h) \overline v/ (\rho + \rho h). \end{align*}

Furthermore, $(\underline u, \underline v, \underline w)$ is also a stationary solution to (P), where

\begin{align*} \underline u & \,:\!=\, (1-h)/ 2 + \sqrt{(1+h)^2 - 4 \gamma \overline v}/2,\\ \underline v & \,:\!=\, \mu \underline u/ (\underline u + h) + \alpha \nu \underline u/ (\rho \underline u + \rho h) + \alpha \theta/ \rho - m,\\ \underline w & \,:\!=\, \nu \underline u \, \underline v \,/ (\rho \underline u + \rho h) + \theta \underline v \,/ \rho. \end{align*}

Theorem 2. Assume that $(u, v, w)$ is a solution to (P). (i) If $\overline v \leq 0$ , and if $u_0 \not \equiv 0$ , then $(u, v, w) \to (1, 0, 0)$ as $t \to \infty$ . Besides, if $u_0 \equiv 0$ , then $(u, v, w) \to (0, 0, 0)$ as $t \to \infty$ . (ii) If $\overline v \gt 0$ , then for any $0 \lt \varepsilon \lt \!\!\lt 1$ , there exists a $T_\varepsilon \geq 0$ such that

\begin{equation*} (u, v, w) \in R_\varepsilon \,:\!=\, [0, 1 + \varepsilon ) \times [0, \overline v + \varepsilon ) \times [0, \overline w + \varepsilon ) \end{equation*}

for $x \in{\mathbb R}^n$ and $t \geq T_\varepsilon$ . Moreover,

\begin{equation*} (u_0, v_0, w_0) \in R_\ast \,:\!=\, [0, 1] \times [0, \overline v] \times [0, \overline w] \,\, \Rightarrow \,\, (u, v, w) \in R_\ast \end{equation*}

for $x \in{\mathbb R}^n$ and $t \gt 0$ . (iii) Let $\overline v \gt 0$ , $\underline u \gt 0$ , $\underline v \gt 0$ , and let $\underline w \gt 0$ . If $u$ , $v$ , $w \geq c_\star$ for $x \in{\mathbb R}^n$ at $t = t_\star \geq 0$ with some $c_\star \gt 0$ , then for $0 \lt \varepsilon \lt \!\!\lt 1$ , there exists a $T_\varepsilon^{\prime} \geq t_\star$ such that

\begin{equation*} ( u, v, w ) \in R_\varepsilon ^\prime \,:\!=\, ( \underline u - \varepsilon, 1 + \varepsilon ) \times ( \underline v - \varepsilon, \overline v + \varepsilon ) \times ( \underline w - \varepsilon, \overline w + \varepsilon ) \end{equation*}

for $x \in{\mathbb R}^n$ and $t \geq T_\varepsilon^{\prime}$ . Moreover,

\begin{equation*} (u_0, v_0, w_0) \in R_\natural \,:\!=\, [\underline u, 1] \times [\underline v, \overline v] \times [\underline w, \overline w] \,\, \Rightarrow \,\, ( u, v, w ) \in R_\natural \end{equation*}

for $x \in{\mathbb R}^n$ and $t \gt 0$ .

The sets $R_\ast$ and $R_\natural$ are invariant regions. The reader may find another (narrower) invariant regions for each individual parameter. Theorem 2 implies that an absorbing set always exists in $R_\ast$ or $R_\natural$ . Our conjecture is that we can also obtain the similar results in several domains with suitable boundary conditions.

3. Semigroups and time-evolution operators

In this section, we recall the definitions of function spaces and properties of the heat semigroup, as well as time-evolution operators.

Let $n \in{\mathbb N}$ , $1 \leq p \lt \infty$ , and let $L^p \,:\!=\, L^p ({\mathbb R}^n)$ be the space of all $p$ th integrable functions in ${\mathbb R}^n$ with the norm $\displaystyle \| f \|_p \,:\!=\, \left( \int _{{\mathbb R}^n} |f(x)|^p dx \right)^{1/p}$ . We often omit the notation of the domain $({\mathbb R}^n)$ , if no confusion occurs. We do not distinguish scalar-valued functions and vector, as well as function spaces. Let $L^\infty$ be the space of all bounded functions with the norm $\| f \| \,:\!=\, \| f \|_\infty \,:\!=\, \textrm{ess}.\textrm{sup}_{x \in{\mathbb R}^n} | f(x) |$ ; $\text{BUC}$ as the space of all bounded uniformly continuous functions. Note that $L^p$ , $L^\infty$ and $\text{BUC}$ are Banach spaces. For $k \in{\mathbb N}$ , let $W^{k, \infty }$ be a set of all bounded functions whose $k$ -th derivatives are also bounded. Furthermore, define

\begin{equation*} \text{BUC}^k \,:\!=\, \big \{ f \in W^{k, \infty }; \, \partial _i^j f \in \text{BUC} \quad \textrm {for} \quad 1 \leq i \leq n, \, 0 \leq j \leq k \big \}. \end{equation*}

In the whole space ${\mathbb R}^n$ , for $\vartheta _0 \in L^\infty ({\mathbb R}^n)$ , the heat equation

