In recent years, the reaction-diffusion equation with nonlocal diffusion effect has gradually become a hot research area in the field of materials science and biological mathematics[1-3]. The introduction of nonlocal diffusion terms in mathematical modeling can, in many cases, better describe some natural phenomena. Although many scholars[4, 5] have studied the traveling wave solutions of reaction-diffusion equations with nonlocal diffusion effect, there is little research on invading phenomenon in reaction-diffusion systems.

Recently, Ducrot et.al[6] have investigated a two-species predator-prey model with nonlocal dispersal as follows

$ \begin{align} \left\{ \begin{array}{l} \frac{\partial U(x, t)}{\partial t} = d_{1}\mathcal{N}_{1} [U(\cdot, t)](x)+r_{1}U(x, t)(1-U(x, t))-aU(x, t)V(x, t), x\in R, t\geq 0, \\ \frac{\partial V(x, t)}{\partial t} = d_{2}\mathcal{N}_{2} [V(\cdot, t)](x)+bU(x, t)V(x, t)-r_{2}V(x, t)(1+V(x, t)), x\in R, t\geq 0, \end{array} \right. \end{align} $

(1)

in which the terms $ d_{1}\mathcal{N}_{1} $ and $ d_{2}\mathcal{N}_{2} $ describe the spatial dispersal of the prey and the predator, other coefficients are nonnegative constants. $ \mathcal{N}_{i} $ is the linear nonlocal diffusion operator defined by

$ \mathcal{N}_{i}[\varphi](x): = (J_{i}*\varphi)(x)-\varphi(x) = \int_{R}J_{i}(x-y)\varphi(y)dy-\varphi(x), \quad i = 1, 2, $

wherein $ J_{i}, i = 1, 2 $ are probability kernel functions and the symbol $ * $ denotes the concolution product relative to the space variable, $ x\in R $ (We also refer readers to specific definitions by [7] while be given later). Under certain conditions, they obtained the spreading speed $ c_{1} $ of predator invading native prey habitat as follow

$ c_{1}: = \inf\limits_{0<\lambda<\bar{\lambda}_{2}}\frac{d_{2}[\int_{R}J_{2}(y)e^{\lambda y}dy-1]+b-r_{2}}{\lambda}. $

In fact, due to the diversity and complexity of ecosystems, it is more practical to study the interaction between multiple species. However, when there are more than two species, it becomes more difficult to study the ecosystem. Chin-Chin Wu [8] has studied the following three-component reaction diffusion system

$ \begin{align} \left\{ \begin{array}{l} \frac{\partial u(x, t)}{\partial t} = d_{1}\Delta u(x, t)+r_{1}u(x, t)[1-u(x, t)-a_{1}v(x, t)-b_{1}w(x, t)], x\in R, t\geq 0, \\ \frac{\partial v(x, t)}{\partial t} = d_{2}\Delta v(x, t)+r_{2}v(x, t))[1-a_{2}u(x, t)-v(x, t)-b_{2}w(x, t)], x\in R, t\geq 0, \\ \frac{\partial w(x, t)}{\partial t} = d_{3}\Delta w(x, t)+r_{3}w(x, t)[-1+b_{1}u(x, t)+b_{2}v(x, t)-w(x, t)], x\in R, t\geq 0, \end{array} \right. \end{align} $

(2)

where $ d_{i}, r_{i}, a_{i}, b_{i}, i = 1, 2, 3 $ are positive constants. When the above system satisfies a series of conditions such as $ a_{1}, a_{2}<1 $ and $ b_{1} = b_{2} = b>1 $, the author has characterized the asymptotic spreading speed $ c_{2} $ of the predator with

$ c_{2}: = 2\sqrt{d_{3}r_{3}(-1+b(\frac{2-a_{1}-a_{2}}{1-a_{1}a_{2}}))}. $

Remark 1.1  In the system $ (1.2) $, the author assumes that the predator has the same predation rate to the two kinds of preys and it is also the conversion rate of the predator's absorption. These hypothetical conditions make the predator-prey model too idealistic. Moreover, the author does not consider the spatial nonlocal diffusion. For the system $ (1.1) $, the authors consider the spatial nonlocal diffusion effect, but the model only has two species. So it is reasonable for us to consider the model of three species interaction. To the author's knowledge, the asymptotic spreading speed of three species model with nonlocal diffusion effect is still a problem to be solved.

