Hopf bifurcation in a diffusive predator–prey model with Smith growth rate and herd behavior

This paper mainly aims to consider the dynamical behaviors of a diffusive delayed predator–prey system with Smith growth and herd behavior subject to the homogeneous Neumann boundary condition. For the analysis of the predator–prey model, we have studied the existence of Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equation. Then we have proved the stability of the periodic solution by calculating the normal form on the center of manifold which is associated to the Hopf bifurcation points. Some numerical simulations are also carried out in order to validate our analysis findings. The implications of our analytical and numerical findings are discussed critically.

by the following ordinary differential equations: Here X(t) represents the prey density and Y (t) the predator density, parameter r is the growth rate of the prey, parameter K is its carrying capacity, parameter s denotes the death rate of the predator in the absence of prey, parameter α is the search efficiency of Y (t) for X(t), and parameter c is viewed as the biomass conversion or consumption rate. This kind of model is also known to us as the predator-prey model with herd behavior, and the existence of the possibility of sustained limit cycles is real; what's more, the solution behavior near the origin shows to be more subtle and interesting. As far as the growth of the prey is concerned, many researchers have considered logistic growth function to be a logically acceptable function so that the average growth rate of N (t) N(t) is a nonlinear function of the density function N(t). This assumption is not realistic for a food-limited population which is under the effect of environmental toxicants. Therefore, the population dynamics with limited growth should be based on the proportion of unused available resources, where r c is viewed as the mass in the population at K , both environmental and food chain effects of toxicants' stress [6][7][8][9] should be taken into consideration.
In the real world, the prey and predator populations are always moving, so, in order to study the dynamics of this model, we should consider the condition that the two populations are spatially disperse. The spatial dynamics of the predator-prey models which include spatial diffusion have already been widely studied in the literature. We will assume the condition that both populations are in an isolated patch, which means that the immigration is neglected using the Neumann boundary conditions. Based on the above discussions, we do rigorously consider the predator-prey system along with Smith growth rate and herb behavior as follows: where u(t) is the prey density and v(t) predator density at time t; βγ is the death rate of the predator in the absence of prey, the conversion or the consumption rate of prey to predator is represented by the parameter γ .
In the real-world systems, delay exist almost everywhere. So in order to reflect the dynamical behavior of models that depend on the past history, a more realistic and better approach is to incorporate the factor time delays into the models. Many interesting and surprising conclusions have been obtained, since incorporating time delays has a great impact and complicated effect on the dynamical behaviors of the systems. In a survey paper [10], Ruan made a conclusion that the delay differential equations show more complex dynamics than ordinary differential equations. Recently, many authors paid more attention to a partial differential system in the field of delay effects; this new kind of diffusion was taken into consideration in [3,[11][12][13]. Some authors put their focus on the study of the delay effects of the reaction-diffusion system, and they seriously investigated the stability/instability of the coexistence equilibrium and related with Hopf bifurcation [4,5,14,15].
Hence, we will try to continue studying the dynamics of the following system: where the delay effect is represented by a nonnegative parameter τ . In this paper, the main aim is to consider the delay-induced Hopf bifurcation for the predator-prey system (1.2) with the proper use of the normal form and the use of center manifold theory. The results of this paper could be summarized as follows. In Sect. 2, we first seriously consider the Hopf bifurcation of the local system (1.1). Secondly, we investigate the existence of the delay-induced Hopf bifurcation for the predator-prey model with strong diffusion. In Sect. 3, we carefully calculate the normal form on the center manifold to further discuss the dynamical behavior around the delay-induced Hopf bifurcation value. In Sect. 5, we present some accurate and vivid numerical simulations in order to precisely illustrate and better expand our theoretical results. In Sect. 8, this paper is ended with some discussions.

