Pattern Formation in a Reaction-Diffusion Predator-Prey Model with Weak Allee Effect and Delay

. In this paper, we establish a reaction-diffusion predator-prey model with weak Allee effect and delay and analyze the conditions of Turing instability. The effects of Allee effect and delay on pattern formation are discussed by numerical simulation. The results show that pattern formations change with the addition of weak Allee effect and delay. More specifically, as Allee effect constant and delay increases, coexistence of spotted and stripe patterns, stripe patterns, and mixture patterns emerge successively. From an ecological point of view, we find that Allee effect and delay play an important role in spatial invasion of populations.


Introduction
Since the Allee e ect was proposed by Allee [1] in 1931, the predator-prey model with Allee e ect has been studied extensively . From the ordinary di erential equation predator-prey model with Allee e ect to the partial differential equation model, many researchers have achieved rich results [4,7,[9][10][11][12][13][14][15][16]28]. Cai et al. [6] established a Leslie-Gower predator-prey model with additive Allee e ect on prey, and they found Allee e ect can increase the risk of ecological extinction. Sen et al. [5] established a twoprey one-predator model with Allee e ect, and the e ects of Allee e ect on the dynamics of predator population are discussed. Of course, the research on reaction-di usion predator-prey model with Allee e ect is also very rich. For example, Wang et al. [7] established a reaction-di usion predator-prey model and found the model dynamics exhibits both Allee e ect and di usion controlled pattern formation growth to holes. ey also studied Allee e ect induced instability in a reaction-di usion predator-prey model [4]. Petrovskii et al. found that the deterministic system with Allee e ect can induce patch invasion [23]. Sun et al. found that predator mortality plays an important role in the pattern formation of populations [13]. It is now believed that the spatial composition of population interactions has been identi ed as an important factor in how ecological communities operate and form. Pattern formation in the predator-prey model is an appropriate tool for understanding the basic mechanism of spatiotemporal population dynamics. We nd that there are few studies on delays in reaction-di usion predator-prey model with Allee e ect. So next we discuss the e ects of Allee e ect and delay on pattern formation. First, we consider a predator-prey model with hyperbolic mortality established by Zhang et al. [10], the model is obtained as follows: where U and V are the population densities of prey and predator, respectively; a is the birth rate, K is the carrying capacity and b is the maximum uptake rate of the prey; c is the prey density at which the predator has the maximum kill rate; m is the birth rate of predator; function h(V) reflects the predator death rate; the habitat Ω ⊂ R n is a bounded domain with smooth boundary zΩ; n is the outward unit normal vector on zΩ; d 1 and d 2 are the diffusion coefficients, respectively; and Δ is the Laplacian operator. e homogeneous Neumann boundary condition implies that the system above is self-contained and there is no host across the boundary. After nondimensionalization, (2) en, considering that the predator-prey model with Allee effect is more realistic, people begin to introduce delay into the predator-prey model and discuss the effects of Allee effect and delay on the dynamics of the model [2,[17][18][19][20][21][22]. We try to introduce weak Allee effect and searching delay into model (1), and then we get where h(v) � cv 2 /e + ηv. For hyperbolic mortality, c is the death rate of the predator, e and η are coefficients of light attenuation by water and self-shading in the context of plankton mortality, and τ is the searching delay. e weak Allee effect term is u/u + A, where A > 0 is described as a weak Allee effect constant.

Turing Instability
First, we consider the model with τ � 0: Obviously, if d 1 � d 2 � 0, without diffusion in model (4), then we can obtain the following ordinary differential equations: We mainly focus on the stability of the positive equilibrium of model (4). Clearly, the positive equilibrium E * � (u * , v * ) of the ordinary differential equation (ODE) or the partial differential equation (PDE) model (4) satisfies f(u * , v * ) � 0 and g(u * , v * ) � 0: For simplicity of discussion, in this paper, we shall concentrate the case of η � c and e � 1. We easily see that model (4) exhibits a positive equilibrium point exhibits two positive equilibrium points E 1 * � (u 1 * , v 1 * ) and E 2 * � (u 2 * , v 2 * ). In this work, we mainly focus on a positive equilibrium point, where We calculate the Jacobian matrix of model (5) at E * , which is given by J * � a 10 a 01 b 10 b 01 , where We can easily know that the characteristic polynomial is us, we have the following conclusions: (a) If T < 0 and D > 0, then the positive equilibrium is locally asymptotically stable (b) If T > 0, then the positive equilibrium is unstable Next, let us consider the PDE model (4); we choose the perturbation function consisting of the following two-dimensional Fourier modes: so, we can find We know, when E * is stable, We can easily find that T k � a 11 + a 22 − k 2 (d 1 + d 2 ) < 0. So, if model (3) changes from stable to unstable, it needs to be that is,

