Bifurcation phenomena in a single-species reaction-di ﬀ usion model with spatiotemporal delay

: In this paper we investigate bifurcation phenomena in a single-species reaction-di ﬀ usion model with spatiotemporal delay under the conditions of the weak and strong kernel functions. We have found that when the weak kernel function is introduced there is Hopf bifurcation but no Turing bifurcation and wave bifurcation to occur, but when the strong kernel function is introduced there exist Hopf bifurcation and wave bifurcation but no Turing bifurcation to occur. Especially, taking the inverse of the average time delay as a bifurcation parameter, we investigate inﬂuences of the time delay on the formation of spatiotemporal patterns through the numerical method. Some spatiotemporal patterns induced by Hopf bifurcation and wave bifurcation are respectively shown to illustrate the mechanism of the complexity of spatiotemporal dynamics.


Introduction
A single-species reaction-diffusion model with spatiotemporal delay which was firstly proposed by Britton [1] is considered as follows: where u = u(x, t) for (x, t) ∈ R×[0, +∞) presents the density of a single-species at space x and time t, ψ is a given function and ψ * u means a convolution in space x and time t, the parameters d, r, α and β are all positive constants and 1 + α − β > 0. For the biological explanations of all terms in the model (1.1), according to references [1,2], we have known that d > 0 is the diffusive coefficient, the term αu is a measure of the advantage to individuals in aggregating or grouping and βu 2 represents competition for space. Besides, the spatio-temporal convolution ψ * u, which represents competition between the individuals for food resources, is interpreted as a weighted average of the values of u over all past time and over all points in space and has the following form where ψ(x, t) is a given positive kernel function which satisfies R +∞ 0 ψ(x, t)dxdt = 1.
For model (1.1), a lot of work had been done by many researchers Britton [1], Gourley [3] and Billingham [4] who had shown that there exist the travelling wave front solution and the periodic travelling wave solution. However, in the recent several decades there has been an increasing interest in the two species reaction-diffusion equations that incorporate spatially and temporally nonlocal terms in the form of the convolution of a kernel function, see [5][6][7][8][9]. Their studied results demonstrated that the predator-prey models with the nonlocal term have many bifurcation phenomena which lead to the complex spatio-temporal patterns to occur. For example, the work of [5] had shown that when the nonlocal term was introduced the more bifurcation phenomena would occur than the system without the nonlocal term. Besides, in order to explore the mechanism of emergence of complex spatiotemporal patterns, the bifurcation behaviours including Hopf bifurcation, Turing bifurcation, wave bifurcation, Turing-Hopf bifurcation an so on, are also investigated in reaction-diffusion equations by many researchers [10][11][12][13][14][15]. The definitions of these bifurcations are introduced in the references [16,17] and have the important relations with roots of the characteristic equation of the studied system. The complex spatiotemporal patterns which include the spatially inhomogeneous periodic solution, homogeneous solution [10], stationary spatial patterns [11] and quasi-periodic solution [12] are also pointed out by them, respectively.
In this paper, according to [1], we take the particular kernel functions on the basis of a random walk argument as follows which are respectively called the weak kernel and strong kernel, where x is the spatial variable, t is the temporal variable, τ = 1 τ andτ called the average time delay means the competition time for food resources. We mainly consider all kinds of bifurcation phenomena including Hopf bifurcation, Turing bifurcation and wave bifurcation for Eq (1.1) with the weak and strong kernel functions. We take the inverse of the average time delay in the kernel function as a bifurcation parameter and explore influences of the time delay on the emergence of the different bifurcation behaviours including Hopf bifurcation, Turing bifurcation and wave bifurcation. The studied results have found that there only exists Hopf bifurcation in Eq (1.1) with the weak kernel and there are Hopf bifurcation and wave bifurcation in Eq (1.1) with the strong kernel. Furthermore, we suppose that Eq (1.1) has the positive initial condition φ(x) for x ∈ R. And because there is zero population flux across the boundary tending to the infinity the following homogeneous Neumman boundary condition ∂u(x,t) ∂n = 0 is supposed for Eq (1.1), where n is the outward unit normal vector of the boundary which tends to infinity.
The rest of this paper is organized as follows. In Section 2, the bifurcation analyses of a singlespecies reaction-diffusion model with spatiotemporal delay including the weak kernel and the strong kernel is given, respectively. The conditions of occurrence of Hopf and wave bifurcation are also obtained. In Section 3, we investigate the effect of the time delay on the spatiotemporal patterns through the numerical method. We finally give some conclusions in Section 4.

