Steady states of a diffusive predator-prey model with prey-taxis and fear effect

In this paper, a diffusive predator-prey system with a prey-taxis response subject to Neumann boundary conditions is considered. The stability, the Hopf bifurcation, the existence of nonconstant steady states, and the stability of the bifurcation solutions of the system are analyzed. It is proved that a high level of prey-taxis can stabilize the system, the stability of the positive equilibrium is changed when χ crosses χ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\chi _{0}$\end{document}, and the Hopf bifurcation occurs for the small s. The system admits nonconstant positive solutions around (u¯,v¯,χi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\bar{u}, \bar{v}, \chi _{i} )$\end{document}, the stability of bifurcating solutions are controlled by ∫ΩΦi3dx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\int _{\Omega} \Phi _{i}^{3} \,\mathrm{d}x$\end{document} and ∫ΩΦi4dx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\int _{\Omega} \Phi _{i}^{4} \,\mathrm{d}x$\end{document}. Finally, numerical simulation results are carried out to verify the theoretical findings.


Introduction
In biomathematics, the interactions among species and the spatial distributions of populations have always been the hot topics of ecosystems, and are important for developing research on economic benefits, pest control, and environmental governance. The predatorprey system has been deeply studied in [1][2][3][4][5][6].
In ecosystems, the prey populations are often affected by the indirect effects of predators, in addition to the direct killing, which is called the fear effect. It shows that the effect of the fear effect on the birth rate of prey is even more significant than the direct effect. In 2016, Wang [7] first introduced the fear effect into a predator-prey model that showed that the fear effect does not work on the system with a linear functional response function. However, for the model with a Holling-II functional response function, a high level of fear makes the system stable, while a low level of fear makes the system oscillate periodically. Wang and Zou [8] proposed a reaction-diffusion convective predator-prey model, which incorporated the fear effect into the local response term of the model. The authors of [9][10][11][12][13][14] studied the effect of the fear effect on the models from different aspects.
Predators prefer the areas with high prey density, in addition to the free diffusion, which is called prey-taxis. According to the positions of the stimuli, motions can be divided into positive taxis and negative taxis. The experiment was first proposed in 1970 by Keller and Segel in [15], and the results have shown that the solutions may blow up in finite time. For predator-prey systems, Kareiva and Odell [16] observed the prey-taxis model in 1987, and described that predators will gather near higher densities of prey to more effectively find prey. The model showed that prey-taxis increased the stability of the predator-prey system. Lee investigated prey-taxis for ρ(u, v) = χ is a constant, the paper showed that preytaxis stabilizes the system and for large prey-taxis sensitivity, no pattern formation was observed in [17]. Wang discussed the effects of prey-taxis on the existence and stability of nonconstant steady states, gave the bifurcation type and stability of the bifurcation curve, and provided a stable wave model selection mechanism for the reaction-diffusion system with prey-taxis in [18] and [19]. Wu considered a more general predator-prey model with prey-taxis in [20], obtained the global existence and boundedness of the equilibria. Moreover, the prey can also adjust the corresponding position to reduce the risk of being caught, which is called predator-taxis [21]. Also, for the systems with three species [22][23][24], the authors investigated the global existence of the system solutions and the local stability conditions of the equilibria. More details about chemotaxis can be seen in [25][26][27][28][29][30][31].
Inspired by the above works, considering that the predator turns to another species when the favorite food is lacking, we introduce the modified Leslie-Gower term to make the model more realistic and established the following system, where is a bounded domain in R N , N ≥ 1, with a smooth boundary ∂ ; ∂ν is the outer flux; and ∇ are Laplace and gradient operators defined in . u, v represent the density of the prey and predator; d 1 and d 2 are self-diffusion coefficients; r 0 and d are the birth and death rate of prey; a is the fear factor; -χ∇(v∇u) stands for the prey-taxis term, χ is called the prey-taxis coefficient, prey-taxis is repulsive for χ < 0 and attractive for χ > 0; s is the intrinsic growth rate of predators, puv u+kv is the functional response; qv u+m is the modified Leslie-Gower term, all the parameters are positive. Furthermore, we discuss the effect of constant efficiency of predation p and the growth rate of predator species s, in addition to the prey-taxis in other articles. We mainly proved the existence of the steady states of the system, the dynamic equilibrium of biological populations helps to prevent species extinction, provides measures to control the population quantity, and forecasts the change of the population quantity, gives theoretical guidance for reasonable control of the population quantity.
The ODE corresponding to the system (1) can be expressed as, We can see that system (2) has two types of equilibria: (1) the prey-free equilibrium e 01 (0, m q ); (2) the predator-free equilibrium e 10 ( r 0 -d c , 0), if r 0 > d; (3) the unique positive equilibrium e 2 (u, v) with v = u+m q , and u is the positive root of the equation kq . The remainder of this paper is structured as follows. In Sect. 2, the stability of the positive equilibrium with and without the prey-taxis χ is verified, and the conditions for the Hopf bifurcation of the system (3) are given. In Sect. 3, the existence of the nonconstant steady states is proved, and the stability of the nonconstant solution is analyzed. Finally, numerical simulations are used to substantiate the theoretical findings in Sect. 4, with some conclusions being drawn in Sect. 5.
Proof The real part of λ is negative if and only if T < 0 and D > 0. First, we discuss the sign of a 11 , When ck 2 m ≥ pq, a 11 < 0 for allū > 0, then T = -(d 1 + d 2 )λ is + a 11 < 0. D is a linear function of χ , denoting we rewrite D as where D > 0 is equivalent as χ i < χ for B(ū,v) < 0 andv > 0. Thus, the positive equilibrium (ū,v) of model (1) is locally asymptotically stable for ck 2 m ≥ pq and χ > χ i according to standard linearized stability analysis. Moreover, T = -sa 11 < 0, D = -(sa 11 + s q a 12 ) > 0 for i = 0, the positive equilibrium (ū,v) is always stable.
Next, we let 0 < ck 2 m < pq in order to further analyze the instability of the system. We give a Lemma about the a 11 first.
Proof View a 11 as a function ofū, Now, a 11 (0) = 0, and there exists one and only one positive solution u 2 > 0 such that a 11 (u 2 ) = 0 for ck 2 m < pq.
From [32], the Hopf bifurcation value u should satisfy the conditions: there exists n ∈ N, such that and for the complex eigenvalue α(u) ± iω(u), Then, any potential Hopf bifurcation point u of model (1) must lie in the interval (0, u 1 ) ∪ (u 1 , u 2 ), and satisfy the condition (5).
Summarizing the above discussion, we have the following theorem.
Remark 2.4 Compared with the system without the prey-taxis, we find that when the Hopf bifurcation occurs in the system, the number and value of the bifurcation points in the case of spatial homogeneous and spatial heterogeneity are no different. When χ > χ 0 = max i∈N + χ i , the value of χ does not affect the number and value of the bifurcation points of the system (1).