(H) \begin{equation} \left \{ \begin {array}{l@{\quad}l} \displaystyle \partial _t \vartheta = \Delta \vartheta \, & \textrm {in} \ \ {\mathbb R}^n \!\times \! (0, \infty ), \\[4pt] \vartheta |_{t=0} = \vartheta _0 \, & \textrm {in} \ \ {\mathbb R}^n \end {array} \right. \end{equation}

admits a time-global unique smooth solution

\begin{align*} \vartheta & \,:\!=\, \vartheta (t) \,:\!=\, \vartheta (x, t) \,:\!=\, (e^{t \Delta } \vartheta _0) (x) \,:\!=\, (H_t \ast \vartheta _0) (x) \\ & \,:\!=\, \int _{{\mathbb R}^n} (4 \pi t)^{-n/2} \exp ({-}|x-y|^2/4t) \vartheta _0 (y) dy \end{align*}

in $C_w ((0, \infty );\, L^\infty ({\mathbb R}^n))$ , that is, $\vartheta \in C ([\tau, \infty );\, L^\infty ({\mathbb R}^n))$ for any small $\tau \gt 0$ . Here, $H_t \,:\!=\, H_t (x) \,:\!=\, (4 \pi t)^{-n/2} \exp ({-}|x|^2/4t)$ is the heat kernel. Since $\| H_t \|_1 = 1$ for $t \gt 0$ , by Young’s inequality we have $\| \vartheta (t) \| \leq \| H_t \|_1 \| \vartheta _0 \| = \| \vartheta _0 \|$ for $t \gt 0$ . In particular, if $\vartheta _0 (x) \geq 0$ for $x \in{\mathbb R}^n$ , then $\vartheta (x, t) \geq 0$ holds true for $x \in{\mathbb R}^n$ and $t \gt 0$ , so-called the maximum principle. Furthermore, if additionally $\vartheta _0 \in \text{BUC} ({\mathbb R}^n)$ and $\vartheta _0 \not \equiv 0$ , then $\vartheta (x, t) \gt 0$ for $x \in{\mathbb R}^n$ and $t \gt 0$ , so-called the strong maximum principle. For $\vartheta _0 \in L^\infty ({\mathbb R}^n)$ , there is a lack of the continuity of solutions to (H) in time at $t = 0$ , in general. Note that $e^{t \Delta } \vartheta _0 \to \vartheta _0$ in $L^\infty$ as $t \to 0$ , if and only if $\vartheta _0 \in \text{BUC} ({\mathbb R}^n)$ . The reader may find its proof in [Reference Giga, Inui and Matsui5]. Indeed, if $\vartheta _0 \in \text{BUC} ({\mathbb R}^n)$ , then $\vartheta \in C ([0, \infty );\, \text{BUC} ({\mathbb R}^n))$ .

We can easily see that for $j \in{\mathbb N}$ , splitting the heat semigroup into $j$ -th parts, there exists a positive constant $C_\sharp \,:\!=\, \pi ^{-1/2} \lt 1$ such that

\begin{equation*} \| \partial _i^j e^{t \Delta } \vartheta _0 \| = \| \left ( \partial _i e^{\frac {t}{j} \Delta } \right ) \cdots \left ( \partial _i e^{\frac {t}{j} \Delta } \right ) \vartheta _0 \| \leq C_\sharp ^j \, j^{j/2} \, t^{-j/2} \| \vartheta _0 \| \end{equation*}

for $t \gt 0$ and $1 \leq i \leq n$ . So, $\vartheta (t) \in \text{BUC}^j ({\mathbb R}^n)$ for any $j \in{\mathbb N}$ and $t \gt 0$ , which implies that $\vartheta (t) \in C^\infty ({\mathbb R}^n)$ for $t \gt 0$ . Moreover, $\vartheta \in C^\infty ({\mathbb R}^n \times (0, \infty ))$ by using (H).

In what follows, we recall some properties and estimates for time-evolution operators. Let us consider the following autonomous Cauchy problem with non-constant coefficients.

\begin{equation*} \quad\qquad\qquad\qquad\qquad\qquad\left \{ \begin {array}{l@{\quad}l} \partial _t \varphi = d \Delta \varphi - \psi (x, t) \varphi \, & \textrm {in} \ \ {\mathbb R}^n \!\times \! (0, \infty ), \qquad\qquad\qquad\qquad\qquad\quad(\textrm {P}_\textrm {A})\\[5pt] \varphi |_{t=0} = \varphi _0 \, & \textrm {in} \ \ {\mathbb R}^n. \end {array} \right. \end{equation*}

Here, $d \gt 0$ is a constant, and $\psi (x, t)$ is a given bounded function. We establish the time-local solvability of $\rm (P_A)$ with upper bounds of $\varphi$ .

Although the proof is written in [Reference Kondo, Novrianti and Tsuge6], we give it here. The idea is to use the standard iteration. Let $\varphi _1 (t) \,:\!=\, e^{d t \Delta } \varphi _0$ , and let

\begin{equation*} \varphi _{\ell + 1} (t) \,:\!=\, e^{d t \Delta } \varphi _0 - \int _0^t e^{d (t - s) \Delta } \left [ \psi \varphi _\ell \right ] (s) \, ds \end{equation*}

for each $\ell \in{\mathbb N}$ , successively. It is easy to see that for $\ell \in{\mathbb N}$ , $\| \varphi _\ell (t) \| \leq \dfrac 43 \| \varphi _0 \|$ for $t \in [0, T_\ast ]$ with some $T_\ast \gt 0$ independent of $\ell$ . We can easily show that $ \{ \varphi _\ell \}_{\ell = 1}^\infty$ is a Cauchy sequence in $C ([0, T_\ast ];\, \text{BUC} ({\mathbb R}^n))$ . So, the limit $\varphi \,:\!=\, \lim _{\ell \to \infty } \varphi _\ell$ exists and satisfies $\rm (P_A)$ , having the estimate $\| \varphi (t) \| \leq \dfrac 43 \| \varphi _0 \|$ for $t \in [0, T_\ast ]$ . It is rather straightforward to obtain the uniqueness and regularity of $\varphi$ . Moreover, the nonnegativity of $\varphi$ easily follows from the maximum principle.

