Spatial patterns for a predator-prey system with Beddington-DeAngelis functional response and fractional cross-diffusion

1.   Introduction

In this paper, we investigate the following predator-prey system with cross-diffusion incorporating the Beddington-DeAngelis functional response subjected to the homogeneous Neumann boundary condition

where $\Omega\subset R^{N}$ is a bounded domain with smooth boundary $\partial\Omega$, $N\geq 1$ is an integer, $\Delta = \sum_{i = 1}^N$ is the Laplace operator in $R^{N}$, $\nu$ represents the outward unit normal vector on the boundary $\partial\Omega$ with $\partial_\nu = \frac{\partial}{\partial \nu}$, and the homogeneous Neumann boundary condition means that the individuals do not cross the habitat boundary, $u$ and $v$ represent the densities of prey and predator, respectively. The parameters $a, b, c$ and $d$ are all positive constants. The interaction between the prey and the predator of system (1.1) is the most usually used the Beddington-DeAngelis functional response

which was introduced by Beddington and DeAngelis, where, the parameters $a, b, c > 0$ are the saturation constant for an alternative prey, the predator interference and the consumption rate, respectively. The term $au$ in the denominator describing mutual interference among the preys while the term $bv$ describing that among the predators. It is well known that the Beddington-DeAngelis functional response has desirable qualitative features of ratio-dependent form but takes care of their controversial behaviors at low densities. Compared with Holling-II functional response, Beddington and DeAngelis response, which considered both the mutual interference among the predator and the handing time of each prey, is more reasonable. One can refer to ^[1,2,3,4] for more details on the background of this functional response.

In view of the inhomogeneous distribution of the predator and prey in different spatial locations with a fixed domain $\Omega$ at any given time and the natural tendency of each species to diffuse, we take into account the predator-prey system (1.1), with self- and cross-diffusions. The role of diffusion into the modelling has been extensively studied. Generally speaking, the diffusion process usually gives rise to a stabilizing effect so that generates a constant equibrium state, namely, the spatial pattern of morphogen or chemical concentration. System (1.1) implies that, in addition to the dispersive force, the diffusion also depends population pressure from other species. The flux of diffusion to the predators of the system is

The part $\frac{d_{4}v}{(1+u)^2}\triangledown u$ of the diffusion flux is directed toward the increasing densities of the prey, which implies the preys respond to attack of team for the movement of predators. The part $-\big(1+\frac{d_{4}}{1+u}\big)\triangledown v$ of the diffusion flux is directed toward the decreasing densities of the predators, which indicates that the predators move towards the preys. The interplay between these diffusion terms and the population dynamics given by (1.1) can lead to complex spatial patterns in the predator-prey system. For example, it may be possible for regions with high predator density to drive local extinction of the prey, leading to further reduction in predator density and eventual recolonization of the area by prey. This kind of spatial dynamics is often observed in natural ecological systems, and can have important implications for conservation efforts and management of ecosystems. For more details about the biological significance, one can see ^{[5,6,7,14,15]} for references, in which the predator-prey system with self- and cross-diffusions were considered.

In ^[8], by employing the Fixed point index theory, the authors studied the existence of the non-constant steady state of a predator-prey with the Beddington-DeAngelis functional response, in which the cross-diffusions are linear. Paper ^[9] considers the stationary problem of the Holling-Tanner prey-predator model with fractional type cross-diffusion terms, and result reveals that the large cross-diffusion can create spatial patterns. Papers ^[10,11] mainly consider the existence of the non-constant positive solutions by making use of the Leray-Schauder degree theory, Furthermore, the authors also discussed the Turing instability of a Gause-type predator-prey system with self-and cross-diffusions mainly by considering the influence of the diffusion terms.