Bifurcation analysis under the different kernel functions
In this section we will employ the linearized analysis to study bifurcation phenomena of Eq (1.1) as the parameter τ varies. The conditions of the emergence of Hopf bifurcation, Turing bifurcation and wave bifurcation in Eq (1.1) under the weak kernel and strong kernel are also given, respectively.

The case of the weak kernel function
When the kernel function has the following weak form according to the results in the references [1,18], we let v(x, t) = ψ * u, (2.1) which leads to transform Eq (1.1) into the following form . Eq (2.2) has three equilibrium points (0, 0), (1, 1) and (− 1 β , − 1 β ). From the viewpoints of population biology, we pay our attention to the equilibrium point E + = (1, 1) which corresponds to the maximum state of the studied species. The linearized equations of system (2.2) are as follows where c 1 and c 2 are coefficients, k denotes the wave numbers in the X direction, and λ is the growth rate of perturbation in time t, i is the imaginary unit and i 2 = −1, r is the spatial vector in one-dimensional space. Thus, substituting Eq (2.4) into Eq (2.3), we get the characteristic equation which is Assume that α > 2β, from the equation T r(0) = a 11 + a 22 = 0, we obtain That is to say, when τ = τ Hc , the characteristic Eq (2.5) for k 2 = 0 becomes which has a pair of purely imaginary roots ± √ a 11 a 22 − a 12 a 21 = ± rτ Hc (1 + β). Besides, the following transversality condition dReλ is also satisfied. When k 2 = 0, the roots of Eq (2.5) will determine the local stability of the equilibrium point E + . If the equilibrium point E + is stable for k 2 = 0, then for k 2 0, it is also stable because T r(k 2 ) = 2dk 2 + T r(0) and ∆(k 2 ) = d 2 k 4 + T r(0)k 2 + ∆(0) are both positive under the conditions of T r(0) and ∆(0) > 0 through using the Routh-Hurwtiz criterion. Connecting the above analyses, the following results are obtained immediately.
(i) When k 2 = 0, the equilibrium point E + is asymptotically stable for τ > τ Hc and unstable for 0 ≤ τ < τ Hc , and Eq (2.2) undergoes Hopf bifurcation at τ = τ Hc which leads to the spatially homogeneous periodic solution to occur; (ii) When k 2 0, the equilibrium point E + of Eq (2.2) is also stable because roots of the characteristic Eq (2.5) have the negative real parts under the conditions of T r(k 2 ) = 2dk 2 + T r 0 , ∆(k 2 ) = d 2 k 4 + T r 0 k 2 + ∆ 0 , T r 0 > 0 and ∆ 0 > 0. That is to say, there are not Turing bifurcation and wave bifurcation to occur.

