Global analysis and Hopf-bifurcation in a cross-di ﬀ usion prey-predator system with fear e ﬀ ect and predator cannibalism

: We investigate a new cross-di ﬀ usive prey-predator system which considers prey refuge and fear e ﬀ ect, where predator cannibalism is also considered. The prey and predator that partially depends on the prey are followed by Holling type-II terms. We ﬁrst establish su ﬃ cient conditions for persistence of the system, the global stability of constant steady states are also investigated. Then, we investigate the Hopf bifurcation of ordinary di ﬀ erential system, and Turing instability driven by self-di ﬀ usion and cross-di ﬀ usion. We have found that the d 12 can suppress the formation of Turing instability, while the d 21 promotes the appearance of the pattern formation. In addition, we also discuss the existence and nonexistence of nonconstant positive steady state by Leray-Schauder degree theory. Finally, we provide the following discretization reaction-di ﬀ usion equations and present some numerical simulations to illustrate analytical results, which show that the establishment of prey refuge can e ﬀ ectively protect the growth of prey.


Introduction
In recent years, many scholars have established the prey-predator models to investigate the dynamic relations, multi-species systems [1][2][3][4] have been studied. All these works are conducive to the analysis of the system, and to achieve the purpose of stabilizing the ecosystem through the modeling research between ecosystems. The problem of ecological species still deserves researchers' attentions. On the other hand, different functional response items play an important role in portraying different situations. For example, Holling type-II function response is used in different systems [5,6] for scholars to investigate different systems. In nature, in order to escape from feeding behavior to avoid species extinction and be better to fit the actual environment, we always consider an effective strategy which is to establish a protection zone in the system. Many scholars are attracted by this interesting topic, and some researchers have established and investigated the predator-prey systems incorporating a prey refuge [7,8]. Thus, adding the prey refuge is significant. We consider a term 1 1+a(1−m)u which can reflect the prey refuge in this article. Applying the idea of prey refuge is a very convincing hypothesis for the prey-predator system and it also adds a new feature in the predator-prey models with prey dependence.
In a recent paper [9], the author investigated a deterministic prey-predator system only with refuge, and proposed the following system, (1.1) However, we should notice that after the establishment of prey refuges, there may be competition among prey due to food shortage and other reasons, so we need to consider the factor of intraspecies competition. On the other hand, when the predator kills the prey, the anti-predator defence behavior may be altered on account of the fear of predators. There is now a popular saying: the prey will change its own behavior and physiological characteristics due to the fear of the predator as soon as the predator appears in front of the prey. In addition, this fear effect affects the normal birth rate of the prey. To a certain extent, it exceeds the impact of direct predation and even in some cases it becomes more influential than the direct predation. Hence it is important to investigate the fear effect. Many scholars have researched the fear effect in different stochastic models or ordinary differential equation(ODE) systems [10][11][12]. Authors in [13] believed that increasing the cost of fear effect or decreasing the strength of refuge can lead to the increase of predators. We multiply the factor 1 1+kv , which aims to better study the fear effect on the population, where k is the fear factor. In [14], the authors proposed a model with prey refuge and fear effect  Moreover, another factor that can not be ignored is predator cannibalism. The predator cannibalism has a strong impact on the dynamics systems [15,16], it may affect the original stability of the system. Yet these behaviors are always omitted in the study of prey-predator systems. In [17], N.A. Schellhorn and D.A. Andow found a phenomenon that the effect of cannibalism was equal to or stronger than that of interspecific predation for both species. Authors in [18] explored a deterministic system as follows Hence, to account for the above factors, we modify the system (1.3) and propose the following ordinary differential equation (ODE) system: (1.4) where r denotes the birth rate, β describes the capture rate. b represents the ability of competition between the prey population. c is the efficiency of food conversion. d is the loss rate of predator. 1 + a(1 − m)u represents the amount of preys which is available to the predators, and m ∈ [0, 1). η is the cannibalism rate between predators. k is the fear factor. All of the coefficients in this article are positive.