Although lots of researchers have investigated the predator-prey system with nonlinear diffusions(see^{[12,13,14,15]} for reference), there are still many open problem on the spatial patterns cased by the linear and nonlinear cross-diffusions. To our knowledge, there are few works focused on system (1.1), which included the Beddington-DeAngelis functional response and a fractional cross-diffusion for the predator. Paper ^[16] considers a strongly coupled partial differential equation model with a non-monotonic functional response—a Holling type-IV function in a bounded domain with no flux boundary condition. The authors proved a number of existence and non-existence results concerning non-constant steady states of the underlying system. The main purpose of this paper is to research into the effect of the self- and cross-diffusions on the non-constant positive solutions of system (1.1), namely, we investigate the existence and non-existence of the non-constant positive solutions to the following elliptic system

In system (1.2), the cross-diffusions implies that the movement of the species at any spatial location is influenced by the gradient of the concentration of the interacting species at that location. By taking these facts into account, the system can capture much more richer phenomena, and this deserves our careful study and discussion.

The organization of this article is as follows. In Section 2, we give a priori estimate for the positive stationary solution by using the maximum principle and the Harnack inequality. In Section 3, we study the non-existence of non-constant positive solutions mainly by employing the energy integral method and the Poincaré inequality. Moreover, we also discuss the existence of non-constant positive steady states for suitable self- and cross-diffusion coefficients by employing the Leray-Schauder degree theory. The results reveal that the diffusion $d_2$ and the fractional type cross-diffusion $d_4$ can create spatial patterns.

2.   The priori estimate of non-constant positive solutions

We know that there exist three non-negative constant solutions $(0, 0)$, $(1, 0)$ and $(u^{*}, v^{*})$ for system (1.1), where

and $u^{*} < 1$ provided by the condition $c > d(1+a)$. Therefore, it is necessary to assume that $c > d(1+a)$ holds throughout this paper so as to $(u^{*}, v^{*})$ is the unique positive constant solution of system (1.2). For convenience, we denote $\Theta = \Theta(a, b, c, d)$ in the sequel.

In this section, in order to obtain a priori estimates of the positive solution of system (1.2), we first present the following lemmas, named the Maximum principle and Harnack inequality^[17,18], respectively.

Lemma 2.1. Let $\phi(u, x)\in C(\Omega\times R^1)$. If $u\in C^{2}(\Omega)\cap C^{1}(\overline\Omega)$ satisfies $\Delta u(x)+\phi(u, x)\geq0$ in $\Omega$, $\partial_{\nu}u = 0$ on $\partial\Omega$ and achieves its maximum at a point $x_{0}\in\overline\Omega$, then $-\Delta u(x_{0})\geq0$.

Lemma 2.2. Assume that $c(x)\in C(\overline\Omega)$ and let $\omega(x)\in C^{2}(\Omega)\cap C^{1}(\overline{\Omega})$ be a positive solution to

Then there exists a positive constant $C^{\ast} = C^{\ast}(||c||_{\infty}, \Omega),$ such that $\max_{\overline{\Omega}}\omega\leq C ^{\ast}\min_{\overline{\Omega}}\omega$.

Theorem 2.3. Let $C_{1}$, $D_{1}, D_2, D_3$ be given positive constants. Then there exists a positive constant $C = C(\Theta, D_{1}, D_2, D_{3}, \Omega)$ which is independent of $d_i (i = 1, 2, 3, 4)$, such that every positive solution $(u, v)$ of system $(1.2)$, satisfies $C^{-1} < u(x), v(x) < C$ for $d_{1}\geq D_{1}$, $d_2\geq D_2$, $d_3\leq D_3$, $C_1 > \frac{1}{a+c}$ and $b < 1$.

Proof. Let $\phi_{1} = d_{1}[(1+d_{3} v)u], \phi_{2} = d_{2}\big[\big(1+\frac{d_{4}}{1+u}\big)v\big]$. Assume that there exists a point $x_{0}\in\overline{\Omega}$ such that $\phi_{1}(x_{0}) = \max_{\overline{\Omega}}\phi_{1}$. According to Lemma 2.1, we obtain $u(x_{0}) < 1, v(x_{0})\leq\frac{1+a}{1-b}$, then

where $b < 1$, $D_{1}$ and $D_{3}$ are given positive constants with $d_{1}\geq D_{1}$, $d_{3}\leq D_{3}$. Let $x_{1}\in\overline{\Omega}$ be a point such that $\phi_{2}(x_{1}) = \max_{\overline{\Omega}}\phi_{2}$. Then, by the maximum principle, we have

where $C_1 > \frac{1}{a+c}$. Therefore, we can obtain

as $\frac{1+d_4}{1+\frac{d_4}{1+C_1}}$ is strictly increasing with respect to $d_4$ and satisfies $\lim_{d_4\to+\infty}\frac{1+d_4}{1+\frac{d_4}{1+C_1}} = 1+C_1$. Thus, we obtain the upper bounds of the solution $(u, v).$