In this paper, we consider the following three-component reaction-diffusion system with nonlocal dispersal

$ \begin{align} \left\{ \begin{array}{l} \frac{\partial u_{1}(x, t)}{\partial t} = d_{1}\mathcal{N}_{1} [u_{1}(\cdot, t)](x)+r_{1}u_{1}(x, t)[1-u_{1}(x, t)-a_{1}u_{2}(x, t)-b_{1}u_{3}(x, t)], x\in R, t\geq 0, \\ \frac{\partial u_{2}(x, t)}{\partial t} = d_{2}\mathcal{N} _{2} [u_{2}(\cdot, t)](x)+r_{2}u_{2}(x, t)[1-a_{2}u_{1}(x, t)-u_{2}(x, t)-b_{2}u_{3}(x, t)], x\in R, t\geq 0, \\ \frac{\partial u_{3}(x, t)}{\partial t} = d_{3}\mathcal{N}_{3}[u_{3}(\cdot, t)](x)+r_{3}u_{3}(x, t)[-1+a_{3}u_{1}(x, t)+b_{3}u_{2}(x, t)-u_{3}(x, t)], x\in R, t\geq 0, \end{array} \right. \end{align} $

(3)

$ \mathcal{N}_{i}[\varphi](x): = (J_{i}*\varphi)(x)-\varphi(x) = \int_{R}J_{i}(x-y)\varphi(y)dy-\varphi(x), \quad i = 1, 2, 3, $

where $ d_{i}, r_{i}, a_{i}, b_{i}, i = 1, 2, 3 $ are positive constants, $ u_{1} $, $ u_{2} $ and $ u_{3} $ represent the population density of preys and predator at space $ x $ and time $ t $, respectively. The parameters $ r_{1} $ and $ r_{2} $ represent the growth rates of $ u_{1} $ and $ u_{2} $, $ r_{3} $ is the death rate of $ u_{3} $, $ a_{1} $and $ a_{2} $ represent the interspecific competition coefficient of two kinds of preys, $ r_{1}b_{1} $ and $ r_{2}b_{2} $ are the predation rates, and $ r_{3}a_{3}, r_{3}b_{3} $ are the conversion rates.

It's noting that system $ (1.3) $ represents a model of a three-species system in which two preys compete with each other and are preyed upon by a predator. Such systems occur frequently in nature. For example, one population could be the predator such as lady beetles, and the second and third could be prey species such as English grain aphid and the oat-bird cherry aphid[9].

In this paper we study a three-species predator-prey model with nonlocal diffusion. Our main concern is the invasion process of the predator into the habitat of two aborigine preys. Under certain conditions, we are able to characterize the asymptotic spreading speed by the use of comparison principle and semigroup theory. Moverover, we can get the same results as system $ (1.2) $ by degenerating our system.

For the sake of describing the process of species invasion, we need to introduce some measures such as the asymptotic spreading speed.

Definition 2.1 [10] Let $ z(x, t) $ be nonnegative for $ x\in R, t>0 $. Then $ s^{*} $ is called the spreading speed of $ z(x, t) $ when

$ \lim\limits_{t\longrightarrow\infty}\sup\sup\limits_{|x|>(s^{*}+\epsilon)t}z(x, t) = 0, \qquad\lim \limits_{t\longrightarrow\infty}\inf\inf\limits_{|x|<(s^{*}-\epsilon)t}z(x, t)>0, \qquad \forall \epsilon\in (0, s^{*}). $

In this paper, we assume that $ J_{i}, i = 1, 2, 3 $ satisfies the following definition.