Delay-Induced Instability
Finally, we consider the PDE model (4) with delay (searching delay), and we get model (3). Considering τ and spatial diffusion, if τ is small enough, the following changes are made [29]: we substitute (16) into model (3) to get Expanding in Taylor series and neglecting the higherorder nonlinearities, we find We can see that if f(u * , v * ) � 0 and g(u * , v * ) � 0 are satisfied at equilibrium point E * � (u * , v * ), then we can get the model: where We consider that the stable equilibrium point Assuming that the solution of the system has the following form, where We can easily find

Complexity
When k � 0, model (3) undergoes Hopf bifurcation at T τ k � 0, so the critical value for undergoing Hopf bifurcation can be obtained: We know, when E * is stable, So, we just need to judge It is easy to know when So, the instability condition caused by delay is as follows:

Amplitude Equations and Pattern Selection
We rewrite the transformed form of system (3) at the positive spatially homogeneous steady state E * � (u * , v * ) as follows and denote by (U, V) T the perturbation solution where X � (U, V) T . en, let the linear operator L be defined as follows: and H be given by Complexity 5 where Next, near the Turing bifurcation threshold, we expand the control parameter τ as where |ε| ≪ 1. Similarly, expand the solution X, linear operator L, and the nonlinear term H into Taylor series at ε � 0: where are terms corresponding to the second and third orders in the expansion of the nonlinear term and for the linear operator We have, Finally, we introduce multiple time scales: en, substituting equations (33)-(44) into equation (32) and expanding it with respect to different orders of ε i , (i � 1, 2, 3), In what follows, we seek the amplitude equations by solving system (45). Since L T has an eigenvector associated with the zero eigenvalue, e general solution of the first system of (45) can be written as where W j is the amplitude of the mode e ik j ·r . Notice that the second system of (45) is nonhomogeneous, and L * T , the adjoint operator of L T , has zero eigenvectors in the form of 1 g e ik j ·r + c.c., j � 1, 2, 3, en, in view of the Fredholm solvability conditions, where F j U and F j V are the coefficients of e ik j ·r in F U and F V , respectively. It follows after some routine calculation that, for j l � 1, 2, 3 and j l ≠ l m , if l ≠ m, Notice the forms of U 1 and V 1 given by (47). We have a particular solution for the second system of (45) as follows: with the coefficients being given below at α T � α: Again, apply the Fredholm solvability condition to the third system of (45). We have, for j � 1, where τ 0 � f + g τ T fm 11 + m 12 + g fm 21 + m 22 , h � H τ T fm 11 + m 12 + g fm 21 + m 22 , (58) Please notice that system (57) is in complex form. Following to reference [30], for the purpose of convenience of discussion, we convert it into the real form by A j � ρ j exp(iφ j ) with ρ j as the real amplitudes and φ j as phase angles: Since we are only interested in the stable steady states and notice the fact that hρ i ≠ 0, from the first equation of (59), we have φ � 0 or π. Also, noticing the fact that τ 0 > 0, it implies that when h > 0, the state corresponding to φ � 0 is stable, but the one corresponding to φ � π when h < 0. en, system of amplitude equation (59) becomes Please notice that generally the amplitude equations are valid only when the control parameter is in the Turing space. It is easy to see that the above system of ordinary differential equation (60) has five equilibria, which corresponds five kinds of steady states [10,30,31]. Noticing the symmetry of the system, we have the following: (1) System (60) always has an equilibrium E 0 � (0, 0, 0), which is stable for μ < μ 2 � 0 and unstable for μ > μ 2 (2) System (60) has an equilibrium E s � ( ���� μ/g 1 , 0, 0) corresponding to stripe patterns, which is stable for μ > μ 3 � h 2 g 1 /(g 2 − g 1 ) 2 and unstable for μ > μ 3 (3) System (60) has an equilibrium E h � (ρ 1 , ρ 2 , ρ 3 ) corresponding to hexagon patterns, with φ � 0 or φ � π, and ρ + 1 � |h| + �������������� � h 2 + 4(g 1 + 2g 2 )μ/2(g 1 + 2g 2 ) is stable for μ < μ 4 � h 2 (2g 1 + g 2 )/(g 2 − g 1 ) 2 and ρ − 1 � |h| − �������������� � h 2 + 4(g 1 + 2g 2 )μ/2(g 1 + 2g 2 ) is unstable, where ρ 1 � ρ 2 � ρ 3 � |h| ± ��������� h 2 + 4(g 1 + 2g 2 )μ/2(g 1 + 2g 2 ) (4) System (60) has an equilibrium E m � (ρ 1 , ρ 2 , ρ 3 ) corresponding to mixed patterns, with g 1 > g 2 , μ > g 1 ρ 2 1 which is unstable, where ρ 1 � |h|/g 2 − g 1 , ρ 2 � ρ 3 � ������������� � μ − g 1 ρ 2 1 /g 2 + g 1