Hereinafter, we show that $(u, v)$ has a lower bound. For convenience, we set

Then, system (1.2) can be written as

Since $||c_{1}(x)||_{\infty} < C_{3}(\Theta, D_{1}, D_2, D_{3}, \Omega)$, according to Lemma 2.2, we know that there exists a positive constant $C_{4} = C_{4}(\Theta, D_{1}, D_2, D_{3}, \Omega)$ and $C_{5} = C_{5}(\Theta, D_{1}, D_2, D_{3}, \Omega)$ such that

and

Similarly, as $||c_{2}(x)||_{\infty} < \frac{1}{D_2}(d+\frac{c}{a})\triangleq C_{6}$, Lemma 2.2 holds for $\phi_{2}$, that is,

for some positive constant $C_{7}$. Therefore, we have

for some positive constant $C_{8}$. Thus, if the positive solution $(u, v)$ does not have positive lower bound, then there exists a sequence $\{(d_{1i}, d_{2i}, d_{3i}, d_{4i})\}$ satisfying $d_{1i}\geq D_{1}, d_{2i}\geq D_3$ and $d_{3i}\leq D_3$ such that the corresponding solutions of system (1.2) satisfy $\min_{\overline{\Omega}}u_{i} \to 0,$ or $\min_{\overline{\Omega}}v_{i} \to 0$ as $i\to\infty$. Due to the Harnack inequality, we have $\max_{\overline{\Omega}}u_{i} \to 0,$ or $\max_{\overline{\Omega}}v_{i} \to 0$ as $i\to\infty$.

By integrating the second equation of system (1.2) over $\Omega$, we obtain $\int_{\Omega}v\big(-d+\frac{cu}{1+au+bv}\big)dx = 0.$ Thus, there must exists a point $x_{1}\in\overline\Omega$ such that $\frac{cu(x_1)}{1+au(x_1)+bv(x_1)} = d$, which implies $u(x_1)\geq\frac{d}{c}$. Thus, combined with (2.2), we have

Similarly, by integrating the first equation of system (1.2) over $\Omega$, we obtain

Thus, there must exists a point $x_{2}\in\Omega$ such that

Furthermore, we get

According to the inequality (2.3), if $\min_{\overline{\Omega}}v\to 0,$ we have $\max_{\overline{\Omega}}v\to 0$. Hence, we get a contradiction. This shows $v$ has a positive lower bound. This completes the proof.

Remark 2.4. Theorem 2.3 shows that if $d_1$ is not too large or not too small, $d_2$ is not too small and $d_3$ is not too large, then the solutions of system (1.2) are bounded, that is to say, there exists a ball $\boldsymbol{B}(C)$ such that all the positive solution $(u, v)$ of (1.2) satisfying $(u, v)\in\boldsymbol{B}(C)$. We can also conclude that the bound of the solution is not constrained by the cross diffusion $d_4.$

3.   Non-existence and existence of the non-constant positive stationary solutions

The purpose of this section is to study the non-existence and existence of non-constant positive stationary solutions of system (1.2) by taking the self- and cross-diffusions as parameters. The main method used to prove the existence of non-constant positive solutions is the Leray-Schauder degree. The results show that the the self-diffusion $d_2$ and the fractional type cross-diffusion $d_4$ can create spatial patterns while $d_1$ and $d_3$ failed.

3.1.   Non-existence of the non-constant positive stationary solutions

This subsection is devoted to investigating the non-existence of non-constant positive solutions of system (1.2). We mainly use the energy integral method and the well-known Poincaré inequality. Let $\mu_{1}$ be the smallest positive eigenvalue of the operator $-\Delta$ subject to the homogeneous Neumann boundary condition. For convenience, we set

Through some calculations and analysis, we obtain the following result.