Definition 2.2 [11] Let $ \overline{\lambda}\in(0, \infty] $ be given. We say that the kernel function $ J:R\longrightarrow R $ belongs to the class $ \tau(\overline{\lambda}) $ if it satisfies the following properties:

(J1) The kernel J is nonnegative and continuous in $ R $;

(J2) For all $ x\in R $ it holds that

$ \int_{R}J(y)dy = 1 \quad and \quad J(y) = J(-y) ; $

(J3) it holds that $ \int_{R}J(y)e^{\lambda y}dy<\infty $ for any $ \lambda\in(0, \overline{\lambda}) $ and

$ \int_{R}J(y)e^{\lambda y}dy\longrightarrow \infty\quad as \quad \lambda\longrightarrow\overline{\lambda}. $

Let $ d>0, r>0, s>0, \overline{\lambda}\in(0, \infty] $ and $ J\in\tau(\overline{\lambda}) $. First, we consider the following nonlocal logistic equation

$ \begin{align} \left\{ \begin{array}{l} \frac{\partial z(x, t)}{\partial t} = d\mathcal{N}[z(\cdot, t)](x)+rz(x, t)[s-z(x, t)], \\ z(x, 0) = z(x), \end{array} \right. \end{align} $

(2.1)

where $ \mathcal{N}[w] = :J*w-w $ and $ 0<z(x)\leq s $ is a bounded and continuous function with nonempty support, for the scalar nonlocal equation enjoys the following comparison principle.

Lemma 2.1 [12] Let $ z(x, t) $ be a solution of $ (2.1) $, $ z(\cdot, t), t>0 $ and $ 0<z(x)\leq s $ are continuous and bounded for $ x\in R $, $ t>0 $. Assume that $ 0<w(x, 0)\leq s $ and $ w(x, t) $ are continuous and bounded for $ x\in R $, $ t>0 $, if they satisfy

$ \left\{ \begin{array}{l} \frac{\partial w(x, t)}{\partial t}\geq(\leq)d\mathcal{N}[w(\cdot, t)](x)+rw(x, t)[s-w(x, t)], x\in R, t>0, \\ w(x, 0)\geq(\leq)z(x), x\in R, \end{array} \right. $

then $ w(x, t)\geq(\leq)z(x, t) $, $ x\in R $, $ t>0 $.

Lemma 2.2 [12] Let $ z(x, t) $ be a solution of $ (2.1) $, $ z(\cdot, t) $ is continuous and bounded for all $ t>0 $ for a given $ 0<z(x)\leq s $. If $ z(x) $ has a nonempty compact support. Then we have

$ \lim\limits_{t\longrightarrow\infty}\sup\sup\limits_{|x|>(c_{3}+\epsilon)t}z(x, t) = 0, \qquad\lim \limits_{t\longrightarrow\infty}\inf\inf\limits_{|x|<(c_{3}-\epsilon)t}z(x, t) = s, \qquad \forall \epsilon\in (0, c_{3}), $

where

$ c_{3}: = \inf\limits_{0<\lambda<\bar{\lambda}}\frac{d[\int_{R}J(y)e^{\lambda y}dy-1]+rs}{\lambda}. $

In this paper, due to the weak competition between two preys, we consider the following initial conditions of the system $ (1.3) $

$ \begin{align} (u_{1}(x, 0), u_{2}(x, 0), u_{3}(x, 0)) = (u_{1}^{*}, u_{2}^{*}, v(x)) = :(\frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{1}}{1-a_{1}a_{2}}, v(x))\quad x\in R, \end{align} $

(2.2)

where $ v(x) $ is a nonnegative continuous function with nonempty compact support. Throughout of this paper, we always assume that $ 0<v(x)\leq\alpha: = a_{3}u_{1}^{*}+b_{3}u_{2}^{*}-1. $

Because of the nonnegativity of $ u_{1} $, $ u_{2} $ and $ u_{3} $, we can easily get $ u_{1}(x, t)\leq1, $ $ u_{2}(x, t)\leq1 $ for all $ (x, t)\in R\times[0, \infty) $ by the comparison principle. Hence $ u_{3}(x, t) $ satisfies the equation

$ \frac{\partial u_{3}(x, t)}{\partial t}\leq d_{3}\mathcal{N}_{3}[u_{3}(\cdot, t)](x) +r_{3}u_{3}(x, t)[-1+a_{3}+b_{3}-u_{3}(x, t)]. $