From the system (1.4), we know that: (1) when k = 0, η = 0, then system (1.4) is similar to the system (1.1), (2) when η = 0, which means there is no predator cannibalism in our system, then the system (1.4) becomes system (1.2), (3) when k = 0, which means we do not consider the fear effect in our system, and the function response terms from β(1−m)uv 1+a(1−m)u to auv h+u+ηv , then the system (1.4) becomes the system (1.3). To sum up, we've improved on previous models. With the development of mathematical theory researches [19][20][21][22][23][24]. In order to describe the relationship of attraction and repulsion among species, a number of predator-prey systems with cross-diffusion was established [25,26]. Many articles have shown that these diffusion coefficients may lead to a Turing instability [27][28][29]. Hence, we consider the effects of diffusion coefficients and give the following reaction-diffusion system, Ω is a connected open region with a smooth boundary ∂Ω. ν is the outward unit normal vector. d u and d v are self-diffusion coefficients. d 12 and d 21 are cross-diffusion coefficients. The other parameters have the same meaning as above. In [30][31][32], researchers investigated Hopf bifurcations of their systems to explain the complex spatiotemporal dynamics. Thus, we need to analyse the dynamics of system (1.5). We establish sufficient conditions where Hopf bifurcation occurs, as well as the stability. Authors in [33] have well combined theoretical results with experiments, we will also give some corresponding numerical simulation results. We mainly analyze from the following aspects: (i) the existence and persistence of the nonnegative solutions, (ii) the local stability of system (1.5) at steady states, (iii) Hopf bifurcation induced by the predator cannibalism in the deterministic system (1.4) and the reaction-diffusion system (1.5), (iv) Turing instability of system (1.5).
The paper is structured as follows. The dynamics of the system, including existence of the solutions and the persistence of the system (1.5) are studied, and the proofs are given in Section 2. Sufficient conditions for the stability and Hopf bifurcation of the deterministic system are given in Section 3. Then, in Section 4, we give some conclusions for the reaction diffusion system: stability analysis, Turing instability. In Section 5, we discuss some well-posedness of the nonconstant steady state. Finally, numerical simulations and conclusion are given.
It is not difficult to find that (ū(t),v(t)) is global existence according to the ordinary differential equations' existence and uniqueness theorem. Then we have There exists T 0 > 0 such that u(x, t) ≤ū(t) according to comparison theorem, similarly, for v(x, t), we have By using the comparison theorem, there exists T 0 ∈ (0, ∞) such that u(x, t) ≤ū(t), similarly, v(x, t) gets the same conclusion. The solution (u(x, t), v(x, t)) satisfies, for all x ∈ Ω, t ≥ 0.
Theorem 2.2. The non-negative solution (u(x, t), v(x, t)) in system (1.5) is bounded [34], if 0 < m < 1 − bd (cβ−ad)r and cβ − ad > 0 hold. Proof. According to the comparison theorem of parabolic equation problems, it is clear that the first where is an arbitrary positive constant, then, .
Define z 1 (t) be a solution with respect to the equation , t ≥ T.
For any arbitrariness of > 0, one has This finishes the proof. is the global attractor for the solutions of system (1.5), and any non-negative solution (u(x, t), v(x, t)) of system (1.5) is attracted to Γ for a sufficiently large t.
bv . Theorem 2.3. Under Assumption 2.1, the system (1.5) is persistent, and one has, where, A = Proof. For the first equation in system (1.5), u(x, t) satisfies An application of comparison of parabolic equation, the first equation of (1.5) holds. Obviously, u(x, t) ≥ A for all x ∈ Ω, when there exists a T 1 > 0 and t ≥ T 1 . Then we obtain Combining Theorems 2.2 and 2.3, we obtain that system (1.5) is persistent. Then the proof is completed.

The system (1.5) without diffusion: stability and Hopf bifurcation analysis
We discuss related properties of various constant equilibrium points. There are three types constant equilibrium points: extinction point E 0 (0, 0), semi-trivia equilibrium point E 1 ( r b , 0), and the positive equilibrium point E 2 (u * , v * ).
U is an open region in R n , for any given ε > 0, and the equilibrium x * ∈ U, there exists a δ > 0 and t 0 > 0 such that x(t) − x * < ε for t ≥ t 0 . We can say the equilibrium point x * is stable. We can find a T > 0 such that when t > T , lim t→∞ x(t) − x * = 0, then the equilibrium point x * is attractive. The equilibrium point x * is said to be asymptotically stable if it is stable and attractive.

