Spatiotemporal dynamics of a diffusive predator–prey model with fear effect

. This paper concerned with a diffusive predator–prey model with fear effect. First, some basic dynamics of system is analyzed. Then based on stability analysis, we derive some conditions for stability and bifurcation of constant steady state. Furthermore, we derive some results on the existence and nonexistence of nonconstant steady states of this model by considering the effect of diffusion. Finally, we present some numerical simulations to verify our theoretical results. By mathematical and numerical analyses, we ﬁnd that the fear can prevent the occurrence of limit cycle oscillation and increase the stability of the system, and the diffusion can also induce the chaos in the system.


Introduction
Since Lotka [11] and Volterra [17] proposed famous Lotka-Volterra equations, the construction and study of models for the population dynamics of predator-prey interactions has been an important topic in theoretical ecology. According to different background, researchers have proposed many types of predator-prey models, and rich dynamics of these systems have been investigated extensively [6,8,18,21]. In the wild, it is easy to observe that the reduction of prey is due to the direct killing of predators, which is reflected by functional responses in the predator-prey model such as Holling type and Beddington-DeAngelis [1,7,9,10,16,24].
However, a new study suggested that the behavior of the prey can be changed by the predator, and it could have a greater impact than direct killing. In fact, Zanette et al. [22] found that the offspring production of the song sparrows reduced by 40% because of the fear from predator. To model the fear effect in predator-prey interactions, Wang et al. [19] proposed a predator-prey model as follows: where r 0 is the birth rate of the prey, d is the natural death rate of the prey, a represents the death rate due to intraspecies competition. The parameter k refers to the level of fear, which reflects the reduction of prey growth rate due to the antipredator behavior. With the increase of k, the growth rate of prey decreases. In [19], the authors consider that the functional response g(u) is the linear (g(u) = pu) or the Holling type II (g(u) = p/(1 + qu)). Their theoretical results show that fear effect could improve the stability of the predator-prey system.
It is considered that the trait effect has reduced the growth rate of the prey due to fear of predators, and the prey has been subjected to a strong Allee effect caused by mating during reproduction. Inspired by this idea, [14] considered a predator-prey model with the trait effect that reduced the growth rate of the prey due to fear of predators, and the prey has been subjected to a strong Allee effect caused by mating during reproduction. Their results showed that the fear effect does not affect the stability of the equilibria, but with the increasing of the cost of fear, the equilibrium density of predator decreases. Sasmal and Takeuchi [15] studied a predator-prey model that incorporates fear effect due to the presence of predator and group defense. Wang et al. [23] investigated a predatorprey model incorporating the fear of predators and a prey refuge, and they found that the fear effect can not only reduce the population density of predator, but also stabilize the system by excluding the existence of periodic solutions. Here, we remark that all models in these papers did not consider the factor of diffusion.
It should be pointed out that in real life, species are heterogeneous in space, so individuals tend to migrate to areas with low population density, which will increase the possibility of survival. Hence, some researchers considered reaction-diffusion predatorprey model by incorporating the fear effect into prey population. Niu et al. [4] investigated a diffusive predator-prey model with the fear effect. Taking the mature delay as bifurcation parameter, they found that the delay can induce Hopf and Hopf-Hopf bifurcations. Wang and Zou [20] proposed and analyzed a reaction-diffusion-advection predator-prey model. [3] investigated a diffusive predator-prey model with fear effect. Their results show that for the Holling type II predator functional response case, there exist no nonconstant positive steady states for large conversion rate.
Motivated by these pioneer work and note that none of the above mentioned models considered the Holling III functional response, we are led to consider a diffusive predatorhttps://www.journals.vu.lt/nonlinear-analysis prey model as follows: where u(x, t), v(x, t) denote the density of the prey and the predator at location x and time t, respectively. r is the birth rate of prey, γ 1 is the natural death rate of prey, a represents the death rate due to intraspecies competition. The parameter k reflects the level of fear, which drives antipredator behaviours of the prey. m 1 u 2 /(b + u 2 ) is Holling type-III function (see [5]). The parameter γ 2 is the death rate of predator. Ω ⊆ R N is a bounded region with smooth boundary ∂Ω, and n denotes the outward normal vector to the boundary ∂Ω. The homogeneous Neumann boundary condition indicates that there is no population flow across the boundary. We also assume that , and X is defined by In this paper, our goal is to investigate the dynamical properties of (1) such as global existence of the solutions, stability and bifurcation of the constant steady state. In addition, we will use energy estimates to obtain of the dynamic and steady state solutions and so to discuss the nonexistence and existence of spatial patterns.
Our paper is organized as follows. In Section 2, we study some basic dynamics of the system. In Section 3, we obtain the stability and bifurcation of the equilibria. In Section 4, we investigate the nonexistence and existence of the nonconstant steady state. In Section 5, numerical results are presented to verify the theoretical results.

