Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition

A diffusive predator-prey model with nonlocal prey competition and the homogeneous Neumann boundary conditions is considered, to explore the effects of nonlocal reaction term. Firstly, conditions of the occurrence of Hopf, Turing, Turing-Turing and double zero bifurcations, are established. Then, several concise formulas of computing normal form at a double zero singularity for partial functional differential equations, are provided. Next, via analyzing normal form derived by utilizing these formulas, we find that diffusive predator-prey system admits interesting spatiotemporal dynamics near the double zero singularity, like tristable phenomenon that a stable spatially inhomogeneous periodic solution with the shape of \begin{document}$ \cos\omega_0 t\cos\frac{x}{l}- $\end{document} like which is unstable in model without nonlocal competition and also greatly different from these with the shape of \begin{document}$ \cos\omega_0 t+\cos\frac{x}{l}- $\end{document} like resulting from Turing-Hopf bifurcation, coexists with a pair of spatially inhomogeneous steady states with the shape of \begin{document}$ \cos\frac{x}{l}- $\end{document} like. At last, numerical simulations are shown to support theory analysis. These investigations indicate that nonlocal reaction term could stabilize spatially inhomogeneous periodic solutions with the shape of \begin{document}$ \cos\omega_0 t\cos\frac{kx}{l}- $\end{document} like for reaction-diffusion systems subject to the homogeneous Neumann boundary conditions.

1. Introduction. Recently, much attention is focused on nonlocal competition of a species for resources (see [1,4,9,15,19,20,39], for example), given that there is no real justification for assuming that the interaction between individuals of a species is local, because individuals may compete for common resource or communicate either visually or by chemical means [14]. And, Britton [6] firstly proposed the following single population model involving nonlocal competition effect, ∂u(x, t) ∂t = d∆u + λu 1 + ϑu − β investigated steady state bifurcation of diffusive system under Neumann boundary conditions. Subsequently, by introducing nonlocal prey competition into the classical diffusive Holling-Tanner predator-prey model and Rosenzweig-MacArthur predator-prey model, Merchant and Nagata [32] proposed the following models and Also, for Ω = (−∞, +∞), it was found that nonlocal competition could induce complex spatiotemporal patterns, by destabilizing spatially inhomogeneous steady states. Lately, by selecting Ω = (0, lπ) and K(x, y) = 1 lπ , Chen and Yu [10] investigated Hopf bifurcation of diffusive system (3) subject to Neumann boundary conditions, to show the existence of complex patterns. It was found that, when the given parameter passes through Hopf bifurcation values, the positive constant steady state loses stability and a spatially inhomogeneous periodic solution emerges, which is greatly different from the local competition case that spatially inhomogeneous periodic solutions arising from Hopf bifurcation are usually unstable, see [3,7,17,35,38,41].
Analogously, spatially inhomogeneous periodic solutions resulting from Hopf bifurcation for system (2) with K(x, y) = δ(x − y) are also unstable. Actually, system (2) is reduced to the classical diffusive Holling-Tanner predator-prey model while K(x, y) = δ(x − y). Investigations on Hopf bifurcation, Turing instability and Turing-Hopf bifurcation for the classical diffusive Holling-Tanner predator-prey model have also been conducted, see [2,11,28,30]. These investigations indicate that stable periodic solutions of the classical diffusive Holling-Tanner predator-prey model subject to Neumann boundary conditions, either are spatially homogeneous or are spatially inhomogeneous and have the shape of cos ω 0 t + cos kx l -like. Then, to explore the effects of nonlocal reaction term, we consider spatiotemporal dynamics of system (2). By selecting Ω = (0, lπ), K(x, y) = 1 lπ and Neumann boundary conditions, system (2) is simplified as where u(x, t) and v(x, t) stand for densities of prey and predator at time t and location x respectively, d 1 and d 2 are diffusion rates of prey and predator respectively, a and c are intrinsic growth rates of prey and predator respectively, K represents carrying capacity of prey, l is the spatial scale, b and e measure the interaction strength between predator and prey, and m measures the prey's ability to evade attack. See [34] for more detailed biological explanations. And, we find that under proper conditions, diffusive system (4) undergoes Hopf bifurcation, Turing bifurcation, Turing-Turing bifurcation, as well as double zero bifurcation, of which the linear matrix of the corresponding normal form reads 0 1 0 0 . In particular, we have great interest in double zero bifurcation, of which Bogdanov-takens bifurcation is the most common type (see [5,13,21,22,25,26,27,31,33,37,43], for example). Actually, double zero bifurcation could help to reveal some complex spatiotemporal patterns. However, we find that spatially inhomogeneous periodic solutions arising from double zero bifurcation for reaction-diffusion systems with local reaction term and Neumann boundary conditions are unstable [29,36,42]. Then, we want to find out the effects of nonlocal reaction term on dynamics near double zero singularity of reaction-diffusion systems.