Numerical Simulations
In this section, we will further study the dynamic behavior of the coexistence equilibrium of the delayed reaction-diffusion model (3) using numerical simulation in two-dimensional space. In this paper, a two-dimensional delay reaction-diffusion model is treated by the finite difference method in the discrete region of 100 × 100. e spatial distance between two lattices is defined as the step size Δx and Δy, using the standard five-point approximation for the 2D Laplacian with the zero-flux boundary conditions, and the time step is expressed as Δt. Take a fixed time step Δt � 0.01. What needs to be further explained is that the concentrations (S n+1 i,j , I n+1 i,j ) at the moment (n + 1)Δt at the mesh position (i, j) are given by S n+1 i,j � S n i,j + Δtd 1 Δ h S n i,j + Δtf S n i,j , I n i,j , I n+1 i,j � I n i,j + Δtd 2 Δ h I n i,j + Δtg S n i,j , I n i,j , with the diffusion term (Laplacian) are defined by Other parameters are fixed as First, we discuss the effect of weak Allee effect on Turing pattern information. We try to take the Allee effect constant to A � 0, A � 0.02, and A � 0.1. Here, we first discuss the situation without delay. When A � 0, Zhang et al. [10] give the condition of Turing instability. Here, we just give the pattern formations. By comparing Figures 1 and 2 (A � 0 and A � 0.02), we find that the initial state is the coexistence of stripes and spots, and the stripes are very long. With the increase of Allee parameters, the length of stripes decreases and some stripes even form a circle. en, we continue to increase the value of the weak Allee parameter like Figure 3 (A � 0.1), and we find that pattern formations have changed again. As time goes on, we find that pattern formations show a cycle when t � 100 and when t � 500 and we find that the cycle diffuses outward (indicating that pattern formations are not stable) to form a butterfly-like shape; and finally, when we increase to t � 2000, we find that the pattern formations are not stable. It was found that pattern formations became stripes and spots again. After we tried to add more time, we found that the pattern formation did not change again.
Next, let us discuss the pattern formation change of the model with time delay and without Allee effect. We change the delay to τ � 0.25. By comparing with Figure 1, we find that the stripes and spots of the original pattern formations change to stripes like Figure 4, and the pattern formations will not change as time goes on.
Finally, we discuss the pattern formations of models with Allee effect and time delay (A � 0.02 and τ � 0.02). We find that pattern formations are spots when t � 100; as time goes on t � 500, we find that pattern formations change again, similar to Figure 3, but the final pattern formations change differently. We can see that pattern formations change into strips surrounded by spots; we increase the time again (t � 2000) to find that the pattern will spread outward in this form, forming the phenomenon of strip pattern surrounded by spots pattern; we further increase the time (t � 5000) to find that the pattern such as in Figure 5 tends to stabilize and does not change again.

Conclusion
is paper is based on a model that considers a predatorprey model with nonlinear mortality and Holling II functional response. e weak Allee effect is introduced and the effect of the Allee effect on pattern formations is considered. Furthermore, we consider a class of reaction-diffusion predator-prey models with searching delay and weak Allee effect, considering the effects of delay on pattern formations. We give the stability and Turing instability of the positive equilibrium point E * . As a result of diffusion, model (3) and model (4) exhibits stationary Turing pattern. Furthermore, through numerical simulation, comparing Figures 1 and 2, we find that the Allee effect will reduce the length of the strip pattern in Figure 1, and there will be some "cycle" pattern as shown in Figure 2. From an ecological point of view, we  know that the Allee effect increases the risk of population extinction, while the effect of the longer stripe pattern in Figure 1 increases the likelihood of predation. However, the shorter stripes and spots in Figure 2 reduce the likelihood of predation. As the Allee effect parameter continues to increase, we find that the pattern has changed again. e type of the pattern is similar to that of Figure 1, but the density and size of the pattern will change slightly as shown in Figure 3. We believe that in order to avoid predator hunting, predators are concentrated in a certain area rather than scattered throughout the habitat, which further reduces the contact area between predator and prey. Over time, prey needs to migrate to new habitats. e aggregation pattern diffuses slowly, the predator follows the pursuit, and the aggregation point enlarges gradually. It is worth noting when the Allee effect parameter is A � 0.1, there are two positive equilibrium points in model (4). Next, we consider the effect of delay on pattern formations. By comparing Figure 1 with Figure 4, we find when the delay is τ � 0.25, the pattern changes from the state where the starting spots pattern and the strip pattern coexist to the case where only the strip pattern exists. Finally, we try to consider the Allee effect and delay to observe the changes in pattern formations, where A � 0.02 and τ � 0.02. We find that when both are present, the spots pattern is surrounded by strip patterns as shown in Figure 5. is reminds us of animals in the natural world at the lower end of the food chain, often with a means of protection. Juvenile animals are surrounded by adult animals to reduce the probability of their juvenile animals being preyed. is may be an interesting finding or not. So, we find that Allee effect and delay play an important role in spatial invasion of populations.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.