Basic dynamics
In this section, we discuss some basic dynamics of system (1) including the existence of solution and the priori bound of the solution.
https://www.journals.vu.lt/nonlinear-analysis (iii) It is noted that Thus, by the comparison principle, one have The maximum principle ensures that u(·, t) C(Ω) K 1 for all t 0. Let Multiplying both sides of Eq. (3) by m 2 /m 1 , then combining with Eq. (4), we obtain Noting that u(·, t) C(Ω) K 1 proved above, we have U (t) K 1 |Ω|. Thus where

Stability
Recall that 0 = µ 0 < µ 1 < µ 2 < · · · < µ n < · · · → ∞ are the eigenvalues of the Laplace operator −∆ on Ω under homogeneous Neumann boundary condition, and S(µ n ) is the space of eigenfunctions corresponding to Assume that (u, v) is a constant solution of system (1), then we have and For each i = 0, 1, . . . , X i is invariant under the operator L, and λ is an eigenvalue of L on X i if and only if λ is an eigenvalue of −µ n D+J(u, v) for all n ∈ {0, 1, 2, . . . } := N 0 .
The direct calculation shows where Proof. (i) For E 0 = (0, 0), the corresponding characteristic equation is Clearly, we obtain Hence, if r < γ 1 , then E 0 is locally asymptotically stable. Note that there is no other constant steady states in this case. This means that E 0 is indeed globally asymptotically stable.
Remark 1. Theorems 3 and 4 show that when r ∈ (0, γ 1 ], system has only trivial constant solution E 0 = (0, 0), and it is globally asymptotically stable; when r increases and enter the interval (γ 1 , γ 1 +a γ 2 b/(m 2 − γ 2 )), E 0 loses its stability to a predator-free constant steady state E 1 ; and when r further passes γ 1 + a γ 2 b/(m 2 − γ 2 ), E 1 loses its stability to a positive steady state E * . We can conclude that as the parameter r increases, the model experiences two bifurcations of constant steady state.

Remark 2.
Obviously, the conditions of Theorem 4 are independent of the diffusion. Consequently, the conclusions of Theorem 4 are still valid for the corresponding ODE model. In addition, we can also conclude that the diffusion cannot destabilize the positive steady state E * . Therefore, the PDE system (1) cannot occur Turing instability/bifurcation.

Hopf bifurcation
In this subsection, we will discuss the bifurcation of system (1). Let the parameters r, k, a, b, γ 1 , γ 2 , m 1 , m 2 and d 1 be fixed, and take d 2 > 0 as a bifurcation parameter.

Nonconstant steady states
In this section, we will discuss nonexistence and existence of nonconstant steady state of system (1). To this end, we consider the following elliptic system:

A priori estimates
To derive some priori estimates for nonnegative solutions of system (12), we need the following technical lemma [12].