In this paper, we investigate spatiotemporal dynamics near the double zero singularity of diffusive predator-prey system (4), to reveal the effects of nonlocal reaction term. It is found that under proper conditions, system admits a large stable spatially inhomogeneous periodic solution with the shape of cos ω 0 t cos x l , which could coexist with a pair of stable spatially inhomogeneous steady states with the shape of cos x l . Therefore, nonlocal reaction term could stabilize spatially inhomogeneous periodic solutions with the shape of cos ω 0 t cos x l −like which are usually unstable for reaction-diffusion systems with local reaction term and Neumann boundary conditions. Furthermore, the spatially inhomogeneous periodic solution near the double zero singularity, is also different from these spatially inhomogeneous periodic solutions with the shape of cos ω 0 t + cos x l [2,7,8,24,38]. And, by linear stability analysis, we also establish conditions of the occurrence of Turing bifurcation, Hopf bifurcation, Turing-Turing bifurcation and double zero bifurcation, from a geometric point of view. In order to further discuss spatiotemporal dynamics of diffusive system (4) near the double zero singularity utilizing normal form theory, we provide some concise formulas to compute the third-order normal form, following the method in [12,23]. Moreover, it's worth noting that these concise formulas apply to the computation of normal form at a double zero singularity for partial functional differential equations, partial differential equations and functional differential equations. And, the procedure of computing normal forms utilizing these concise formulas, could be implemented by computer programs. This paper is organized as follows. In Section 2, conditions for the occurrence of Turing bifurcation, Hopf bifurcation, Turing-Turing bifurcation and double zero bifurcation, are established, by linear stability analysis. And in Section 3, several concise formulas for computing normal forms at a double zero singularity of partial functional differential equations are derived, following normal form method in [12,23]. Then by analyzing the third-order normal form derived by using these concise formulas, spatiotemporal dynamics near the double zero singularity of diffusive predator-prey system (4) are explored, in Section 4. At last, conclusions and discussions are in Section 5.
and dropping the tildes, diffusive predator-prey system (4) becomes where parameters d 1 , d 2 , β, l, b, c are all positive.
Obviously, system (5) admits a unique positive constant equilibrium denoted by Then, the linearized system of (5) at E * is Therefore, the sequence of characteristic equations of system (6), reads where N denotes the set of positive integers, and and for k ∈ N, After some analysis, we have conclusions about the occurrence of Hopf bifurcation and Turing bifurcation for system (5).
Proof. We present the argument from a geometric point of view. That is, regard conditions of the occurrence of Hopf bifurcation and Turing bifurcation as bifurcation curves in d 1 -c parameter plane.
By further analyzing characteristic equations (7), we could also derive conditions of the occurrence of other bifurcations, like Turing-Turing bifurcation [7,24,40] and double zero bifurcation. Next, we discuss conditions of the occurrence of these bifurcations respectively, from the geometric point of view.
Then, in d 1 -c plane, define Hopf bifurcation curves, Turing bifurcation curves and piecewise smooth curve, And, T is called the first Turing bifurcation curve [7,24], because diffusive system (5) undergoes Turing bifurcation for the first time, when (d 1 , c) crosses T from top to bottom. That is, Proof. According to Theorem 2.2, Turing bifurcation curves T i and T j intersect and some calculations yield, That is, when (d 1 , c) goes from top to bottom in d 1 -c plane, it will meet with T firstly. The conclusion follows.