Stability and bifurcation analysis 2.1 Stability and bifurcation analysis for system (1.1) without diffusion
In order to further study the very complex dynamics of system (1.1), firstly, we discuss the dynamics of system (1.1) with no diffusion as follows: As for the system (1.2), when 0 < β < 1, the unique positive equilibrium point E * (u * , v * ) exists, where The linearization of (2.1) at the unique positive equilibrium point E * (u * , v * ) is where a 11 = 1 -3β 2cβ 2 (1 + β 2 ) 2(1 + cβ 2 ) 2 , a 12 = -β, The characteristic equation is where To study the stability of the positive equilibrium E * for system (2.1) more accurately, the mathematical relation between c and β which appear in the preceding equations is needed. Denote then T 0 (β, c 0 (β)) = 0. In the following, we choose parameter c to be the bifurcation parameter and analyze the existence of Hopf bifurcation at the interior equilibrium E * . In fact, parameter c can also be regarded as the Smith growth rate of prey, which plays a very important role in determining the stability of the interior equilibrium and also influences the existence of Hopf bifurcation.
If we choose to consider parameter c as a bifurcation parameter, then (2.4) will have a pair of opposite purely imaginary eigenvalues ω = ± √ D 0 when the value of the parameter c is c = c 0 . System (2.1) should have a small amplitude nonconstant periodic solution bifurcated from the positive E * , when the parameter c crosses through c 0 if the transversality condition is satisfied.
Let λ(c) = α(c) + iω(c) be the root of (2.4), then Hence, α(c 0 ) = 0 and From the above discussions, we can imply that system (2.1) will undergo Hopf bifurcation at E * as c crosses through c 0 if the transversality condition (2.5) is satisfied.
Next, we will continue to study the delay-induced Hopf bifurcation for the predatorprey model with diffusion.

Spatial-temporal dynamics for the diffusive predator-prey model
Let The linearization of (1.2) at the positive equilibrium E * is where a 11 , a 12 , a 21 , a 22 are already given in (2.3). Hence, the characteristic equation of (2.6) is where I is the 2 × 2 identity matrix and M k = -k 2 diag{d 1 , d 2 }, k ∈ N 0 , from which we can conclude that When τ = 0, (2.10) According to the result of [15], we always assume that the positive equilibrium E * of system (1.2) without delay is asymptotically stable, which is equivalent to the condition T k > 0, D k > 0 for any k ∈ N 0 , Hence, we can obtain a long series of Hopf bifurcation lines H k given by When τ = 0, we assume that λ = iω, and substitute iω into (2.8), to obtain Separating the real and imaginary parts, we have which is equivalent to For 0 < k < N 1 , there is a unique positive root ω k of (2.13), (2.14) From (2.14), we can obtain that
Then, we can have the following transversality condition.
Proof Differentiating the two sides of (2.8), we get Thus, by (2.12) and (2.13), we have . . , N 1 }, and from the above lemmas, we already know that τ 0

Normal form of the Hopf bifurcation for a diffusive model
We denote c * = c j , and then introduce a new parameter ε ∈ R by setting the parameter c = c * + ε such that ε = 0 obviously becomes the bifurcation value. Then, we rewrite the positive equilibrium just as a parameter-dependent form E * ε (u * (ε), v * (ε)) with Settingũ(·, t) = u(·, t)u * (ε),ṽ(·, t) = v(·, t)v * (ε),Ũ(t) = (ũ(·, t),ṽ(·, t)) and then dropping the tildes for the simplification of notation, system (1.1) can be written as the following equation: where By a direct computation, we obtain that f 020 = f 120 = f 030 = 0. Assuming that there does exist a parameter k ∈ N 0 such that k = 0 with c = c * which has a pair of purely imaginary roots ±iω k , the remaining roots of the characteristic Eq. (3.1) will actually have nonzero real parts, where In term of M k p k = iω k p k and M T k q k = iω k q k , we choose p k and q k such that q T k , p k = 1, where with D = [1 + (iω k +d 1 k 2 -a 11 ) 2 a 12 a 21 ] -1 . By (3.1) and a very direct computation, we get where A k20 = f 200 p 2 k1 + 2f 110 p k1 p k2 , A k02 =Ā k20 , A k11 = 2f 200 |p k1 | 2 + 4f 110 Re{p k1pk2 }.
Thus, we obtain It is easy to verify that ). The calculation of Proj S [(D z f 1 2 )(z, 0, 0)U 1 2 (z, 0)] requires where there is a straightforward calculation which shows that , and then The calculation of Proj S [(D w f 1 2 )(z, 0, 0)U 2 2 (z, 0)] is as follows: In order to obtain D k21 , we compute h kj20 and h kj11 as follows: h 0020 = -c kj A 011 , k = 0, j = 0, 2k.
Thus, the normal form of Hopf bifurcations on the center manifold iṡ which can be written down in real coordinates w by letting z 1 = w 1iw 2 , z 2 = w 1 + iw 2 , and transforming to polar coordinates w 1 = ρ cos ξ , w 2 = ρ sin ξ . Then, this normal form becomes It's know that the direction of the bifurcation is determined by the sign of ν k1 ν k2 (supercritical if ν k1 ν k2 < 0, subcritical ν k1 ν k2 > 0), and the stability of the nontrivial periodic orbits can be determined by the sign of ν k2 (stable if ν k2 < 0, unstable if ν k2 > 0).