The case of the srong kernel function
When the kernel function has the following strong form we use the same symbols as the subsection 2.1 for the convenience of our studies and let according to the results in the references [1,18], which lead to (1.1) to transform into the following form Similar to Eq (2.2), Eq (2.8) has three equilibrium points (0, 0, 0), (1, 1, 1) and (− 1 β , − 1 β , − 1 β ). The linearized equations of (2.8) at E + = (1, 1, 1) are Similarly, we can get the characteristic equation as follows where According to the Routh-Hurwitz criterion, the conditions of the emergence of Hopf bifurcation, Turing bifurcation and wave bifurcation have the relation with the distribution of roots of Eq (2.10) at k 2 = 0 and k 2 0, respectively. That is to say, we need to discuss whether the following results hold.
For k 2 0, when the characteristic Eq (2.10) has a zero root and the other roots with negative real parts under the conditions of m 1 (0) > 0, m 2 (0) > 0 and m 3 (0) > 0, there is possible Turing bifurcation to occur in Eq (2.8).
For k 2 0, when the characteristic Eq (2.10) has a pair of purely imaginary roots and the another root with negative real part under the conditions of m 1 (0) > 0, m 2 (0) > 0 and m 3 (0) > 0, there possibly exist wave bifurcation in Eq (2.8).
According to the above discussions of the root distributions of Eq (2.10) the following results are obtained immediately.
(i) For k 2 = 0, the constant equilibrium point E + is asymptotically stable for 0 ≤ τ < τ shc− and τ > τ shc+ and unstable for τ shc− < τ < τ shc+ , and Eq (2.8) undergoes the Hopf bifurcation at τ = τ shc± ; ( Through the detailed calculations about Eq (2.12), under the condition of 17β − 9α − 1 < 0 the critical value of the bifurcation parameter τ is as follows And the another root of Eq (2.11) is λ = −m 1 (0). In order to obtain the transversality condition, we need to differentiate about the parameter τ on the two sides of Eq (2.11) and get (2.14) Connecting And the transversality condition of occurrence of Hopf bifurcation (ii) For k 2 0, the conditions of Eq (2.8) undergoing Turing bifurcation are that Eq (2.10) has a zero root and the equilibrium point E + is asymptotically stable for k 2 = 0. These conditions need the following results simultaneously hold The derivative function of m 3 (z) about the variable z is .
(iii) For k 2 0, the conditions of Eq (2.8) undergoing wave bifurcation are that Eq (2.10) has a pair of purely imaginary roots and a root with the negative real and the equilibrium point E + is asymptotically stable for k 2 = 0. These conditions need also the following results simultaneously hold In order to get the critical value of the bifurcation parameter τ about the wave bifurcation, we firstly let z = k 2 and F(z) = m 1 (z)m 2 (z) − m 3 (z). By the detailed calculations, we get the expression of function F(z) about the variable z as follows F(z) = 8d 3 z 3 + 8d 2 (2τ − b 11 )z 2 + 2d(τ − b 11 )(5τ − b 11 )z + τ(2τ − b 11 )(τ − 2b 11 ) + τ 2 (b 11 + b 12 ).
The derivative function of F(z) about the variable z is Letting dF(z) dz = 0, we obtain that the zero points of the derivative function dF(z) dz are z min = b 11 −5τ 6d and z max = b 11 −τ 2d which correspond the local extreme maximum value and local minimum value of the function F(z), respectively. Let the local extreme minimum F(z max ) of the function F(z) be equal to zero, we get the critical value of the bifurcation parameter 19) and the corresponding critical wave number k 2 wc = z max = b 11 −τ swc 2d . In order to obtain the transversality condition, we need to differentiate about the parameter τ on the two sides of Eq (2.10) and get we have dλ dτ = p 1 3λ 2 + 2m 1 (k 2 )λ + m 2 (k 2 ) , So when τ = τ swc and k 2 = k 2 wc , dλ is also satisfied. The proof is completed.

Numerical simulations of spatiotemporal patterns
In this section we will give some numerical results when this single-species reaction-diffusion model with the strong kernel function. According to the results in Theorem 2.2 of subsection 2.2, we fix d = 1, r = 2, β = 0.2 and take the parameters τ and α as variables. The pictures of Hopf bifurcation curve and wave bifurcation curve are given in the parameter plane spanned by τ and α, see Figure 1. Because of τ = 0.01 < 0.333939 = τ shc− , Figure 2 (A) presents that the state of u will tend to the stable state E + = (1, 1, 1) as the time varies.
Because of τ = 0.2 > 0.066667 = τ swc , Figure 2 (C) shows that there exists the temporally oscillation pattern induced by the interactions of Hopf bifurcation and wave bifurcation. Because the Hopf bifurcation is much more prominent than the wave bifurcation, we would obtain the seriously oscillatory pattern in the temporal direction.
Besides, if we take the parameters α = 0.8 and τ = 0.05 corresponding to the point D, we get the spatiotemporal pictures about wave bifurcation which leads to the spatially heterogeneous pattern to occur, see Figure 2 (D).

Conclusions
The bifurcation behaviours resulting in the occurrence of all kinds of patterns in the reaction-diffusion equations have been focused in the mathematical biology. It is well known that these bifurcation behaviors including Hopf bifurcation, Turing bifurcation, wave bifurcation and interactions of them are reported by many researchers. Recently, the spatiotemporal patterns in the reaction-diffusion equations due to the presence of the nonlocal reaction have been a fascinating subject on the study of the pattern formation [5,8]. What's more, the effect of the kernel functions on the spatial pattern formation are also discussed by [19]. In this paper, we mainly pay our attention to influences of the different kernel function, which are respectively called weak kernel and strong kernel, on pattern formation in a single-species reaction-diffusion equation. We have found that when the weak kernel function is introduced there only exists Hopf bifurcation which induces the spatially homogeneous pattern to occur, but when the strong kernel is introduced there are Hopf bifurcation and wave bifurcation which can induce the spatially homogeneous pattern and the spatially heterogeneous pattern to occur, respectively. It is shown that the rich spatiotemporal patterns occur when the different kernel functions are introduced.