Moreover, we have the following conclusions about k−mode double zero bifurcation, that is, k−mode bifurcation from a double zero eigenvalue, where k ∈ N 0 . Here, we regard a k−mode double zero bifurcation point as the intersection of k−mode steady state bifurcation curve and k−mode Hopf bifurcation curve. And when k ∈ N, k−mode steady state bifurcation curve is actually the k−mode Turing bifurcation curve. Theorem 2.4. If ξ < β and l ∈ l * 1 , ∞ , 1−mode double zero bifurcation occurs for system (5) when (d 1 , c) = d * 1 ,c * 1 , satisfying that characteristic equations (7) have a double zero root with the remaining roots having negative real parts, wherẽ Given that T is piecewise smooth continuous curve, we apply Intermediate Value Theorem proving the theorem. Define function δ Otherwise, assume that there exist at least two pointd j ∈ d * 1,2 ,d 1 , j = 1, 2, 3, · · · satisfying δ d j = 0. Then, there exist at least two pointd 1 > 0. Thus, by solvingc 1 (d 1 ) =c 1 (d 1 ), we derivedd * 1 andc * 1 . Then, 1−mode double zero bifurcation occurs. Also, we havec 1 (d 1 ) >c k (d 1 ) , k ≥ 2 on 0,d 1 , which means that system (5) undergoes Hopf bifurcation for the first time when (d 1 , c) goes from top to bottom in d 1 -c plane. Therefore, except a double zero root, the remaining roots of characteristic equations (7) have negative real parts. Then, the conclusion follows.
Analogously, we also have Theorem 2.5. If ξ > β, l > d2 λβ and l ∈ l * 1 ,l * , 1−mode double zero bifurcation occurs for system (5) when (d 1 , c) = d * 1 ,c * 1 , satisfying that characteristic equations (7) have a double zero root with the remaining roots having negative real parts, wherel * The proof is similar to proof of Theorem 2.4, and we omit it here.
Remark 1. For the classical diffusive Holling-Tanner predator-prey model, it is easy to verify that double zero bifurcation doesn't occur if ξ < β. However, by Theorem 2.4, double zero bifurcation could occur for model (5) with nonlocal prey competition when ξ < β and l ∈ l * 1 , ∞ . Therefore, nonlocal reaction term makes it easier for double zero bifurcation to occur. Moreover, following similar discussions in [41], one can easily find that spatially inhomogeneous solutions near the double zero singularity of model (8) are unstable. According to Theorem 2.4 and Theorem 2.5, model (5) with nonlocal prey competition could generate stable spatially inhomogeneous solutions (see Figure 4) near the double zero singularity under proper conditions. Then, we conclude that nonlocal reaction term could also stabilize unstable spatially inhomogeneous solutions near a double zero singularity of the classical diffusive Holling-Tanner predator-prey model. 3. The third-order normal form at a double zero singularity. Assume that Ω ⊂ R n (n ∈ N) is a bounded open set with smooth boundary, and X is Hilbert space of functions defined onΩ with inner product ·, · . Let {β k } k∈N * be eigenfunctions of −∆ on Ω, with the corresponding eigenvalues {µ k } k∈N * satisfying µ k ≥ 0 and µ k → +∞ as k → ∞, where N * N 0 for the homogeneous Neumann boundary conditions, N for the homogeneous Dirichlet boundary conditions.
Then, {β k } k∈N * form an orthonormal basis of X.
be the Banach space of continuous maps from [−r, 0] to X m with the sup norm. Then in phase space C, consider the following parameterized abstract partial functional differential equations (PFDEs), Then, characteristic equation of the linearized system of (9) at the origin is equivalent to the sequence of "characteristic" equations where Then, the adjoint bilinear form (·, ·) k on C Moreover, we have the hypothesis about a double zero bifurcation point, (H1) There exists a neighborhood (0, 0) ∈ V 0 ⊂ R 2 such that for α = (α 1 , α 2 ) ∈ V 0 , characteristic equation of the linearized system of (9) at the origin has a double real root γ(α) associated with k 1 ∈ N * , satisfying that γ(0) = 0, ∂ ∂αj γ(0) = 0, ∂ 2 ∂α 2 j γ(0) = 0, j = 1, 2, and the remaining roots have non-zero real parts.