Note that if $\| \varphi _0 \| \leq L$ and $\textrm{sup}_{0 \leq t \leq T} \| \psi (t) \| \leq L$ with some $L \gt 0$ , then we may derive the estimate $T_\ast \geq C/L$ with $C \gt 0$ . The solution to $({\textrm{P}}_{\textrm{A}})$ can be rewritten as $\varphi (t) = U (t, 0) \varphi _0$ , using time-evolution operators $ \{ U (t, s) \}_{t \geq s \geq 0}$ associated with $A \,:\!=\, A(x, t) \,:\!=\, d \Delta - \psi (x, t)$ , see the book by Tanabe [Reference Tanabe8]. The boundedness of solutions $\varphi$ implies that $\| U (t, 0) \|_{L^\infty \to L^\infty } \leq 4/3$ for $t \in [0, T_\ast ]$ , and then $\| U (t, s) \|_{L^\infty \to L^\infty } \leq 4/3$ for $0 \leq s \leq t \leq T_\ast$ . Here, we have used the notation of an operator-norm $\|{\mathcal O} \|_{X \to Y} \,:\!=\, \textrm{sup}_{x \in X} \|{\mathcal O} x \|_Y/ \| x \|_X$ .

4. Time-local solvability

We give a proof of the time-local solvability on (P) in this section. Recall $\| \cdot \| \,:\!=\, \| \cdot \|_\infty$ , and put $M \,:\!=\, \max \{ \| u_0 \|, \| v_0 \|, \| w_0 \|, \| \partial _i w_0 \| \}$ .

Proof. For the sake of simplicity, we assume that all parameter is positive. Making the approximation sequences, we begin with (7). For $\ell \in{\mathbb N}$ , we successively define $u_{\ell + 1}$ , $v_{\ell + 1}$ and $w_{\ell + 1}$ by (4)–(6). So, $u_{\ell + 1}$ , $v_{\ell + 1}$ and $w_{\ell + 1}$ also satisfy their abstract equations for $x \in{\mathbb R}^n$ and $t \gt 0$ with nonnegative functions $u_0$ , $v_0$ , $w_0$ , $u_\ell$ , $v_\ell$ and $w_\ell$ , formally.

In what follows, we estimate $u_\ell$ , $v_\ell$ , $w_\ell$ , $\partial _i u_\ell$ , $\partial _i v_\ell$ and $\partial _i w_\ell$ . Put

\begin{equation*} \begin {array}{l@{\quad}l} K_{1, \ell } \,:\!=\, \textrm {sup}_{0 \leq t \leq T} \|u_\ell (t)\|, & \quad K_{2, \ell } \,:\!=\, \textrm {sup}_{0 \leq t \leq T} \|v_\ell (t)\|, \\[5pt] K_{3, \ell } \,:\!=\, \textrm {sup}_{0 \leq t \leq T} \|w_\ell (t)\|, & \quad K_{4, \ell } \,:\!=\, \textrm {sup}_{0 \leq t \leq T} t^{1/2} \|\partial _i u_\ell (t)\|, \\[5pt] K_{5, \ell } \,:\!=\, \textrm {sup}_{0 \leq t \leq T} (dt)^{1/2} \|\partial _i v_\ell (t)\|, & \quad K_{6, \ell } \,:\!=\, \textrm {sup}_{0 \leq t \leq T} \|\partial _i w_\ell (t)\| \end {array} \end{equation*}

for $T \gt 0$ , $\ell \in{\mathbb N}$ and $1 \leq i \leq n$ . To derive uniform estimates, we argue the induction of $\ell$ , taking $T$ small.

$\ell = 1$ For $0 \leq u_0(x), v_0(x), w_0(x) \leq M$ , by the maximum principle and the fact that $e^{t(d \Delta -m)} = e^{-mt} e^{dt \Delta }$ , we can easily see that

\begin{equation*} 0 \leq u_1 (x, t) \leq \| u_0 \|, \quad 0 \leq v_1 (x, t) \leq \| v_0 \|, \quad 0 \leq w_1 (x, t) \leq \| w_0 \| \end{equation*}

for $x \in{\mathbb R}^n$ and $t \gt 0$ by $m, \rho \geq 0$ . In addition, it is easy to obtain that

\begin{equation*} t^{1/2} \| \partial _i u_1 (t) \| \leq \| u_0 \|, \quad (dt)^{1/2} \| \partial _i v_1 (t) \| \leq \| v_0 \|, \quad \| \partial _i w_1 (t) \| \leq \| \partial _i w_0 \| \end{equation*}

for $t \gt 0$ and $1 \leq i \leq n$ by the estimate of the heat kernel. Here and hereafter, we replace the constant $C_\sharp \,:\!=\, \pi ^{-1/2} \lt 1$ by $1$ , for the sake of simplicity. Thus, we have

(8) \begin{equation} K_{j, 1} \leq M \quad \textrm{for} \quad T \gt 0, \quad 1 \leq j \leq 6 \quad \textrm{and} \quad 1 \leq i \leq n. \end{equation}

$\ell = 2$ Before estimating $u_2$ and $v_2$ , we will confirm bounds for time-evolution operators $U_1$ and $V_1$ . By $u_1 \geq 0$ and (8), it holds that

\begin{equation*} \| \eta _1 (t) \| \leq M + \frac {\gamma M}h \,=\!:\, \overline \eta _1 \quad \text {with} \quad \eta _1 (x, t) \,:\!=\, u_1 (x, t) + \frac {\gamma v_1 (x, t)}{u_1 (x, t) + h} \end{equation*}

for $t \gt 0$ . By Lemma 1, for $ \{ U_1 (t, s) \}_{t \geq s \geq 0}$ with $A_1 (x, t) \,:\!=\, \Delta - \eta _1 (x, t)$ , we thus see that $\displaystyle 0 \leq U_1 (t, s) u_0 \leq \frac{4}{3} \| u_0 \|$ for $x \in{\mathbb R}^n$ and $0 \leq s \leq t \leq T^{\prime}_2$ with some $T^{\prime}_2 \gt 0$ depending only on $\overline \eta _1$ . By (4) with $\ell = 1$ , we have

\begin{equation*} 0 \leq u_2 (t) \leq \| U_1 (t, 0) u_0 \| + \int _0^t \| U_1 (t, s) \zeta _1 (s) \| ds \leq 2 M \end{equation*}