Putting (3.4) into (3.2), yields
From the Descarte's rule of sign, A < 0 always exists and the Equation (3.5) has one unique positive solution u * , if any case holds The Equation (3.5) has two positive solutions or no positive solution if The Equation (3.5) has three positive solutions or one positive solution if

The stability at E
We only consider the condition that B < 0, C > 0, D > 0. First, we investigate the stability of ODE system. The Jacobian matrix at E 2 (u * , v * ) can be expressed as where For convenience, we have utilized the following notations The characteristic polynomial can be expressed as Proof. a 11 + a 22 = a 1 u 3 * + a 2 u 2 * + a 3 u * + a 4 , using Cardanor's formula, we have a 11 + a 22 = u 3 Moreover, when Det(J) = a 11 a 22 − a 12 a 21 > 0, the characteristic polynomial has two strictly negative real parts.
For the sake of convenience, we introduce the following notations.
For the proof, we give the following assumptions.
, and A has the same meaning as before.
3.4. Hopf bifurcation of ordinary differential system at E 2 (u * , v * ) We discuss the Hopf bifurcation points at the E 2 (u * , v * ), we make a translation which is to translate Applying Taylar series expansion theorem about the (u * , v * ) = (0, 0), and the system (3.9) is rewritten as J has the same meaning as J ω and The characteristic equation of J is When p = p 0 , both of the two roots are a pair of imaginary. As we , where x expresses the real part, and y expresses the imaginary part. We have and when p = p 0 , we can obtain dx dp Next, we set a matrixD By the transformation, then, the Equation (3.10) yields, where and Using the polar coordinate form, (3.14) becomes Then, when p = p 0 , according to the Taylor expansion that We analyse the sign of the coefficient a(p 0 ), Therefore, we get

The system (1.5) with diffusion: stability analysis
Linearizing system (1.5) at (u * , v * ) which yields Define A i = −µ i D w + J. and the characteristic equation of A i at E 2 is and λ is the eigenvalue of the matrix A i .
respectively. When all eigenvalues have negative real parts, we have the conclusion that E 2 (u * , v * ) is locally asymptotically stable. In other words, when Tr(A i ) < 0, and Det(A i ) > 0, E 2 (u * , v * ) is locally asymptotically stable. Then we analyze the numerator of a 11 , let , then a 11 < 0 always holds, , and one positive Combining with the image, if A < u * < x 1 or x 2 < u * < u, then we obtain a 11 < 0. Combining with (4.7) and (4.8), it is not difficult to see, Tr(A i ) < 0 and Det(A i ) > 0 hold, the system (1.5) is locally asymptotically stable.
We get the following conclusions: , which means∆ < 0, and Det(A i ) > 0 for all i ∈ N 0 .
The proof is completed.
Proof. We first give the Liapunov function: According the maximum principle in [36], let M be a positive constant, such that where (u(x, t), v(x, t)) is the solution of (1.4). Moreover, following the Theorem in [35] that On the basis of (4.6), that which means E 2 (u * , v * ) is global asymptotically stable.

Turing instability
With the conclusions we have already drawn in Section 3.3, the ordinary differential equation system is stable ifã 11 +ã 22 < 0. As we all known, diffusion can make a stable system be unstable, we call this kind of instability as Turing instability [37]. We assumeã 11 Then there exists positive real part in the following equation under the condition (4.2), Tr(A i ) < 0 is always true. For a fixed d v > 0, It implies the same conclusion when d u > 0 is fixed. Then, Turing instability occurs.

The system (1.5) with cross-diffusion: stability analysis
In this subsection, we take the cross-diffusion coefficients into our system, we can analyze the stability and give the following equations there exists a d * 21 , when d 21 > d * 21 , then the system (1.5) becomes unstable. We obtain that d 21 can arise a region of the system, in which Turing instability may appear. The cross-diffusion coefficient d 12 is almost the complete opposite to the role of the d 21 . In the Figure 2, it is clear that as d 21 increases, the system (1.5) becomes unstable. By comparing two columns in Figure 2, we also obtain that the ability of cannibalism can affect the stability of system (1.5). The smaller the cannibalism rate, the more likely instability is to occur. modes, and that's almost the complete opposite to the role of the cross-diffusion d 21 .