By Lemma 2.1, we have $ u_{3}(x, t)\leq-1+a_{3}+b_{3}. $

Further by a comparison, we obtain

$ \begin{align} u_{1}(x, t)\geq1-a_{1}-b_{1}(-1+a_{3}+b_{3}) = :\underline{u}_{1}, \end{align} $

(2.3)

$ \begin{align} u_{2}(x, t)\geq1-a_{2}-b_{2}(-1+a_{3}+b_{3}) = :\underline{u}_{2}. \end{align} $

(2.4)

Since the asymptotic spreading involves long time behavior, we have

$ \begin{align} u_{3}\geq a_{3}\underline{u}_{1}+b_{3}\underline{u}_{2}-1 = :\underline{u}_{3}. \end{align} $

(2.5)

Next, we discuss the spreading speed of the predator $ u_{3}(x, t) $ in system $ (1.3) $ with following conditions

$ \begin{align} \underline{u}_{i}>0, \quad i = 1, 2, 3, \quad 1-a_{1}\underline{u}_{2}-b_{1}\underline{u}_{3}\leq u_{1}^{*}, \quad1-a_{2} \underline{u}_{1}-b_{2}\underline{u}_{3}\leq u_{2}^{*}. \end{align} $

(2.6)

We now state our main result on the spreading speed of the predator as follows.

According to the Definition 2.2, we know there exists $ \lambda_{3}\in(0, \infty] $ such that $ J_{3}\in \tau(\overline{\lambda_{3}}) $. Then we define the quantity

$ c^{*}: = \inf\limits_{0<\lambda<\bar{\lambda}_{3}}\frac{d_{3}[\int_{R}J_{3}(y)e^{\lambda y}dy-1]+r_{3}\alpha}{\lambda}. $

Theorem 2.1  Assume that conditions of $ (2.6) $ are enforced. Let $ (u_{1}(x, t), u_{2}(x, t), u_{3}(x, t)), $ $ x\in R $, $ t>0 $ be a solution of $ (1.3) $ with initial data $ (2.2) $. As long as $ v(x) $ is a non-zero compactly supported continuous function with $ 0\leq v(x)\leq \alpha $, then the density of the predator $ u_{3}(x, t) $ satisfies

$ \begin{align} \lim\limits_{t\longrightarrow\infty}\sup\limits_{\mid x\mid>ct} u_{3}(x, t) = 0\quad, c>c^{*}, \end{align} $

(2.7)

$ \begin{align} \lim\limits_{t\longrightarrow\infty}\inf\inf\limits_{\mid x\mid<ct} u_{3}(x, t)>0, c\in(0, c^{*}). \end{align} $

(2.8)

Now we first prove $ (2.7). $ On the base of the results of $ (2.3)-(2.6) $, we have

$ 0\leq u_{1}(x, t)\leq u_{1}^{*}, \quad 0\leq u_{2}(x, t)\leq u_{2}^{*}. $

Further, $ u_{3}(x, t) $ satisfies

$ \frac{\partial u_{3}(x, t)}{\partial t}\leq d_{3}\mathcal{N}_{3}[u_{3}(\cdot, t)](x) +r_{3}u_{3}(x, t)[\alpha-u_{3}(x, t)], \quad x\in R, t>0. $

By Lemma 2.1 we obtain $ u_{3}(x, t)\leq\widetilde{u}_{3}(x, t), (x, t)\in R\times[0, \infty), $ where $ \widetilde{u}_{3}(x, t) $ is the solution to

$ \left\{ \begin{array}{l} \frac{\partial \widetilde{u}_{3}(x, t)}{\partial t} = d_{3}\mathcal{N}_{3}[\widetilde{u}_{3}(\cdot, t)](x)+r_{3}\widetilde{u}_{3}(x, t)[\alpha-\widetilde{u}_{3}(x, t)], x\in R, t\geq0, \\ \widetilde{u}_{3}(x, 0) = v(x), x\in R. \end{array} \right. $

It follows from the classical result of Lemma 2.2 that

$ 0\leq\lim\limits_{t\longrightarrow\infty}\sup\limits_{\mid x\mid>ct} u_{3}(x, t)\leq\lim\limits_{t\longrightarrow\infty}\sup\limits_{\mid x\mid>ct} \widetilde{u}_{3}(x, t) = 0, \quad c>c^{*}. $

