Spatiotemporal Dynamics Induced by Nonlocal Competition in a Diffusion Predator-Prey System with Habitat Complexity

In this paper, we study a delayed diﬀusive predator-prey model with nonlocal competition in prey and habitat complexity. The local stability of coexisting equilibrium are studied by analyzing the eigenvalue spectrum. Time delay inducing Hopf bifurcation is investigated by using time delay as bifurcation parameter. We give some conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution by utilizing the normal form method and center manifold theorem. Our results suggest that only nonlocal competition and diﬀusion together can induce stably spatial inhomogeneous bifurcating periodic solutions.


Introduction
Predator-prey model is an important model in biomathematics, which is used to study the growth law of two populations with predator-prey relationship [1][2][3]. In the ecological environment, habitat complexity plays an important role, which can be used to control the size and growth trend of the population. Many studies have shown that habitat complexity has a stabilizing effect on the predator-prey model [4][5][6]. In [8], Z. Ma and S. Wang studied a predator-prey model with habitat complexity and time delay, that is (1.1) All parameters are positive. The biological description of parameters is in Table 1. The authors assume there is a gestation delay in predator. In particular, parameter β is used to measure the strength of habitat complexity, such as oyster and coral reefs, mangroves, sea grass beds and salt marshes [7]. The habitat complexity can be used to control the population growth. If α = 1 (α = 2), the functional response is Holling type I (II). In [8], Z. Ma and S. Wang studied the positivity, boundedness, stability, and Hopf bifurcation of the model (1.1). They observed the stabilizing and destabilizing effects of habitat complexity and periodic oscillation caused by time delay under some parameters.
In the ecological environment, the space is often inhomogeneous, and the spatial diffusion often occurs in population. Therefore, the reaction-diffusion equation may be more realistic when we study the predator-prey model. In addition, many scholars have studied the predator-prey model with reaction diffusion, and suggest that diffusion can induce spatial pattern, inhomogeneous periodic solution [9,10]. However, spatially homogeneous periodic solutions often appear in numerical simulation, which means the inhomogeneous periodic solutions are usually unstable. Therefore, we want to study the effect of diffusion term on the model (1.1), and whether inhomogeneous periodic solutions will appear? In fact, the resources is limited in nature, and competition within the population always exist.
This competition is usually nonlocal. In [11,12], the authors suggest that the consumption of resources in spatial location is related not only to the local population density, but also to the number of nearby population density. They measure this effect by weighting and integrating, modifing the u K as 1 Where d 1 and d 2 are diffusion coefficients of prey and predator. For the convenience of calculation, we choose Ω = (0, lπ), where l > 0. 1 K ∫ Ω G(x, y)u(y, t)dy represents the nonlocal competition effect. The kernel function is This is based on the assumption that the competition strength among prey individuals in the habitat is the same, that is the competition between any two prey is the same [14]. The boundary condition is Newman boundary, which is be explained by the fact that the habitat of the population is closed and no prey or predator can entering or leaving the habitat.
The aim of this paper is to study the effect of diffusion term and nonlocal competition on the model (1.2). The rest of this paper is organized as follows. In Sec. 2, we study the stability of coexisting equilibrium and existence of Hopf bifurcation. In Sec. 3, we study the Property of Hopf bifurcation. In Section 3, we give some numerical simulations to illustrate the theoretical results. In Sec. 4, we give a short conclusion.

Stability analysis
In this section, we will study the stability and existence of Hopf bifurcation at coexisting equilibrium. Denote N as positive integer set, and N 0 as nonnegative integer set. (0, 0) and (K, 0) are boundary equilibria of system (1.2). Make the following hypothesis If (H 0 ) holds, then system (1.2) has a unique coexisting equilibrium E * (u * , v * ), where The characteristic equation are (2.4)

The case of τ = 0
We will first discuss the stability of the coexisting equilibrium when the delay τ = 0. Make the following hypothesis Proof. When τ = 0, the characteristic equations (2.3) are and If (H 1 ) holds, we can obtain that the characteristic root of (2.5) and (2.6) all have negative real parts. Then E * (u * , v * ) is locally asymptotically stable. If (H 2 ) holds, we can obtain that the characteristic roots of (2.5) all have negative real parts. And the characteristic equation (2.6) at least have one root with positive real part. Then E * (u * , v * ) is Turing unstable.

The case of τ > 0
Now, We will discuss the stability of the coexisting equilibrium when the delay τ > 0.
3) has a pair of purely imaginary roots ±iω n at τ j n , j ∈ N 0 , n ∈ S, where
Then eq. (2.3) has a pair of purely imaginary roots ±iω n at τ j n , j ∈ N 0 , n ∈ S.
Next, we verify the transversal condition for the existence of Hopf bifurcation.
Denote τ * = min{τ 0 n | n ∈ S}. According to the above analysis, we have the following theorem.

Property of Hopf bifurcation
We give detailed computation about property of Hopf bifurcation using the method in [16,17].
(3.13) Theorem 3.1. For any critical value τ j n (n ∈ S, j ∈ N 0 ), we have the following results.

Numerical simulations
To analyze the effect of diffusion term and nonlocal competition, we compare our results with the work [8] and the following model without nonlocal competition. (4.1)

α = 1
Choose the same parameters with [8]. It shows that the habitat complexity has stabilizing effect which is consistent with [8].
However, with the increase of the habitat complexity, spatially inhomogeneous periodic solutions will appear.

Conclusion
In this paper, we study a delayed diffusive predator-prey model with nonlocal competition in prey and habitat complexity. We mainly study the local stability of coexisting equilibrium and existence of Hopf bifurcation. We also studied the property of bifurcating periodic solutions by the normal form method and center manifold theorem. Our work show that the habitat complexity has stabilizing effect when α = 1, 2, 3, which is consistent with [8]. Under the   same parameters with [8], when β cross some values, the stably spatial inhomogeneous periodic solutions will appear first when α = 1, 2. Under some parameters, the inhomogeneous oscillation curve is always below the homogeneous oscillation curve when α = 3. Compared with the model (4.1), we conclude that only nonlocal competition and diffusion together can induce stably spatial inhomogeneous periodic solutions. In addition, the increase of time delay will increase the amplitude of the periodic solution.

Statements and Declarations
Funding This research is supported by the National Nature Science Foundation of China Competing Interests The authors have no relevant financial or non-financial interests to disclose.
Author Contributions All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Ruizhi Yang. Numerical simulations were performed by Ruizhi Yang and Chenxuan Nie. All authors read and approved the final manuscript.
Data Availability Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.