with $\zeta _1 (x, t) \,:\!=\, u_1 (x, t)$ and $0 \leq \zeta _1 (x, s) \leq \overline \zeta _1 \,:\!=\, M$ , provided $0 \leq s \leq t \leq T_2^\dagger$ with $T_2^\dagger \,:\!=\, \min \{ T^{\prime}_2, 1/2 \}$ . Similarly, since

\begin{equation*} \| \xi _1 (t) \| \leq m + M \,=\!:\, \overline \xi _1 \quad \textrm {with} \quad \xi _1 (x, t) \,:\!=\, m + v_1 (x, t) \end{equation*}

for $x \in{\mathbb R}^n$ and $t \gt 0$ , we may define the time-evolution operator $ \{ V_1 (t, s) \}_{t \geq s \geq 0}$ associated with $B_1 (x, t) \,:\!=\, d \Delta - \xi _1 (x, t)$ , having a uniform bound. Applying Lemma 1, we see that $\displaystyle 0 \leq V_1 (t, s) v_0 \leq \frac{4}{3} \| v_0 \|$ for $0 \leq s \leq t \leq T_2^\sharp$ with some $T_2^\sharp \gt 0$ depending only on $\overline \xi _1$ . By (5)

\begin{equation*} 0 \leq v_2 (t) \leq \| V_1 (t, 0) v_0 \| + \int _0^t \| V_1 (t, s) \chi _1 (s) \| ds \leq 2 M \end{equation*}

hold with $\chi _1 (x, t) \,:\!=\, \mu u_1 (x, t) v_1 (x, t)/ \{ u_1 (x, t) + h \} + \alpha w_1 (x, t)$ and $0 \leq \chi _1 (x, s) \leq \overline \chi _1 \,:\!=\, (\mu M/h + \alpha ) M$ , provided if $0 \leq s \leq t \leq T_2^\flat$ with $T_2^\flat \,:\!=\, \min \{ T_2^\dagger, T_2^\sharp, h/(2 \mu M + 2 \alpha h) \}$ . For the estimate of $w_2$ , we obtain

\begin{equation*} 0 \leq w_2 (t) \leq \| e^{- \rho t} w_0 \| + \int _0^t e^{- \rho (t-s)} \| \nu u_1 v_1/ (u_1 + h) + \theta v_1 \| ds \leq 2 M \end{equation*}

for $0 \leq s \leq t \leq T_2^\natural$ with $T_2^\natural \,:\!=\, \min \{ T_2^\flat, h/(\nu M + h \theta ) \}$ . To derive the estimate for $\partial _i u_2$ , we use the heat semigroup expression:

\begin{equation*} u_2 (t) = e^{t \Delta } u_0 + \int _0^t e^{(t-s) \Delta } \left [ \zeta _1 - \eta _1 u_2 \right ] (s) \, ds \end{equation*}

by rewriting (4). Hence, it holds that

\begin{equation*} t^{1/2} \| \partial _i u_2 (t) \| \leq \| u_0 \| + t^{1/2} \int _0^t (t-s)^{-1/2} \left [ \overline \zeta _1 + \overline \eta _1 \| u_2 \| \right ] ds \leq 2M \end{equation*}

for $t \in (0, T_2^\heartsuit ]$ with $T_2^\heartsuit \,:\!=\, \min \left\{ T_2^\natural, h/ (2 h + 4 h M + 4 \gamma M) \right\}$ . As the similar way, for $\partial _i v_2$ , we appeal to the heat semigroup expression again:

\begin{align*} (dt)^{1/2} \| \partial _i v_2 (t) \| & \leq (dt)^{1/2} \| \partial _i e^{d t \Delta } v_0 \| \\ & \quad + (dt)^{1/2} \int _0^t \| \partial _i e^{d (t-s) \Delta } \left [ \chi _1 - \xi _1 v_2 \right ] (s) \| ds \\ & \leq \| v_0 \| + t^{1/2} \int _0^t (t-s)^{-1/2} \left [ \overline \chi _1 + \overline \xi _1 2 M \right ] ds \\ & \leq 2M \end{align*}

for $t \in \big(0, T_2^\diamondsuit \big]$ with $T_2^\diamondsuit \,:\!=\, \min \left\{ T_2^\heartsuit, h/ (2 \mu M + 2 \alpha h + 4 hm + 4 h M) \right\}$ . Furthermore,

\begin{align*} \partial _i w_2 (t) = e^{- \rho t} \partial _i w_0 + \int _0^t e^{- \rho (t-s)} \left [ \frac{\nu h (\partial _i u_1) v_1 + \nu u_1 (\partial _i v_1) (u_1 + h)}{(u_1 + h)^2} + \theta \partial _i v_1 \right ] ds \end{align*}

holds true, and this implies that

\begin{align*} \| \partial _i w_2 (t) \| & \leq M + \int _0^t \left \{ \frac{\nu h \sqrt{d} M + \nu M (M + h)}{h^2} + \theta \right \} M (d s)^{-1/2} ds \\ & \leq 2 M \end{align*}

for $t \in [0, T_2]$ with

\begin{equation*} T_2 \,:\!=\, \min \left\{ T_2^\diamondsuit, d h^4/\left[4 \nu h \sqrt {d} M + 4 \nu M^2 + 4 \nu h M + 4 h^2 \theta \right]^2 \right\}. \end{equation*}

Therefore, it is shown that $u_2, v_2, w_2 \geq 0$ and

(9) \begin{equation} K_{j, 2} \leq 2 M \quad \textrm{for} \quad t \in (0, T_2], \quad 1 \leq j \leq 6 \quad \text{and} \quad 1 \leq i \leq n. \end{equation}

$\ell = 3$ We stand for the time-evolution operator $ \{ U_2 (t, s) \}_{t \geq s \geq 0}$ associated with $A_2 (x, t) \,:\!=\, \Delta - \eta _2 (x, t)$ and

\begin{equation*} \eta _2 (x, t) \,:\!=\, u_2 (x, t) + \gamma v_2 (x, t)/ \{ u_2 (x, t) + h \}. \end{equation*}