Theorem 3.1. Let $d_1, d_2, d_4$ be fixed positive constants and $d_3 = 0$. If there exists a positive constant $\tilde {C} = \tilde {C}(\Theta, D_{1}, \Omega)$, such that $d_{1} > \max\{D_1, \tilde {C} (1+d_{2}^{2}d_{4}^{2})\}, \; d_2 > D_2$ and $d_4 < D_4$, then system (1.2) has no non-constant positive solution.

Proof. Assume that $(u, v)$ is a positive solution of system $(1.2)$. We multiply $\frac{u-\tilde{u}}{u}$ and $\frac{v-\tilde{v}}{v}$ to the equations of system $(1.2)$, respectively, and integrate the equation by parts in $\Omega$. Then, we have

According to Theorem 2.3, for $d_{1}$ with a fixed small $D_1$, there exists a large enough positive constant $C_1 = C_1(\Theta, D_{1}, D_2, D_{4}, \Omega)$, such that

Therefore, we have

It follows the Young's inequality^[19,20,21],

where $K$ is a arbitrary small positive constant. For convenience, we take $d_{3} = 0$ and $K = \min\{\frac{d_{2}}{C_{1}^{3}}, \frac{bc}{(1+aC_{1}+bC_{1})^{2}C_{1}}\}$. By employing the Cauchy inequality and Poincaré inequality^[22,23], we have

Therefore, we can assert that $u\equiv \tilde{u}, v\equiv \tilde{v}$ if $d_{1} > \max\{D_1, \tilde {C} (1+d_{2}^{2}d_{4}^{2})\}, \; d_2 > D_2$ and $d_4 < D_4$.

3.2.   Existence of non-constant positive solutions

In this subsection, we mainly consider the existence of non-constant positive solution of system (1.2) by taking the self- and cross- diffusion coefficients as parameters. Particularly, combing with Theorems 2.3 and 3.1, we consider the cases that the self-diffusion $d_2$ or the cross-diffusion $d_4$ is large enough. The key method used in this subsection to prove the existence of non-constant positive solutions is the well-known Leray-Schauder degree theory ^[24,25,26], which has been extensively used in many different papers. In order to establish the existence of stationary patterns of system (1.2), for convenience, we first introduce some notations and definitions. We define

and set

where $C$ is a positive constant provided by Theorem 2.3. Let $0 = \mu_0 < \mu_1 < \mu_2 < \cdots$ be the eigenvalues of the operator $-\Delta$ and $\{\mu_{i}, \psi_{i}\}_{i = 0}^{\infty}$ be a complete set of eigenpairs for the operator $-\Delta$ in $\Omega$ under homogeneous Neumann boundary condition. Moreover, we can decompose $\boldsymbol{X} = \oplus_{i = 0}^{\infty}\boldsymbol{X}_{i}$ and $\boldsymbol{X_i} = \oplus_{j = 1}^{\dim\boldsymbol{E}(\mu_i)}\boldsymbol{X}_{ij}$, where $\boldsymbol{X}_{i}$ is the eigenspace corresponding to the eigenvalue $\mu_i$.

Therefore, system (1.2) can be rewritten as

It is clear that system (3.1), as well as system (1.2), has a constant positive equilibrium point, denoted by $\boldsymbol{\omega}^{\ast} = (u^{\ast}, \ v^{\ast})^T$, where $u^{\ast}, \ v^{\ast}$ are given in Section 2.

By direct computation, we obtain $\frac{\partial\boldsymbol{\Phi}(u, v)}{\partial(u, v)} = \left[\begin{array}{cc} d_1+d_1d_3v & d_1d_3u \\ \frac{-d_{2}d_{4}v}{(1+u)^2} & d_{2}+\frac{d_{2}d_4}{1+u} \\ \end{array}\right]$ and it is easy to show that $\det \left[\frac{\partial\boldsymbol{\Phi}(u, v)}{\partial(u, v)}\right] > 0$ for all non-negative solutions $(u, v)^T$. Therefore, we know that $\boldsymbol{\Phi}_{\boldsymbol{\omega}}^{-1}$ exists and $\det \left[\frac{\partial\boldsymbol{\Phi}(u, v)}{\partial(u, v)}\right]^{-1}$ is positive. Then, system $(1.2)$ can also be rewritten as