This proves (2.7) and then we will prove $ (2.8) $. From the first formula of system $ (1.3) $ and $ u_{2}(x, t)\leq u_{2}^{*} $, $ u_{1}(x, t) $ satisfies

$ \frac{\partial u_{1}(x, t)}{\partial t}\geq d_{1}\mathcal{N}_{1} [u_{1}(\cdot, t)](x) +r_{1}\underline{u}_{1}[u_{1}^{*}-u_{1}(x, t)]-r_{1}b_{1}u_{3}(x, t), \quad x\in R, t>0. $

Set $ \overline{u}_{1}(x, t) = u_{1}^{*}-u_{1}(x, t), $ $ x\in R $, $ t>0 $, then previous formula can be rewritten as

$ \begin{eqnarray*} \frac{\partial \overline{u}_{1}(x, t)}{\partial t}\leq d_{1}\mathcal{N} [\overline{u}_{1}(\cdot, t)](x)+r_{1}b_{1}u_{3}(x, t)-r_{1}\underline{u}_{1}\overline{u}_{1}(x, t), \quad x\in R, t>0. \end{eqnarray*} $

Next, by using $ \overline{u}_{1}(x, t) = 0 $ at $ t = 0 $ and the theory of semigroup, we have

$ \begin{align} \overline{u}_{1}(x, t)\leq r_{1}b_{1}\int_{0}^{t}e^{-r_{1}\underline{u}_{1}(t-s)}\{exp((t-s)d_{1}\mathcal{N}_{1})[u_{3}(\cdot, s)]\}(x)ds, \quad x\in R, t>0. \end{align} $

(2.9)

Set $ \overline{u}_{2}(x, t) = u_{2}^{*}-u_{2}(x, t), $ similarly, we have

$ \begin{align} \overline{u}_{2}(x, t)\leq r_{2}b_{2}\int_{0}^{t}e^{-r_{2}\underline{u}_{2}(t-s)}\{exp((t-s)d_{2}\mathcal{N}_{2})[u_{3}(\cdot, s)]\}(x)ds, \quad x\in R, t>0. \end{align} $

(2.10)

It follows from the third formula of system $ (1.3) $ that

$ \begin{align} \frac{\partial u_{3}(x, t)}{\partial t} = d_{3}\mathcal{N}_{3}[u_{3}(\cdot, t)](x)+r_{3}u_{3}(x, t)[\alpha-a_{3}\overline{u}_{1}(x, t)-b_{3}\overline{u}_{2}(x, t)-u_{3}(x, t)], \quad x\in R, t\geq0. \end{align} $

(2.11)

For any given constant $ \epsilon\in(0, c^{*}) $, we can choose constant $ \delta\in(0, \alpha) $ small enough such that

$ \inf\limits_{0<\lambda<\bar{\lambda}_{3}}\frac{d_{3}[\int_{R}J_{3}(y)e^{\lambda y}dy-1]+(\alpha - \delta) r_{3}}{\lambda}>c^{*}-\epsilon. $

We first assume that there exist $ M_{1}>0, M_{2}>0 $ and $ \tau_{1}>0, \tau_{2}>0 $ large enough such that

$ \begin{align} a_{3}\overline{u}_{1}(x, t)\leq\frac{\delta}{2}+M_{1}u_{3}(x, t)\quad x\in R, t\geq\tau_{1}, \end{align} $

(2.12)

$ \begin{align} b_{3}\overline{u}_{2}(x, t)\leq\frac{\delta}{2}+M_{2}u_{3}(x, t)\quad x\in R, t\geq\tau_{2}. \end{align} $

(2.13)

Then, combining $ (2.12) $, $ (2.13) $ with the third formula, we obtain

$ \frac{\partial u_{3}(x, t)}{\partial t}\geq d_{3}\mathcal{N}_{3}[u_{3}(\cdot, t)](x)+r_{3}u_{3}(x, t) [\alpha-\delta-(M_{1}+M_{2})u_{3}(x, t)], x\in R, t\geq \tau: = \max\{\tau_{1}, \tau_{2}\}. $