Normal form of the delay-induced Hopf bifurcation for a diffusive model
In this subsection, we shall study the directions, stability and period of bifurcating periodic solutions by moderately and scrupulously applying the normal formal theory and the center manifold theory of partial functional differential equations which are presented in [15][16][17][18] in detail. For fixed j ∈ N 0 , 0 ≤ k ≤ N 1 , we denote τ * = τ j k , and then introduce a new parameter ε ∈ R by setting ε = ττ * such that ε = 0 becomes the Hopf bifurcation value obviously. τ t),ṽ(·, τ t)) and C = C ([-1, 0], X), and then dropping the tildes for the simplification of notation, system (1.2) can be written as follows: ∂u i ∂v j ∂w l , n = 1, 2, and By direct computation, we can obtain f 020 = f 210 = f 120 = f 030 = 0.
(b) When κ k1 κ k2 > 0, the Hopf bifurcation that the system undergoes at the critical value τ = τ * is a subcritical bifurcation. Moreover, if κ k2 < 0, then the bifurcating periodic solution is stable; if κ k2 > 0, then the bifurcating periodic solution is an unstable one.
In the next section, we perform some accurate numerical simulations, together with dynamical analysis for Hopf bifurcation, of systems (1.1) and (1.2).

Numerical simulations
In this section, by using mathematical software Matlab, we present some numerical simulations to support and extend our analytical results.
simulations have been carried out in order to depict our theoretical analysis. For system (1.1), we have chosen to set d 1 = 0.05, d 2 = 0.1, γ = 1, β = 0.5048, and, together with a direct computation, we have obtained the critical value c 0 = 0.7366, and have also gotten the values κ k1 = -0.0567, κ k2 = 0.0186. We can conclude from the result that the positive equilibrium (u * , v * ) = (0.4094, 0.3517) is asymptotically stable for the parameter c = 0.8 > 0.7366, and there exist unstable spatially homogenous periodic solutions that are bifurcating from the positive equilibrium (u * , v * ) when the parameter c = 0.75 > 0.7366, which is shown in Figs. 1 and 2. For system (1.2), choosing to set the values d 1 = 0.05, d 2 = 0.1, γ = 1, c = 0.75, β = 0.6, together with a very direct computation, we have also obtained the critical value of delay for the parameter k = 0, that is, τ * = 0.9363, κ k1 = 0.0625, κ k2 = -0.1026. We have concluded that the positive equilibrium (u * , v * ) = (0.4094, 0.3517) is asymptotically stable for the parameter τ = 0.9063 < 0.9363 and unstable for τ = 0.9463 > 0.9363. So, system (1.2) undergoes Hopf bifurcation near the positive equilibrium (u * , v * ) at the time when the delay τ increasingly exceeds the critical value τ * . By computing κ 01 = 0.0625, κ 02 = -0.1026, and combining with the results in Sect. 3, we know that the direction of Hopf bifurcation is supercritical, and the bifurcating periodic solution is stable, and there also exist stable spatially homogenous periodic solutions, which is shown in Figs. 3 and 4.