Nonconstant positive steady states
We derive the properties of nonconstant positive steady state of system (1.5). A priori estimate and a standard approach based on the Leray-Schauder degree theory are used to obtain the corresponding conclusions.

A priori estimate
then the reaction-diffusion system has upper bounds.

It follows from (2.3), that
Proof. Let u m = min u(x), according to Maximum principle in [39], we have Applying the Maximum principle [39], we have the following conclusion whereL is a positive constant. On the contrary, we suppose that there is a sequence of positive solution (5.5) By the regularity theorem for the elliptic equations, we can find some and two nonnegative functionsũ,ṽ ∈ C 2 (Ω), which can make (u i (x), v i (x)) → (ũ,ṽ) in [C 2 (Ω)] 2 as i → ∞. By (5.4), we haveũ ≡ 0 orṽ ≡ 0. Integrating (5.5), we obtain (5.6) Ifṽ 0, andũ ≡ 0, then It is a contradiction to the second equation of (5.6).

Non-existence of non-constant positive steady states
We first give the steady-state problem with Proof. Define (u(x), v(x)) be any positive solution of system (5.7), denoteū = 1 |Ω| Ω u(x)dx. Let multiply the first equation of system (5.7) by u −ū, we arrive at By similar arguments as (5.8), one yields By the Poincare inequality, we obtain , we obtain u =ū and v =v, (u, v) is a constant solution has been proved.

Non-constant positive steady states
We always require Assumption 3.1 or Assumption 3.2 holds. And the definition of ω in this section is different from the previous section, d 21 Denote (I − ∆) −1 be the inverse of (I − ∆). The system (5.7) can be rewritten as and ω is the positive solution of system (5.9), when it satisfies the equation (5.10). Theorem 5.2 suggests that F(d u , d v , ω) has a unique positive solution (u * , v * ). Then In fact, it is clear that the eigenvalue λ(1 + µ i ) satisfies the following matrix where A =Θ −1 G ω (u, v). Then, Let m(µ j ) be the multiplicity of m(µ j ), we find that Clearly, On the basis of the following Lemma, in order to obtain the index of F((·), ω * ), we only need to determine the range of µ i for which S (d u , d v , µ i ) 0. From (5.11) andã 11 d u + d 12 v * ∈ (µ q , µ q+1 ), we get that when d v is large enough, then And there exists a sufficient large d 0 such that S (µ) > 0. We choosed u > d * such thatã Assume the conclusion is not true, there is no non-constant positive solution for somed ≥d v in the system (5.7) . Fixing d v =d, for t ∈ [0, 1], we define where t ∈ [0, 1], and when t = 0,D The solution ω in the system (5.9) is equivalent to finding the fixes point of Ψ(ω, 1) with t = 1. We obtain that Let define (u * , v * ) ∈ Θ for any solution of system (5.10) on Ω.