where $\boldsymbol{I}$ is the identity operator and $(\boldsymbol{I}-\Delta)^{-1}$ is the inverse of the operator $\boldsymbol{I}-\Delta$ on $\boldsymbol{X}$ with the homogeneous Neumann boundary condition. Since $\boldsymbol{F}(\Lambda; \cdot)$ is a compact perturbation of the identity operator, the Leray-Schauder degree $\deg(\boldsymbol{F}(\Lambda; \cdot), \boldsymbol{0}, \boldsymbol{B})$ is well-defined if $\boldsymbol{F}(\Lambda; \boldsymbol{\omega})\neq0$ for all $\boldsymbol{\omega}\in\partial \boldsymbol{B}$. Furthermore, we notice that the linearizition of the operator $\boldsymbol{F}(\Lambda; \boldsymbol{\omega})$ at the equilibrium point $\boldsymbol{\omega}^{\ast}$ is

and $\boldsymbol{X}_{i}$ is invariant under $D_{\boldsymbol{\omega}}\boldsymbol{F}(\Lambda; \boldsymbol{\omega}^{\ast})$ for every integer $i\geq0$. What's more, one can check that $\lambda$ is an eigenvalue of the operator $D_{\boldsymbol{\omega}}\boldsymbol{F}(\Lambda; \boldsymbol{\omega}^{\ast})$ on $\boldsymbol{X}_{i}$ if and only if $\lambda$ is an eigenvalue of the matrix

Denote

As the sign of the

is determined by the number of negative eigenvalue of the matrix

then both $H(\Lambda, \boldsymbol{\omega}^{\ast}; \mu_{i})$ and $\det\{\boldsymbol{I}-\frac{1}{1+\mu_{i}}[\boldsymbol{\Phi_{\boldsymbol{\omega}}}^{-1}(\boldsymbol{\omega}^{\ast})\boldsymbol{G_{\boldsymbol{\omega}}}(\boldsymbol{\omega}^{\ast})+\boldsymbol{I}]\}$ have the same sign. Hence, if $H(\Lambda, \boldsymbol{\omega}^{\ast}; \mu_{i})\neq0$, the number of eigenvalues with negative real parts of $\boldsymbol{D_{\boldsymbol{\omega}}F}(\Lambda; \boldsymbol{\omega}^{\ast})$ on $\boldsymbol{X_{i}}$ is odd if and only if $H(\Lambda, \boldsymbol{\omega}^{\ast}; \lambda_{i}) < 0.$

If $H(\Lambda, \boldsymbol{\omega}^{\ast}; \mu_{i})\neq0$ for all integer $i\geq0$, then $0$ is not an eigenvalue of the operator $\boldsymbol{D_{\boldsymbol{\omega}}F}(\Lambda; \boldsymbol{\omega}^{\ast}).$ This indicates that $\boldsymbol{D_{\boldsymbol{\omega}}F}(\Lambda; \boldsymbol{\omega}^{\ast})$ is a homeomorphism operator from the space $\boldsymbol{X}$ to $\boldsymbol{X}$. Then the implicit function theorem shows that the equilibrium point $\boldsymbol{\omega} = \boldsymbol{\omega}^{\ast}$ is an isolated solution of equation $\boldsymbol{F}(\Lambda; \boldsymbol{\omega}) = 0$. In summary, according to Leray-Schauder degree theory, we present the following results (One can refer to ^[27,28]).