By Lemma 1, $U_2 (t, s) u_0 \geq 0$ holds and $\| U_2 (t, s) \|_{L^\infty \to L^\infty } \leq 4/3$ for $0 \leq s \leq t \leq T^{\prime}_3$ with some $T^{\prime}_3 \gt 0$ , since $0 \leq \eta _2 (x, t) \leq \overline \eta \,:\!=\, 2 M + 2 \gamma M/h$ by (9). So, we get

\begin{equation*} 0 \leq u_3 (x, t) \leq \| U_2 (t, 0) u_0 \| + \int _0^t \| U_2 (t, s) \zeta _2 (s) \| ds \leq 2 M \end{equation*}

for $x \in{\mathbb R}^n$ and $t \in [0, T_3^\dagger ]$ with $T_3^\dagger \,:\!=\, \min \{ T^{\prime}_3, 1/4 \}$ . Here, we used that

\begin{equation*} 0 \leq \zeta _2 (x, t) \,:\!=\, u_2 (x, t) \leq \overline \zeta \,:\!=\, 2 M. \end{equation*}

Similarly, we denote the time-evolution operator by $ \{ V_2 (t, s) \}_{t \geq s \geq 0}$ associated with $B_2 (x, t) \,:\!=\, d \Delta - \xi _2 (x, t)$ , where

\begin{equation*} 0 \leq \xi _2 (x, t) \,:\!=\, m + v_2 (x, t) \leq \overline \xi \,:\!=\, m + 2 M. \end{equation*}

We seek that $V_2(t, s) v_0 \geq 0$ and $\| V_2 (t, s) \|_{L^\infty \to L^\infty } \leq 4/3$ for $0 \leq s \leq t \leq T_3^\sharp$ with some $T_3^\sharp \gt 0$ by Lemma 1. Hence, we can see that

\begin{equation*} 0 \leq v_3 (x, t) \leq \| V_2 (t, 0) v_0 \| + \int _0^t \| V_2 (t, s) \chi _2 (s) \| ds \leq 2 M \end{equation*}

for $x \in{\mathbb R}^n$ and $t \in [0, T_3^\flat ]$ with $T_3^\flat \,:\!=\, \min \{ T_3^\dagger, T_3^\sharp, h/(8 \mu M + 4 \alpha h) \}$ . Here, we have used

\begin{align*} 0 & \leq \chi _2 (x, t) \,:\!=\, \mu u_2 (x, t) v_2 (x, t)/ \{ u_2 (x, t) + h \} + \alpha w_2 (x, t)\\ & \leq \overline \chi \,:\!=\, 4 \mu M^2/h + 2 \alpha M \end{align*}

by (9). It is also easy to show that

\begin{equation*} 0 \leq w_3 (x, t) \leq \| w_0 \| + \int _0^t \| \nu u_2 v_2/ (u_2 + h) + \theta v_2 \| ds \leq 2 M \end{equation*}

for $x \in{\mathbb R}^n$ and $t \in [0, T_3^\natural ]$ with $T_3^\natural \,:\!=\, \min \{ T_3^\flat, h/(4 \nu M + 2 h \theta ) \}$ . By the heat semigroup expression, we obtain that

\begin{equation*} t^{1/2} \| \partial _i u_3 (t) \| \leq \| u_0 \| + t^{1/2} \int _0^t (t-s)^{-1/2} \left [ \| \zeta _2 \| + \| \eta _2 u_3 \| \right ] ds \leq 2M \end{equation*}

for $t \in (0, T_3^\heartsuit ]$ with $T_3^\heartsuit \,:\!=\, \min \{ T_3^\natural, h/ (4 h + 8 h M + 8 \gamma M) \}$ . As the similar way, we derive

\begin{equation*} (dt)^{1/2} \| \partial _i v_3 (t) \| \leq \| v_0 \| + t^{1/2} \int _0^t (t-s)^{-1/2} \left [ \| \chi _2 \| + \| \xi _2 v_3 \| \right ] ds \leq 2M \end{equation*}

for $t \in (0, T_3^\diamondsuit ]$ with $T_3^\diamondsuit \,:\!=\, \min \{ T_3^\heartsuit, h/ (4 h m + 8 h M + 8 \mu M + 4 \alpha h) \}$ . For $\partial _i w_3$ , see

\begin{align*} \| \partial _i w_3 (t) \| & \leq M + \int _0^t \left \| \frac{\nu h (\partial _i u_2) v_2 + \nu u_2 (\partial _i v_2) (u_2 + h)}{h^2} + \theta \partial _i v_2 \right \| ds\\ & \leq 2 M \end{align*}

for $t \in (0, T_0]$ with

\begin{equation*} T_0 \,:\!=\, \min \left\{ T_3^\diamondsuit, d h^4/\left[8 \nu h \sqrt {d} M + 16 \nu M^2 + 8 \nu h M + 4 h^2 \theta \right]^2 \right\}. \end{equation*}

Note that the estimate $T_0 \geq C/(M^4+1)$ is yielded with some $C \gt 0$ .

Therefore, we see that $u_3, v_3, w_3 \geq 0$ and

\begin{equation*} K_{j, 3} \leq 2M \quad \textrm {for} \quad t \in (0, T_0], \quad 1 \leq i \leq n \quad \text {and} \quad 1 \leq j \leq 6. \end{equation*}

$\ell = 4, 5, \ldots$ Let $\ell \geq 4$ . We assume that $u_\ell$ , $v_\ell$ , $w_\ell \geq 0$ and

(10) \begin{equation} K_{j, \ell } \leq 2 M \quad \textrm{for} \quad t \in (0, T_0], \quad 1 \leq j \leq 6 \quad \text{and} \quad 1 \leq i \leq n \end{equation}

hold true. We will compute estimates for $u_{\ell + 1}$ , $v_{\ell + 1}$ and $w_{\ell + 1}$ . Note that $\eta _\ell \leq \overline \eta$ , $\zeta _\ell \leq \overline \zeta$ , $\xi _\ell \leq \overline \xi$ and $\chi _\ell \leq \overline \chi$ hold, independently of $\ell \geq 3$ . So, as the same discussion in the case $\ell = 3$ above, we can see that $u_{\ell +1}$ , $v_{\ell +1}$ , $w_{\ell +1} \geq 0$ and

\begin{equation*} K_{j, \ell +1} \leq 2 M \quad \textrm {for} \quad t \in (0, T_0], \quad 1 \leq j \leq 6 \quad \text {and} \quad 1 \leq i \leq n. \end{equation*}

The detail is omitted here. Hence, the nonnegativities of approximations and (10) hold true for all $\ell \in{\mathbb N}$ .