Numerical simulations and conclusions
First, we give the following discretization equations: the time increment ∆t > 0, and space increment ∆x > 0. The Discrete initial value u 0 j = ϕ 1 (x j ) = 0, v 0 j = ϕ 2 (x j ) = 0, and the bounded conditions are  In Figure 3a and b, we choose m = 0.5, the solution of system (1.5) tends to the E 2 , while m = 0.95 in Figure 3c and d, the solution tends to the E 1 . At the same time, we can obtain that the variation of m does not make spatial inhomogeneous solution appear. We change the value of η. η = 0.075 in Fig.4a and b, the system exhibits temporal periodic patterns. η = 0.2 in Figure 4c and d, the system becomes stable. η = 0.8 in Figure 4e and f, the system keep stable, and the density of predator v(x, t) is decreasing. Combining Figure 4 and Figure 1, we are not difficult to find that the cannibalism rate can influence the stability of systems (1.4) and (1.5). To sum up, too small cannibalism rate can destabilize the system, but too large cannibalism rate can make the predator go extinct. Cannibalism has certain positive effects on the stability of the system which is different from our previous cognition. Increasing k = 0.2 further to k = 0.9, we can see in Figure 5 that the stability do not change, but the density of the population is decreasing, we know large fear effect can cause the prey and predator decrease but not be extinct. we keep k = 0.1, η = 0.2, and increase m = 0.2 to m = 0.9 in Figure 6, and we observe that the density of the prey is increasing, while the predator is decreasing. When the m is large enough, the predator may go extinct. From Figure 7a to Figure 7d, we can obtain that the diffusion ratio of predator to prey plays important roles, Turing instability occurs when d v d u at a large value, it can break stability of the coexisting state. In Figure 8a, for prey, as m increases, the number of prey presents an increasing trend which means the prey refuge can protect the prey, the greater the number of prey refuge, the stronger protection the prey will accept. In Figure 8b, for predator, the number of predators present a state that the number of predator is decreasing. We can see obviously from the image that the adding a prey refuge will affect the normal predation behavior of the predator, and reduce the number of predators. It is possible that too much prey refuge could upset the ecological balance and affect the species. In Figure 9a and Figure 9b, prey refuge is beneficial to the prey, and the number of predators is reduced by the protection. In Figure 10, when η is small, the system may be unstable, and if we increase the value of m, it can promote the stability of the system. In Figure 11, we give the bifurcation diagrams for prey and predator for spatial system, it is not difficult to find that the diffusion can extend the time of unstable.         Figure 12. We investigated the effects of cannibalism behavior among predators. By the same way, we fix the other parameters, we take different values from 0.01 to 0.5, and plot the pictures which can present the changes of population density. From the image, we can observe that cannibalism behavior can increase the number of prey. The predator is increasing first and then decreasing, we can see the number of predator gets a top at 0.1 to 0.15. After that, the number of predators will continue to decline. From a practical point of view, putting in the right amount of predators can increase the number of predators and preys, thus increasing yields.

Conclusions
This is a diffusive predator-predator system considering predator cannibalism and taking the effects of predator refuge and fear effect as practical matters. To begin with, we modify the previous systems and establish a new reaction-diffusion system. We give a theoretical analysis for the existence of the positive solutions and stability of constant steady states from Theorem 2.1 to Theorem 2.3. We obtain the sufficient criteria for stability of each steady state. Considering the influence of diffusion factors, we study the Turing instability caused by diffusion, which also produces spatial inhomogeneous patterns when the diffusion coefficients satisfy the condition We discuss the Hopf bifurcation of ODE system, we investigate the Hopf point p 0 , and get the solution that when p 0 = p, the Tr(A i ) = 0, then Hopf bifurcation could occur. We also verify the results by numerical simulation. In addition, Leray-Schauder degree theory is applied to discuss the constant steady states.
The results in this paper show that: (1) If the system is based on the assumption of spatially homogeneous and we do not consider the fear effect, predator cannibalism, we can obtain the same conclusion with the system (1.1). When we only consider the prey refuge and fear effect, the fear of predator can affect the density of prey and predator, which is different from the conclusion in [14], the decrease of the predator in our system may be due to the predator cannibalism. In non-spatial systems, small predator cannibalism can cause Hopf-bifurcation while a large value of predator cannibalism can make the system be stable. However, the density of prey and predator may decrease.
(2) Both diffusion and cannibalism can disturbance the stability of the spatiotemporal system. From the numerical results that small cannibalism can induce the periodic solution, and large cannibalism can cause the decrease of predator population and prey population; large diffusion rate can make stable system be unstable, and cause the occurrence of Turing instability.
From the images obtained from the numerical simulation, we can conclude that in real life, the purpose of protecting the prey can be achieved by appropriately establishing a prey refuge. For the stability of the ecosystem, we should not build prey refuges indefinitely which may lead to the extinction of predators. Putting a proper amount of predators and increasing the competition between predators are beneficial to the growth of prey and predators. However, if the amount exceeds a certain amount, it will cause the reduction of predators and increase of prey, which may eventually lead to the extinction of predators.
In traditional papers, some scholars considered prey-predator systems without diffusion-reaction, and others considered the systems without predator cannibalism or prey refuge. We improve the previous systems and take more factors into account, we can obtain from the conclusion that diffusion factors, prey refuge and cannibalism do have a huge impact on a prey-predator system. In the models [42,43], the authors have considered the effect of delay for their system, which made the system more realistic. In [44], the authors explored the stochastic partial differential prey-predator system, and in [45], Hopf bifurcation has been studied. In the future research, we hope to consider a time delay in the diffusion. We will investigate more interesting and actual systems.