Lemma 3.3. Assume that the matrix $\mu_{i}\boldsymbol{I}-\boldsymbol{\Phi_{\boldsymbol{\omega}}}^{-1}(\boldsymbol{\omega}^{\ast})\boldsymbol{G_{\boldsymbol{\omega}}}(\boldsymbol{\omega}^{\ast})$ is non-singular for each $i > 0$. Then

In order to calculate $\mbox{index}(\boldsymbol{F}(\Lambda; \cdot), \boldsymbol{\omega}^{\ast})$, we will consider the sign of $H(\Lambda, \boldsymbol{\omega}^{\ast}; \mu_{i})$ in detail. Notice that $\det\{\boldsymbol{\Phi_{\boldsymbol{\omega}}}(\boldsymbol{\omega}^{\ast})^{-1}\} > 0$, so we only need to consider $\det[\mu\boldsymbol{\Phi_{\boldsymbol{\omega}}}(\boldsymbol{\omega}^{\ast})-\boldsymbol{G_{\boldsymbol{\omega}}}(\boldsymbol{\omega}^{\ast})]$. Review that

and

Then, by direct calculation, we obtain

and

where

Denote $\mu_1(\Lambda)$ and $\mu_2(\Lambda)$ be the two roots of the equation $A\mu^2+B\mu+C = 0$ with $\mbox{Re}\mu_1(\Lambda)\leq\mbox{Re}\mu_2(\Lambda)$. We notice that $C > 0$ guaranteed by $2u^{\ast}+(1-u^*)^2\Big(b+\frac{1}{v^*}\Big) > 1$, and we obtain

Moreover, we have

where

We notice that $A_1, A_2$ are always positive while $B_1, B_2$ may change the sign. In conclusion, we can obtain the following lemmas.

Lemma 3.4. Assume that $d_{i} (i = 1, 3, 4)$ are all fixed and $B_1 < 0$. Then there exists a positive constant $\hat{d}_{2}$, such that for $d_{2}\geq\hat{d}_{2}$, the two roots $\tilde{\mu}_{1}(d_2)$ and $\tilde{\mu}_{2}(d_2)$ of the equation $\psi(\mu) = 0$ are all real and satisfy

where

Lemma 3.5. Assume that $d_{i} (i = 1, 2, 3)$ are all fixed and $B_2 < 0$. Then there exists a positive constant $\hat{d}_{4}$, such that for $d_{4}\geq\hat{d}_{4}$, the two roots $\tilde{\mu}_{1}(d_4)$ and $\tilde{\mu}_{2}(d_4)$ of the equation $\psi(\mu) = 0$ are all real and satisfy

where

In the following, by using Leray-Schauder degree theory ^[29,30], we investigate the existence of non-constant positive solutions to system (1.2) with respect to the diffusion coefficients $d_i, i = 1, 2, 3, 4$. We mainly consider the cases that $d_2$ or $d_4$ is large enough and in view of Lemmas 3.4 and 3.5, we can obtain the following theorems. We only prove Theorems 3.6 and 3.7 can be finished similarly.

Theorem 3.6. Assume that $d_{i} (i = 1, 2, 3)$ are all fixed and $B_1 < 0$. For $\tilde{\mu}$ be given by the limit of Lemma 3.5, if $\tilde{\mu}\in(\mu_n, \mu_{n+1})$ for some integer $n\geq1$ and the sum $\tau_n = \sum_{i = 1}^{n}\dim \boldsymbol{E}(\mu_{i})$ is odd, then there exists a positive constant $\hat{d}_{4}$ such that system (1.2) has at least one non-constant positive solution when $d_4\geq\hat{d}_{4}.$

Proof. According to Lemma 3.4, there exists a positive constant $\hat{d}_{4}$, such that (3.4) holds and $0 = \mu_0 < \tilde{\mu}_1 < \tilde{\mu}_2, \ \tilde{\mu}_2\in(\mu_n, \mu_{n+1})$ when $d_4\geq\hat{d}_{4}$.

We shall prove the result by contradiction. Suppose on the contrary that the result is not true for some $d_4 = \tilde{d}_4\geq\hat{d}_{4}$, that is, system (1.2) does not have any positive non-constant positive solution and $\mbox{index}(\boldsymbol{F}(\Lambda; \cdot), \boldsymbol{\omega}^{\ast}) = 1$ when $d_4\geq\hat{d}_{4}.$ If we take $\Lambda^* = (\hat{d}_1, d_2, d_3, \hat{d}_4)$, then,

where, $\hat{d}_1$ is a moderately large constant provided by Theorem 2.3.