By the comparison principle, we have $ u_{3}(x, t+\tau)\geq\underline{U}_{3}(x, t), x\in R, t\geq0, $ where $ \underline{U}_{3}(x, t) $ is the solution to

$ \left\{ \begin{array}{l} \frac{\partial\underline{U}_{3}(x, t)}{\partial t} = d_{3}\mathcal{N}_{3}[\underline{U}_{3}(\cdot, t)](x)+r_{3}\underline{U}_{3}(x, t)[\alpha-\delta-(M_{1}+M_{2})\underline{U}_{3}(x, t)], \\ \underline{U}_{3}(x, 0) = v(x). \end{array} \right. $

Finally, combining Lemma 2.2 and Lemma 2.3, we obtain

$ \lim\limits_{t\longrightarrow\infty}\inf\inf\limits_{\mid x\mid<ct} u_{3}(x, t)\geq\lim\limits_{t\longrightarrow\infty}\inf\inf\limits_{\mid x\mid<ct}\underline{U}_{3}(x, t) = \frac{\alpha-\delta}{M_{1}+M_{2}+1}>0, c\in(0, c^{*}-\epsilon). $

Hence, this completes the proof of $ (2.8) $ since $ \epsilon $ is arbitrary.

To complete the proof of Theorem 1.1, it remains to derive (2.12) and (2.13). In the following, we only prove the formula $ (2.12) $. In order to achieve our goal, we need to know more detailed properties about the strongly positive semigroup $ T_{i}(t): = exp(td_{i}\mathcal{N}_{i}) $.

Let $ Z(x, t) $ be a solution of the following problem

$ \left\{ \begin{array}{l} \frac{\partial Z(x, t)}{\partial t} = d_{i}\{[J_{i}Z(\cdot, t)](x)-Z(x, t)\}, \\ Z(x, 0) = \delta_{0}(x), \end{array} \right. $

wherein $ \delta_{0}(x) $ denotes the Dirac mass at $ x = 0 $. According to[12], $ Z(x, t) $ can be decomposed into the following forms

$ Z(x, t) = e^{-d_{1}t}\delta_{0}(x)+K(x, t), \quad x\in R, t\geq0. $

where $ K(x, t) $ is a nonnegative smooth function and $ \int_{R}K(x, t)dx\leq2 $, $ t\geq0. $

For any bounded and continuous function $ 0\leq\phi(x) $, the semigroup $ T_{i}(t) $, $ t\geq0 $ can be expressed as

$ T_{i}(t)[\phi](x) = e^{-d_{i}t}\phi(x)+\int_{R}K(x-y, t)\phi(y)dy, \quad t\geq0. $

Next we start to prove the formula of $ (2.12). $ By $ (2.9) $, we have

$ \begin{align} a_{3}\overline{u}_{1}(x, t) &\leq r_{1}b_{1}a_{3}[\int_{0}^{t}\{e^{-(d_{1}+r_{1}\underline{u}_{1})(t-s)}u_{3}(x, s)+ \int_{R}e^{-r_{1}\underline{u}_{1}(t-s)}K(x-y, t-s)u_{3}(y, s)dy\}ds]\\ & = :V_{1}(x, t)+V_{2}(x, t)\quad x\in R, t>0. \end{align} $

(2.14)

Set $ \beta_{1} = d_{1}+r_{1}\underline{u}_{1}, \beta_{2} = r_{1}\underline{u}_{1}, $ then $ V_{1}(x, t), V_{2}(x, t), x\in R, t>0 $ can be rewritten as

$ \begin{align} V_{1}(x, t) = &r_{1}b_{1}a_{3}\int_{0}^{t}e^{-\beta_{1}(t-s)}u_{3}(x, s)ds, \\ V_{2}(x, t) = &r_{1}b_{1}a_{3}\int_{0}^{t}\int_{R}e^{-\beta_{2}(t-s)}K(x-y, t-s)u_{3}(y, s)dyds. \end{align} $

(2.15)

For the fixed positive constant $ \delta $, there is a large enough constant $ \tau_{1} $ such that