XUN CAO AND WEIHUA JIANG
Also, write G(·, α) in Taylor expansion in φ at α = 0, where Q(·, ·) and C(·, ·, ·) are symmetric multilinear forms. Then, we have the following conclusion about normal form at a double zero singularity for PFDEs (9), of which the proof is put in Appendix.

XUN CAO AND WEIHUA JIANG
where satisfy ψ j , h 0 2000 (θ) 0 = 0 and ψ j , h 0 1100 (θ) 0 = 0, j = 1, 2. As for k 1 = 0, we have Lemma 3.3. For k 1 = 0, it is not difficult to verity that, According to Lemma 3.3, we have, Proposition 2. For k 1 = 0, on spatial domain Ω = (0, lπ), l > 0 and under Neumann boundary conditions, formulas (15) and (16) in Theorem 3.1, are simplified as follows, g 11 qs =0, |q| + |s| = 2, 3; g 12 2000 = 0; g 12 1100 = 0; We emphasize that Proposition 1 and Proposition 2 also apply to computing normal forms at a double zero singularity of partial differential equations and functional differential equations. Moreover, these concise formulas could be calculated on computer, that is, the process of computing normal forms at a double zero singularity utilizing these concise formulas, could be implemented by computer programs. 4. Spatiotemporal patterns near the double zero singularity. In this section, we investigate complex spatiotemporal patterns of system (5) near the double zero singularity, via analyzing normal form derived by applying Proposition 1 or Proposition 2.
Near 1−mode double zero bifurcation point d * 1 ,c * 1 , the d 1 -c parameter plane is divided into six regions. And dynamics of normal form (25) can be described by the corresponding phase portraits respectively, when (d 1 , c) is chosen in these regions. Then based on center manifold theory, dynamics near the double zero singularity of diffusive system (5) is locally topologically equivalent to dynamics of normal form (25). Thus, we could reveal dynamical behaviors of diffusive system (5), when parameters are chosen in these six regions, respectively. And, dynamics of diffusive system (5) are concluded as follows.
For given parameters β = 1, b = 1, d 2 = 1, l = 4, diffusive predator-prey system (5) supports complex spatiotemporal patterns, when parameter point (d 1 , c) is chosen near 1−mode double zero bifurcation point d * 1 ,c * 1 = (1.07475, 0.01623). Here are the complete dynamics: 1. When (d 1 , c) ∈ D 1 , the coexistence equilibrium E * of system (5) is asymptotically stable (See Figure 2). Otherwise, the coexistence equilibrium E * is unstable while (d 1 , s) / ∈D 1 . 2. When (d 1 , c) crosses D 1 into D 2 , the coexistence equilibrium E * becomes unstable and a stable spatially inhomogeneous periodic solution with the shape of cos ω 0 t cos x l −like arises, through Hopf bifurcation, where 2π ω0 is the temporal period. 3. When (d 1 , c) crosses D 2 into D 3 , a pair of unstable spatially inhomogeneous steady states with the shape of cos x l −like emerges through Turing bifurcation, and the stable spatially inhomogeneous periodic solution with the shape of cos ω 0 t cos x l −like, persists and becomes a large periodic solution (See Figure  3). 4. When (d 1 , s) crosses D 3 into D 4 , the pair of spatially inhomogeneous steady states with the shape of cos x l −like becomes stable and a pair of unstable spatially inhomogeneous periodic solution emerges, via Hopf bifurcation. Furthermore, the large spatially inhomogeneous periodic solution with the shape of cos ω 0 t cos x l −like, is still stable. Therefore, system (5) has three different stable solutions, that is, system exhibits tristability. 5. When (d 1 , s) crosses D 4 into D 5 , the pair of unstable spatially inhomogeneous periodic solutions collides with a saddle-node point and a pair of homoclinic orbits emerges and soon breaks up, then another unstable large spatially inhomogeneous periodic solution with the shape of cos ω 0 t cos x l −like arises inside the stable larger spatially inhomogeneous periodic solution with the shape of  cos ω 0 t cos x l −like. The pair of spatially inhomogeneous steady states with the shape of cos x l −like is still stable. Therefore, system admits tristability (See Figure 4). 6. When (d 1 , s) crosses D 5 into D 6 , these two large spatially inhomogeneous periodic solutions with the shape of cos ω 0 t cos x l −like collide and disappear, through saddle-node bifurcation of closed orbits, leaving the pair of spatially inhomogeneous steady states with the shape of cos x l −like still stable (See Figure 5).