For $t\in[0, 1],$ we define a homotopy as

Thus, $\boldsymbol{\omega}$ is a non-constant positive solution of system (3.7) if and only if $\boldsymbol{\omega}$ is a non-constant positive solution of the following problem

on $\boldsymbol{X}$, where $\boldsymbol{\tilde{\Phi}} = \Big((td_1+(1-t)\hat{d}_1+td_1 d_3v)u, \big(td_2+((1-t)\hat{d}_{4}+td_4)\frac{d_2}{1+u}\big)v \Big)$. It is clear that

Theorem 3.1 shows that $\boldsymbol{\tilde{F}}(0, \Lambda; \boldsymbol{\omega}) = 0$ only has the constant positive solution $\boldsymbol{\omega}^*$ in $\boldsymbol{X}$. Through some calculation, we obtain

Furthermore, in view of Lemma 3.5, for $t = 1$, we have

Then, 0 is not an eigenvalue of the matrix $\mu_i \boldsymbol{I}-[\boldsymbol{\Phi_{\boldsymbol{\omega}}}]^{-1}\boldsymbol{G_{\boldsymbol{\omega}}}(\boldsymbol{\omega}^{\ast}),$ and $\tau_n = \sum_{i = 1}^{n}\dim \boldsymbol{E}(\mu_{i})$ is odd. Moreover, in view of Lemma 3.3, we know that

According to Theorem 2.3, we know that all the positive solutions $(u, v)$ of the system (1.2) are in $\boldsymbol{B}(C)$ for large enough constant $C$. Therefore, system (3.6) has no solution on $\partial\boldsymbol{B}$ for any $t\in[0, 1]$, and $\deg(\boldsymbol{\tilde{F}}(t, \Lambda; \cdot), 0, \boldsymbol{B})$ is well defined. According to the homotopy invariance of the Leray-Schauder degree, we obtain

Therefore, we have

From (3.9)–(3.11), we obtain a contradiction. Hence, system (1.2) has at least one non-constant positive solution and the proof is finished.

Theorem 3.7. Assume that $d_{i} (i = 1, 3, 4)$ are all fixed and $B_2 < 0$. For $\tilde{\mu}^{'}$ be given by the limit of Lemma 3.4, if $\tilde{\mu}^{'}\in(\mu_n, \mu_{n+1})$ for some integer $n\geq1$ and the sum $\tau_n = \sum_{i = 1}^{n}\dim \boldsymbol{E}(\mu_{i})$ is odd, then there exists a positive constant $\hat{d}_{2}$ such that system (1.2) has at least one non-constant positive solution when $d_2\geq\hat{d}_{2}.$

4.   Conclusions

This paper investigates the existence of positive stationary solutions of a predator-prey system with Beddington-DeAngelis functional response and fractional cross-diffusion $d_4$ subjected to the homogeneous Neumann boundary condition. The priori estimate result shows that if $d_1$ is not too large or not too small, $d_2$ is not too small and $d_3$ is not too large, then the solutions of system (1.2) are bounded and we can assert that the bound of the solution is not constrained by the cross diffusion $d_4.$ Moreover, from the proof the non-existence of non-constant positive solution, we obtain the sufficient condition for the non-existence, that is, $d_2 d_4$ is small enough. Finally, we discuss the existence of non-constant positive solution and the results indicate that the system admits a non-constant positive solution provided by the self-diffusion $d_2$ or the cross-diffusion $d_4$ is large enough, which means the diffusion $d_2$ and the fractional type cross-diffusion $d_4$ can create spatial patterns.

Furthermore, this study contributes to the growing field of fractional diffusion models in ecology. Fractional diffusion is a generalization of classical diffusion that allows for non-local interactions and has been shown to better capture the long-range effects observed in many ecological systems. The results of this paper demonstrate the potential for fractional cross-diffusion to play an important role in determining the existence and properties of positive solutions in predator-prey systems.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

We would like to give our sincere thanks to the anonymous referees and the handing editors for their valuable comments and helpful suggestions on this manuscript.

This work is supported in part by the Basic Research Program of Natural Science of Shaanxi Province (No. 2022JM-055), and also by the Science Research Program of Xi'an Eurasia University (No. 2022GCPY01).

Conflict of interest

No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication. I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part. All the authors listed have approved the manuscript that is enclosed.