$ \begin{align} r_{1}b_{1}a_{3}\alpha\int_{0}^{t-\tau_{1}}e^{-\beta_{1}(t-s)}ds\leq\frac{\delta}{8}, \quad 2r_{1}b_{1}a_{3}\alpha\int_{0}^{t-\tau_{1}}e^{-\beta_{1}(t-s)}ds\leq\frac{\delta}{8}. \end{align} $

(2.16)

For all $ x\in R, t\geq\tau_{1}, $if $ a_{3}\overline{u}_{1}(x, t)\leq\frac{\delta}{2}, $ then it is obvious that the formula $ (2.12) $ is tenable. Otherwise, there is a point $ (x_{0}, t_{0}) $ with $ t_{0}\geq \tau_{1} $ such that $ a_{3}\overline{u}_{1}(x_{0}, t_{0})>\frac{\delta}{2}. $

In the circumstances, we deduce from $ (2.14)-(2.16) $ that

$ \begin{align} a_{3}\overline{u}_{1}(x_{0}, t_{0})\leq\frac{\delta}{4}+r_{1}b_{1}a_{3}[\int_{t_{0}-\tau_{1}}^{t_{0}}\{ e^{-\beta_{1}(t-s)}u_{3}(x_{0}, s)+\int_{R}e^{-\beta_{2}(t_{0}-s)}K(x_{0}-y, t-s)u_{3}(y, s)dy\}ds]. \end{align} $

(2.17)

Further

$ r_{1}b_{1}a_{3}\int_{0}^{\tau_{1}}e^{-\beta_{1}l}u_{3}(x_{0}, t_{0}-l)dl+r_{1}b_{1}a_{3}\int_{0}^{\tau_{1}} \int_{R}e^{-\beta_{2}l}K(y, l)u_{3}(x_{0}-y, t_{0}-l)dydl\geq\frac{\delta}{4}, $

where $ l = t_{0}-s. $ Next, we choose $ R>0 $ such that

$ r_{1}b_{1}a_{3}\int_{0}^{\tau_{1}}e^{-\beta_{1}l}u_{3}(x_{0}, t_{0}-l)dl+r_{1}b_{1}a_{3}\int_{0}^{\tau_{1}} \int_{-R}^{R}e^{-\beta_{2}l}K(y, l)u_{3}(x_{0}-y, t_{0}-l)dydl\geq\frac{\delta}{8}. $

Note that $ R $ is independent of $ (x_{0}, t_{0}). $ Then we choose $ \eta>0 $ small enough such that

$ \begin{align} r_{1}b_{1}a_{3}\alpha\int_{0}^{\eta}e^{-\beta_{1}l}dl+r_{1}b_{1}a_{3}\alpha\int_{0}^{\eta} \int_{-R}^{R}e^{-\beta_{2}l}K(y, l)dydl\leq\frac{\delta}{16}. \end{align} $

(2.18)

It follows from $ (2.17) $ that

$ r_{1}b_{1}a_{3}\int_{\eta}^{\tau_{1}}e^{-\beta_{1}l}u_{3}(x_{0}, t_{0}-l)dl+r_{1}b_{1}a_{3} \int_{\eta}^{\tau_{1}}\int_{-R}^{R}e^{-\beta_{2}l}K(y, l)u_{3}(x_{0}-y, t_{0}-l)dydl\geq\frac{\delta}{16}. $

Then choose $ \theta>0 $ such that

$ \frac{\delta}{16} = \theta[r_{1}b_{1}a_{3}\int_{\eta}^{\tau_{1}}e^{-\beta_{1}l}dl+r_{1}b_{1}a_{3} \int_{\eta}^{\tau_{1}}\int_{-R}^{R}e^{-\beta_{2}l}K(y, l)dydl], $

there exist $ l_{0}\in[t_{0}-\tau_{1}, t_{0}-\eta] $ and $ y_{0}\in[x_{0}-R, x_{0}+R] $ such that $ u_{3}(y_{0}, l_{0})\geq\theta. $ Moreover, since the function $ u_{3}(x, t) $is uniformly continuous on $ R\times[0, \infty) $, there exists $ \rho>0 $ independent of $ (y_{0}, l_{0}) $ such that $ u_{3}(y, l_{0})\geq\frac{\theta}{2} $, $ \forall y\in[y_{0}-\rho, y_{0}+\rho]. $