Usually, spatially inhomogeneous solutions near a double zero singularity are unstable for reaction-diffusion systems with local reaction term and Neumann boundary conditions, see [29,36,42]. However, spatially inhomogeneous solutions in the paper are stable, because of the existence of nonlocal prey competition. Therefore, nonlocal reaction term makes it easier for reaction-diffusion systems subject to Neumann boundary conditions to generate stable spatially inhomogeneous periodic solutions with the shape of cos ω 0 t cos x l − like, see Remark 1 and [10]. We also say  that nonlocal reaction term could stabilize spatially inhomogeneous periodic solutions with the shape of cos ω 0 t cos x l − like for the classical diffusive Holling-Tanner predator-prey model subject to Neumann boundary conditions. Furthermore, we remark that, spatially inhomogeneous periodic solution near the double zero singularity in the paper, which has the shape of cos ω 0 t cos x l −like, is different from these spatially inhomogeneous periodic solutions with the shape of cos ω 0 t + cos x l −like resulting from Turing-Hopf bifurcation [2,7,24,38], although they are all spatially inhomogeneous periodic solutions. Actually, Baurmann et al. [3] firstly showed that the interaction of Turing and Hopf instabilities is very likely to result in spatially inhomogeneous periodic solutions for a predator-prey model. Recently, some researchers [2,7,24,38] have also theoretically investigated spatiotemporal periodic solutions arising from Turing-Hopf bifurcation, utilizing normal form method. It was found that these spatially inhomogeneous periodic solutions resulting from Turing-Hopf bifurcation have the shape of cos ω 0 t+cos x l −like, which is a linear combination of temporal pattern cos ω 0 t and spatial pattern cos x l . And, spatially inhomogeneous periodic solution with the shape of cos ω 0 t cos x l −like in this paper, which is the product of cos ω 0 t and cos x l , is essentially different from spatially inhomogeneous periodic solution with the shape of cos ω 0 t + cos x l −like. Figure 6 shows the differences between cos ω 0 t cos x l and cos ω 0 t + cos x l . Obviously, spatially inhomogeneous periodic solution of system (5) has the shape of cos ω 0 t cos x l −like.

Conclusion.
To study the effects of nonlocal reaction term, we investigate spatiotemporal dynamics near the double zero singularity for diffusive predator-prey system (5), which involves nonlocal prey competition. By linear stability analysis,    conditions of the occurrence of Hopf bifurcation, Turing bifurcation and Turing-Turing bifurcation are established, besides these conditions of the occurrence of double zero bifurcation. Moreover, to investigate spatiotemporal dynamics of system (5) utilizing normal form theory, some concise formulas of computing normal forms at a double zero singularity for partial functional differential equations are provided, following normal form method in [12,23]. It is worth noting that, the procedure of computing normal forms at a double zero singularity utilizing these concise formulas, could be implemented by computer programs, which indicates that we could get rid of complicated manual calculations when calculating normal forms. Moreover, these concise formulas also apply to the computation of normal forms at a double zero singularity for partial differential equations and functional differential equations. Then, by analyzing normal form at the double zero singularity, it is found that under proper conditions, system (5) has a large stable spatially inhomogeneous periodic solution with the shape of cos ω 0 t cos x l −like, which could coexist with a pair of spatially inhomogeneous steady states with the shape of cos x l −like. These investigations indicate that, nonlocal reaction term makes it easier for reaction-diffusion systems with Neumann boundary conditions to generate complex spatiotemporal patterns through Hopf bifurcation [10] and double zero bifurcation (see Remark 1), like spatially inhomogeneous periodic solution with the shape of cos ω 0 t cos x l −like which is usually unstable for reaction-diffusion systems with local reaction term and Neumann boundary conditions, see [7,17,29,35,36,38,41,42]. Furthermore, this kind of spatially inhomogeneous periodic solution is also different from these with the shape of cos ω 0 t + cos x l −like arising from Turing-Hopf bifurcation [2,7,24,38].