Lemma 1 [Maximum principle].
Suppose that Ω is a bounded domain in R n and g ∈ C(Ω) × R. If z ∈ H 1 (Ω) is a weak solution of the inequalities and if there is a constant K such that g(x, z) < 0 for z > K, then z K a.e. in Ω.
Further, from Lemma 1 we obtain that u(x) (r − γ 1 )/a := M 1 , and by the strong maximum principle, we have u(x) < M 1 for all x ∈ Ω. Then It can be obtained from the maximum principle that Theorem 7. Let d * be a fixed positive constant. Then for d 1 , d 2 d * , there exists two positive constants C and C with C < C depending possibly on ∧ such that any solutions (u(x), v(x)) of system (12) satisfies C u(x), v(x) C.
Next, we shall prove u(x), v(x) C. Let Thus, Lemma 2 shows that there exists a positive constant C 2 such that Hence, now it remains to prove that there exists C 3 > 0 such that https://www.journals.vu.lt/nonlinear-analysis Contrariwise, let us assume that (14) does not hold. Then there exists a sequence (u n (x), v n (x)) such that max Ω u n → 0 or max Ω v n → 0 as n → +∞. (15) By the regularity theory for elliptic equations, there exists a subsequence of {(u n , v n )}, which will be denoted again by Note that u 0 (r − γ 1 )/a and from (15) either u 0 ≡ 0 or v 0 ≡ 0. Therefore, we have that Also, {(u n , v n )} satisfy (13), so do u and v. Letting n → ∞, we get that {(u n , v n )} is a positive solution of (12). Therefore, by integrating Eq. (12) for u n and v n over Ω, we have (i) In this case, u 0 ≡ 0, then and v n > 0, then for sufficiently large n. So, we obtain a contradiction.
(ii) If u 0 ≡ 0, v 0 ≡ 0, using the first equation of (12). So, u 0 ≡ (r −γ 1 )/a for large n. Thus for a sufficiently large n, which is a contradiction. This completes the proof.

Nonexistence of nonconstant positive steady states
In this subsection, we show the nonexistence of positive steady state solutions when the diffusion coefficients d 1 and d 2 are large.
Proof. Assume that (u(x), v(x)) is nonnegative solution of (12). Denotē Obviously, Ω (u −ū) dx = 0 and Ω (v −v) dx = 0. For the purpose of discussions, let H(u, v) = u 2 v/(b + u 2 ). By the mean value theorem of bivariate functions, we have Obviously, Multiplying both sides of the first equation of (12) by u −ū and using Theorem 6, we get Applying Theorem 6 and by multiplying v −v to the second equation in (12) and then integrating on Ω, we have Using the Poincaré inequality, where µ 1 is the second eigenvalue of the Laplace operator −∆ on Ω under homogeneous Neumann boundary condition. Combining (16) and (17) leads to This implies that then we can conclude that
Consider the following problem: It is easy to see that solving (12) is equivalent to find a fixed point of A t (w) with t = 1. w * is the unique constant solution of (19) for any t ∈ [0, 1]. By the definition of d * in Theorem 8, one have that E * is the only fixed point of A 0 .
Since F = I − H(·, 1) and if (12) has no nonconstant positive solution, then we have In addition, by the homotopy invariance of the topological degree, which is a contradiction.

Numerical results and discussions
In this section, we take some numerical simulations to discuss the effect of diffusion and the cost of fear.
We further find that different initial conditions with the same diffusion rate d 2 = 1.2 can lead to different spatial patterns that can be stationary or periodic (Fig. 5).  . According to Theorem 4, we observe that the positive steady state E * of system (1) is locally asymptotically stable, and the dynamic behaviors of system (1) is illustrated graphically in Fig. 6. From above discussions we can obtain that fear can affect the stability of the positive steady state, and it can induce the Hopf bifurcation, which is different from the results found in [14,19] with linear functional response (see Fig. 9). Figure 9 shows that there exists a threshold value k 0 such that when k ∈ (0, k 0 ], system (1) has a periodic solution. But when k passes the threshold value, then system becomes stable.    If we choose k = 20.37, while other parameters do not change, according to Theorem 5, system (1) undergoes spatial homogeneous Hopf bifurcation (see Fig. 7). It is shown that system (1) has spatially homogeneous periodic solutions emerged from the positive steady state E * .
In addition, we find that the positive steady state can be changed by the different value of the cost of fear. Figure 8 shows that the positive steady state v * decreases with increasing of the cost of fear.

Conclusion
A diffusive predator-prey model with the fear effect is studied in our paper. We derive some basic dynamics of the system and give condition for the existence of the positive steady state. According to eigenvalue analysis method, we investigate the stability and bifurcation of the positive constant steady state. We also give some conditions for the nonexistence and existence of nonconstant solutions of the system. Theorems 3 and 4 show that the birth rate of prey r can not only induce the static bifurcation, but also can induce saddle-node bifurcation.
Theorem 4 indicates that the diffusion can not induce the Turing instability/bifurcation. However, Theorem 5 provides that the diffusion can induce the inhomogeneous Hopf bifurcation, which can lead to the formation of spatial patterns. Furthermore, Theorem 9 shows that system (12) has at least one nonconstant positive solution under the effect of the diffusion. From Section 5.1 we can obtain that the different diffusion rate d 2 can