We then consider the solution $ w(x, t) $ to

$ \left\{ \begin{array}{l} \frac{\partial w(x, t)}{\partial t} = d_{3}\mathcal{N}_{3}[w(\cdot, t)](x)-r_{3}(1+\alpha)w(x, t), \\ z(x, 0) = w(x), \end{array} \right. $

where $ w(x) $ is a uniformly continuous function with $ w(x)\leq\frac{\rho}{2} $ such that

(i) $ w(x) = \frac{\theta}{2}, \quad |x|\leq\frac{\rho}{2}; $

(ii) $ w(x) = 0, \quad |x|\geq\rho; $

(iii) if $ x\in[\frac{\rho}{2}, \rho] $, then $ w(x) $ is decreasing, if $ x\in[-\rho, -\frac{\rho}{2}] $, then $ w(x) $ is increasing.

By the comparison principle, we have $ w(x, t)\leq u_{3}(y_{0}+x, l_{0}+t) $, $ x\in R $, $ t\geq0. $

Moreover, set $ w(x, t) = e^{r_{3}(1+\alpha)t}exp(d_{3}t\mathcal{N}_{3})[w](x) $, we have $ w(x, t)>0 $, $ x\in R $, $ t\geq0. $ Thus one obtains that

$ \begin{align} u_{3}(x_{0}, t_{0})\geq w(x_{0}-y_{0}, t_{0}-l_{0})\geq\gamma: = \min\limits_{t\in[\eta, \tau_{1}]\quad|x|\in[-R, R]}w(x, t)>0. \end{align} $

(2.19)

Further then it follows from $ (2.18) $ and $ (2.19) $ that

$ a_{3}\overline{u}_{1}(x_{0}, t_{0})\leq\frac{\delta}{2}+\frac{\alpha r_{1}b_{1}a_{3}(2\beta_{1}+\beta_{2})}{\beta_{1}\beta_{2}}\leq\frac{\delta}{2}+M_{1}u_{3}(x_{0}, t_{0}), $

by setting $ M_{1} = \frac{\alpha r_{1}b_{1}a_{3}(2\beta_{1}+\beta_{2})}{\beta_{1}\beta_{2}\gamma}. $ Since $ (x_{0}, t_{0}) $ is arbitrary, $ (2.12) $ is proved.

Example 3.1  Suppose $ J_{3} = \frac{4}{3}(1-x^{2}), \mid x\mid<1 $ and $ J_{3} = 0, \mid x\mid\geq1 $. If the coefficient of system (1.3) with $ a_{1} = a_{2} = 0.2 $, $ a_{3} = 2.4 $, $ b_{1} = b_{2} = 0.1 $, $ b_{3} = 2.6 $, $ d_{i} = 2 $ and $ r_{i} = 0.3 $, then we can obtain a three-species predator-prey competition system with nonlocal diffusion. By Theorem 2.1, we can calculate the asymptotic spreading speed $ c_{4} $ of the predator with

$ c_{4}: = \inf\limits_{0<\lambda<\infty}\frac{16(\lambda-1)e^{\lambda}+16(\lambda+1)e^{-\lambda}-1.05\lambda^{3}}{3\lambda^{4}}\approx 2.3820. $

Example 3.2  Suppose $ J_{3} = \frac{1}{2}e^{-\mid x\mid} $, $ a_{1} = a_{2} = 0.2 $, $ a_{3} = b_{3} = 2.5 $, $ b_{1} = b_{2} = 0.1 $, $ d_{i} = 2 $ and $ r_{i} = 0.3 $, then we can obtain

$ c_{5}: = \inf\limits_{0<\lambda<1}\frac{-1.05\lambda^{2}-0.95}{\lambda(\lambda-1)(\lambda+1)}\approx 9.7959. $

Remark 3.1  System $ (1.1) $ is a special case of system $ (1.3) $. Assume the coefficient $ a_{1} = a_{2} = b_{2} = b_{3} = 0 $, $ b_{1} = \frac{a}{r_{1}} $ and $ a_{3} = \frac{b}{r_{2}} $, we can use the Theorem 2.1 to get the same result as [6].