We can see that $u_\ell$ , $v_\ell$ and $w_\ell$ are continuous in $t \in [0, T_0]$ for $\ell \in{\mathbb N}$ . And also, it is easy to see that $ \left\{ u_\ell, v_\ell, w_\ell, t^{1/2} \partial _i u_\ell, t^{1/2} \partial _i v_\ell, \partial _i w_\ell \right\}_{\ell =1}^\infty$ are Cauchy sequences in $C ([0, T_0];\, \text{BUC})$ , choosing $T_0$ small again, if necessary. Let

\begin{equation*} (u, v, w, \hat u, \hat v, \hat w) \,:\!=\, \lim _{\ell \to \infty } \left(u_\ell, v_\ell, w_\ell, t^{1/2} \partial _i u_\ell, t^{1/2} \partial _i v_\ell, \partial _i w_\ell \right) \end{equation*}

in the topology of $C ([0, T_0];\, \text{BUC})$ . Obviously, the coincidences $\hat u = t^{1/2} \partial _i u$ , $\hat v = t^{1/2} \partial _i v$ and $\hat w = \partial _i w$ hold by construction. Furthermore, it is also ensured that

\begin{equation*} 0 \leq u (x, t), v (x, t), w (x, t) \leq 2M \quad \textrm {for} \,\,\, x \in {\mathbb R}^n \,\,\, \text {and} \,\,\, t \in [0, T_0]. \end{equation*}

The uniqueness follows from (1)–(3) and Gronwall’s inequality, directly. If fact, let $(u, v, w)$ and $(u^\ast, v^\ast, w^\ast )$ be solutions to (P) in $[0, T_0]$ with the same initial data, then $u \equiv u^\ast$ , $v \equiv v^\ast$ and $w \equiv w^\ast$ simultaneously hold. Thanks to the boundedness of the first derivatives, it is easy to control the second derivatives in $x$ of $u$ and $v$ for $t \in (0, T_0]$ , as well as the first derivatives in $t$ of solutions. So, we see that $( u, v, w )$ is a time-local unique classical solution to (P). This completes the proof of Proposition 1.

5. Global well-posedness

In this section, we will derive a priori bounds of solutions and their derivatives. To do so, our first task is to obtain upper bounds of solutions to (P) with large initial data. For the case when $\| u_0 \| \leq 1$ , we will discuss in Remark 3 (ii) below and Section 6.

If $v_0 \equiv 0$ and $w_0 \equiv 0$ , then $v \equiv w \equiv 0$ for $t \gt 0$ . Assume either $v_0 \not \equiv 0$ or $w_0 \not \equiv 0$ . So, as seen in Remark 2 (iii), we have $u$ , $v$ , $w \gt 0$ . For observing the behaviour of $u$ , we consider the following logistic equation:

(11) \begin{equation} \kappa^{\prime} = (1 - \kappa ) \kappa, \quad \kappa (0) = \kappa _0 \gt 1, \end{equation}

where $\kappa _0 = \| u_0 \|$ . By maximum principle, $u (x, t) \leq \kappa (t)$ holds for $x \in{\mathbb R}^n$ and $t \gt 0$ , as long as the classical solution $u$ exists. Since

\begin{equation*} \kappa (t) = \kappa _0/ \left ( \kappa _0+e^{-t}-\kappa _0e^{-t} \right ) \lt \kappa _0 \end{equation*}

for $t \gt 0$ , it is clear that $u \lt \kappa _0$ .

Next, we investigate on upper bounds of $v$ and $w$ . We will use renormalising arguments to ODE. Let a pair $\sigma = \sigma (t)$ and $\omega = \omega (t)$ be solutions to

(12) \begin{align} \left \{ \begin{array}{l@{\quad}l} \sigma^{\prime} = \alpha \omega - (m_\star + \sigma ) \sigma, & \sigma (0) = \sigma _0 \,:\!=\, \| v_0 \|, \\[5pt] \omega^{\prime} = \theta _\star \sigma - \rho \, \omega, & \omega (0) = \omega _0 \,:\!=\, \| w_0 \|. \end{array} \right. \end{align}

Here, $m_\star \,:\!=\, m - \mu \kappa _0/ ( \kappa _0 + h )$ and $\theta _\star \,:\!=\, \theta + \nu \kappa _0/ ( \kappa _0 + h )$ . Since $(e^{\rho t} \omega )^{\prime} = \theta _\star e^{\rho t} \sigma$ , we have

\begin{equation*} \omega (t) = e^{-\rho t} \omega _0 + e^{-\rho t} \theta _\star \int _0^t e^{\rho s} \sigma (s) ds \leq \omega _0 + \left ( \theta _\star/ \rho \right ) \textrm {sup}_{0 \leq s \leq t} \, \sigma (s) \end{equation*}

for $t \gt 0$ . Inserting it into the first equation of (12), it holds that

\begin{equation*} \sigma^{\prime} \leq \alpha \left \{ \omega _0 + (\theta _\star/\rho ) \textrm {sup}_{0 \leq s \leq t} \sigma (s) \right \} - m_\star \sigma - \sigma ^2 \end{equation*}

for $t \gt 0$ . Therefore, we can see that $\sigma (t) \lt \widetilde v \,:\!=\, \max \{ \sigma _0, \bar \sigma \} + 1$ for $t \gt 0$ , where

\begin{equation*} \bar \sigma \,:\!=\, \alpha \theta _\star/2 \rho - m_\star/2 + \sqrt {\alpha \omega _0 + (\alpha \theta _\star/\rho - m_\star )^2/4}. \end{equation*}