Steady states 3.1 The existence of nonconstant steady-state solutions
We know that the stability of (ū,v) is changed when χ passed max i∈N + χ i . Therefore, we want to further analyze the existence of nonconstant solutions of the following equation: Treat χ as the bifurcation parameter to investigate the effect of prey-taxis, and introduce Rewrite A(ū,v) = a 11 , B(ū,v) = a 12 , and (6) as For any fixed (û,v) ∈ X × X , the Fréchet derivative of F is given by in particular, for (û,v) = (ū,v), In order to apply the bifurcation theory from Crandall and Rabinowitz [33], we need to verify the following four conditions of the operator F : The first two conditions have already been proved. Next, we examine that we suppose that there exists (u, v) that is the solution of the following system where T i and S i are constants, i is the eigenfunction corresponding to λ i , λ i is the ith simple eigenvalue of (1) and i is normalized with 2 i dx = 1, i ∈ N + . Substituting (9) into (7), we obtain The case of T 0 and S 0 = 0 is ruled out since λ 0 = 0 for i = 0. For i ∈ N + , we obtain condition (8) holds at χ = χ i through direct calculation which is the same as the (4). Moreover, we can see that dim N (D (u,v) F(ū,v, χ)) = 1 and Also, D (u,v) F(û,v, χ) is a Fredholm operator with index 0 according to Corollary 2.11 in [34]. Finally, we prove the transversality condition where R denotes the range and We argue by contradiction and suppose that there exists a nontrivial pair ( u, v) such that (12) fails, thus the ( u, v) satisfies multiplying both sides of (13) by i and integrating it over , since 2 i dx = 1, This is a contradiction in light of (10) and the transversality condition is verified. Moreover, to ensure χ i = χ j we require d 1 d 2 qλ i λ j = -sqA(ū,v) + sB(ū,v), for i = j, i, j ∈ N + . All the above four conditions are satisfied then, according to Shi [34], we have the following result. where and (ū i ,v i ) is given in (11).

The stability of bifurcation solutions
Since F is C 4 -smooth and (u i , v i , χ i ) are C 3 -smooth functions of h, in order to further study the stability of the nonconstant solutions, we expand (14) as follows: where Q i is given by (11) and (ϕ(x), ψ(x)) ∈ Z. It is easy to obtain through direct calculation that Moreover, from the Taylor's expansion, we have and From Wang [35], the stability of (u i (h, x), v i (h, x), ) is determined by the sign of K 1 (K 1 = 0). For K 1 = 0, we need to further discuss the sign of K 2 . Next, we discuss K 1 first. Theorem 3.2 Suppose all conditions in Theorem 3.1 hold. Then, for each i ∈ N + , the sign of K 1 is given by (21).
Next, we continue to analyze the stability of bifurcating solutions. (25) is satisfied, we have that the sign of K 2 is decided by (27).

multiplying (26) by i and integrating it over ,
From (25), we have ϕ 1 (x) i dx = 0. Then, we need to calculate
Remark 3.5 By Theorem 3.2 of Wang [35], we know that sgn(λ) = sgn(K 2 ) for h ∈ (-δ, δ), we know that 0 is a simple eigenvalue of D (u,v) F(ū,v, χ i ) with an eigenspace span{(1, Q i ) cos iπ x L }. For each i = i 0 , (37) has the eigenvalue with a positive real part when h = 0, then from the standard eigenvalue perturbation theory in [36], it also holds for small h.

Conclusion
In this paper, we consider a diffusive predator-prey system with prey-taxis. We show that the positive equilibrium (ū,v) is stable for χ > max χ i and ck 2 m ≥ pq, the stability changes for χ < max χ i or ck 2 m < pq. For χ > max χ i , ck 2 m < pq, the stability is determined by s, the system undergoes a spatially homogeneous Hopf bifurcation atū = u 0,± and a spatially nonhomogeneous Hopf bifurcation atū = u j,± for small s. When χ > passes χ 0 , the system loses stability, also we examined the existence of nonconstant steady states by the bifurcation theory from Crandall and Rabinowitz. The stability of the solution (ū,v, χ i ) is investigated, the sharpnesses of the bifurcation branches are determined by the value of