Note that $\bar \sigma$ satisfies $\alpha ( \omega _0 + (\theta _\star/ \rho ) \bar \sigma ) - m_\star \bar \sigma - \bar \sigma ^2 = 0$ . Indeed, if there exists some $t_\star \gt 0$ such that $\sigma (t_\star ) = \widetilde v \geq \bar \sigma + 1$ and $\sigma (t) \lt \widetilde v$ for $t \in [0, t_\star )$ , then $\sigma^{\prime} (t_\star ) \geq 0$ . This contradicts $\sigma^{\prime} (t_\star ) \lt 0$ . We can similarly deduce $\omega (t) \leq \widetilde w$ holds for $t \gt 0$ , where $\widetilde w \,:\!=\, \max \{ \omega _0, \theta _\star \widetilde v/ \rho \} + 1$ .

We will use enclosing arguments, that is, applying the comparison principle between solutions to PDE and those to ODE. Put $V \,:\!=\, \sigma - v$ and $W \,:\!=\, \omega - w$ . Hence, $V(0) \geq 0$ and $W(0) \geq 0$ . Also, we see

\begin{align*} \partial _t V & = d \Delta V + \alpha W - m V + \mu \kappa _0 \sigma/ \left ( \kappa _0 + h \right ) - \mu u v/ (u + h) - \sigma ^2 + v^2\\ & = d \Delta V + \alpha W - \left ( m + \sigma + v \right ) V \\ & \quad + \frac{\mu }{(\kappa _0+h)(u+h)} \big [ (u + h) \kappa _0 V + h v (\kappa _0-u) \big ] \end{align*}

and

\begin{align*} \partial _t W & = \theta V - \rho W + \nu \kappa _0 \sigma/ \left ( \kappa _0 + h \right ) - \nu u v/ (u + h)\\ & = \theta V - \rho W + \frac{\nu }{(\kappa _0+h)(u+h)} \big [ (u + h) \kappa _0 V + h v \left ( \kappa _0 - u \right ) \big ]. \end{align*}

We thus find the fact that $V \geq 0$ and $W \geq 0$ for $t \gt 0$ , as the same discussion in the proof of Proposition 1. This implies that

(13) \begin{equation} v (x, t) \leq \sigma (t), \quad w (x, t) \leq \omega (t) \end{equation}

for $x$ and $t$ . Therefore, we conclude that $0 \leq v \leq \widetilde v$ and $0 \leq w \leq \widetilde w$ .

In what follows, we give the a priori estimate for $\| \partial _i w (t) \|$ , which may grow in $t$ . As seen in Proposition 2, and by using definitions of $\overline v$ and $\overline w$ in Theorem 2, we prove that $0 \leq u, v, w \leq N$ as long as the classical solutions exist, if $N$ is chosen as

(14) \begin{equation} N \,:\!=\, \max \big \{ 1, \| u_0 \|, \overline v, \widetilde v, \| v_0 \|, \overline w, \widetilde w, \| w_0 \| \big \}. \end{equation}

Proposition 3. Let $T$ , $N \gt 0$ . If $0 \leq u, v, w \leq N$ for $x \in{\mathbb R}^n$ and $t \in [0, T]$ , then there exists a $C \gt 0$ independent of $N$ and $T$ such that

\begin{equation*} \| \partial _i w (t) \| \leq \| \partial _i w_ 0 \| + C \left ( N^4 + N \right ) \left ( t^{1/2} + t^{3/2} \right ), \quad t \in [0, T], \,\, 1 \leq i \leq n. \end{equation*}

We first derive the estimate for $\partial _i u$ . By (1), we have

\begin{align*} \| \partial _i u (t) \| & \leq \| u_0 \| t^{-1/2} + \int _0^t (t-s)^{-1/2} \left \| (1-u) u - \frac{\gamma u v}{u+h} \right \| ds \\ & \leq C \left ( N^2 + N \right ) \left ( t^{-1/2} + t^{1/2} \right ) \end{align*}

for $t \in [0, T]$ and $1 \leq i \leq n$ with some $C$ . Similarly, by (2), we seek

\begin{align*} \| \partial _i v (t) \| & \leq \| v_0 \| (d t)^{-1/2} \!+\! \int _0^t (dt-ds)^{-1/2} \left \| \frac{\mu u v}{u + h} \!+\! \alpha w - (m + v) v \right \| \! ds \\ & \leq C \left ( N^2 + N \right ) \left ( t^{-1/2} + t^{1/2} \right ) \end{align*}

with some $C$ . Finally, by (3) and estimates above, it turns out that

\begin{align*} \| \partial _i w (t) \| & \leq \| \partial _i w_0 \| + \int _0^t \left \| \frac{\nu h (\partial _i u) v + \nu u (\partial _i v) (u + h)}{(u + h)^2} + \theta \partial _i v \right \| ds \\ & \leq \| \partial _i w_0 \| + C \left ( N^4 + N \right ) \int _0^t \left ( s^{-1/2} + s^{1/2} \right ) ds \\ & \leq \| \partial _i w_0 \| + C \left ( N^4 + N \right ) \left ( t^{1/2} + t^{3/2} \right ) \end{align*}

for $t \in [0, T]$ and $1 \leq i \leq n$ with some positive constant $C$ depending on parameters, however, independent of $N$ and $T$ .

Note that the proof of Theorem 1 is now complete. In fact, Theorem 1 follows from Propositions 1, 2, 3 and $T_0 \geq C_\ast/(M^4+1)$ in Proposition 1, since we can extend the obtained unique classical solutions time-globally, repeating the construction.

6. Invariant regions

This section will be devoted to observing invariant regions. The proof of Theorem 2 (i) is easy, since $(1, 0, 0)$ is only one stable constant state. So, we skip it here.

We are now in position to give a proof of Theorem 2 (ii). The key step is to deduce a priori bounds of solutions, due to the maximum principle and comparison with solutions to the system of corresponding ordinary differential equations of $\kappa$ , $\sigma$ and $\omega$ given by (11) and (12). Let us recall the assumptions:

\begin{align*} \overline v & \,:\!=\, \mu/(1+h) + \alpha (\nu + \theta + \theta h)/ (\rho + \rho h) - m \gt 0,\\ \overline w & \,:\!=\, (\nu + \theta + \theta h) \overline v/ (\rho + \rho h) \gt 0 \end{align*}

and $R_\ast \,:\!=\, [0, 1] \times [0, \overline v] \times [0, \overline w]$ .

Proof. We first show that $R_\ast$ is an invariant region. Let $(u_0, v_0, w_0) \in R_\ast$ . By construction of time-local solutions in Proposition 1, the nonnegativity of solutions is clarified. Note that $(0, 0, 0)$ and $(1, 0, 0)$ are classical solutions in $R_\ast$ . If $u_0 \equiv 0$ , then $u \equiv 0$ , in addition, $v \in [0, \overline v]$ and $w \in [0, \overline w]$ , since $v^\flat \,:\!=\, \alpha \theta/ \rho - m \leq \overline v$ and $w^\flat \,:\!=\, \theta (\alpha \theta - m \rho )/ \rho ^2 \leq \overline w$ . Also, it is easy to see that $v \equiv 0$ and $w \equiv 0$ hold for $t \gt 0$ , provided if $v_0 \equiv 0$ and $w_0 \equiv 0$ .

Let $u_0 \not \equiv 0$ and either $v_0 \not \equiv 0$ or $w_0 \not \equiv 0$ . As seen in Remark 2 (iii), it is clear that the classical solutions $u$ , $v$ , $w$ never touch $0$ , as long as they exist. Moreover, with $u_0 \leq 1$ , we observe that $u(\tau ) \lt 1$ for small $\tau \gt 0$ by the strong maximum principle. Similarly, it turns out that $v(\tau ) \lt \overline v$ by $v_0 \leq \overline v$ , as well as $w(\tau ) \lt \overline w$ . So, regarding $\tau$ as the initial time, we can assume $(u_0, v_0, w_0) \in R_\ast ^\circ \,:\!=\, (0, 1) \times (0, \overline v) \times (0, \overline w) = R_\ast \setminus \partial R_\ast$ , without loss of generality.

Put $\hat t \in (0, T_0]$ is the first time when $u$ touches $1$ at $\hat x \in{\mathbb R}^n$ . We may assume $|\hat x| \lt \infty$ by Oleinik’s argument on the maximum principle; see e.g. [Reference Giga, Giga and Saal4]. Since $u (\hat x, \hat t) = 1$ is the local maximum, at $(\hat x, \hat t)$ we see that $\partial _t u \geq 0$ , $\Delta u \leq 0$ , $(1 - u) u = 0$ and $-\gamma uv/(u+h) \lt 0$ by $v \gt 0$ . This contradicts to that $u$ is a solution to (P). Hence, $u$ never touches $1$ .

The same argument works on $v$ and $w$ . Indeed, let $0 \lt u \lt 1$ , $0 \lt w \lt \overline w$ , and if there exists $(\check x, \check t) \in{\mathbb R}^n \times (0, T_0]$ such that $\check t$ is the first time when $v$ touches $\overline v$ at $\check x$ . So, at $(\check x, \check t)$ , we see that $\partial _t v \geq 0$ , $d \Delta v \leq 0$ and

\begin{equation*} \frac {\mu u v}{u+h} + \alpha w - (m + v) v \lt \frac {\mu \overline v}{1+h} + \alpha \overline w - \left ( m + \overline v \right ) \overline v = 0. \end{equation*}

So, $v$ never touches $\overline v$ . As the same as above, we can confirm that $w$ never touches $\overline w$ as long as classical solutions exist. This means that the solutions always remain in $R_\ast ^\circ$ .

Next, we show the asymptotic behaviour of solutions, briefly. Even if $\| u_0 \| \gt 1$ , by $u (x, t) \leq \kappa (t)$ , then there exists a $T_\varepsilon ^\ast \gt 0$ such that $\| u (t) \| \lt 1 + \varepsilon$ for $t \gt T_\varepsilon ^\ast$ . From this and the comparison $v(x, t) \leq \sigma (t)$ , there exists $T_\varepsilon ^\sharp \gt T_\varepsilon ^\ast$ such that $\| v (t) \| \lt \overline v + \varepsilon$ for $t \gt T_\varepsilon ^\sharp$ . Finally, we can also show that there exists $T_\varepsilon \gt T_\varepsilon ^\sharp$ such that $\| w (t) \| \lt \overline w + \varepsilon$ for $t \gt T_\varepsilon$ , by the similar way. This completes the proof of Theorem 2 (ii).

The proof of Theorem 2 (iii) is essentially similar to above. So, we omit it here.

Remark 4. The stability of non-trivial constant states to the system of corresponding ODE can be easily obtained. For example, if

\begin{equation*} \mu = \nu = \frac {\gamma }2, \, m = \theta = 0, \, \alpha = \rho = \frac 14, \, \gamma = h + \frac 12 \end{equation*}

are chosen, then the bifurcation occurs, that is, the stability of a constant state $(u, v, w) = (1/2, 1/2, 1/2)$ is changed in $h$ at $0$ . Indeed, the constant state $(1/2, 1/2, 1/2)$ is stable for any $h \gt 0$ , while this is unstable for any $-1/2 \lt h \lt 0$ . The authors believe that such stability is still valid for solutions to (P). For studying the Turing instability, we need to deal with more complicated situation, for example, when $\mu$ and $\nu$ are sigmoid functions of $u$ .