Stability and Hopf bifurcation for a delayed di ﬀ usive competition model with saturation e ﬀ ect

: This paper presents an investigation on the dynamics of a delayed di ﬀ usive competition model with saturation e ﬀ ect. We ﬁrst perform the stability analysis of the positive equilibrium and the existence of Hopf bifurcations. It is shown that the positive equilibrium is asymptotically stable under some conditions, and that there exists a critical value of delay, when the delay increases across it, the positive equilibrium loses its stability and a spatially homogeneous or inhomogeneous periodic solution emerges from the positive equilibrium. Then, we derive the formulas for the determination of the direction of Hopf bifurcation and the properties of the bifurcating periodic solutions. Finally, some numerical simulations are performed to illustrate the obtained results.


Introduction
The competition model is an important part in the study of population dynamics and has been discussed widely [1][2][3][4][5][6][7][8]. The classical two-dimension Lotka-Volterra competition model is constructed based on the law of mass action, that is, the competitive capacity of one species is assumed to be proportional to the number or density of its competitive species. Though the classical Lotka-Volterra competition model and its various extensions have been extensively studied in the literature [9][10][11][12][13][14][15][16], it has a deficiency that the competitive capacity of one species may increase and go to infinity with the increase of its competitors density.
Let u = u(t) and v = v(t) be respectively the densities of two competitive species. The following model with saturation competitive capacity has properly overcome the above deficiency of the classical Lotka-Volterra competition model: where a and d are the two species' respective intrinsic growth rates; b, e and c, f are their respective intra-and inter-specific interaction coefficients. The parameters here are all assumed to be positive. We refer the readers to references [17][18][19] for some related extension works. Due to the ubiquitousness of spatial diffusion in the real world, the spatiotemporal dynamics of the reaction-diffusion equation have recently been increasingly discussed by many authors [20,[22][23][24][25][26]. For example, one may focus on the pattern formation of biological tissues to reveal the mechanism behind the forms [26]. In addition, in order to better understand the interaction between prey and the predators, some diffusive population models are discussed to illustrate the influence of diffusion and the way in which different individuals interact (cooperation, competition, herd behaviors etc.) [20,[22][23][24][25]. To observe how the diffusion affects the dynamical behaviour of model (1), in a recent paper [21], we proposed and analyzed a diffusive competition model with saturated interaction terms, and showed that cross-diffusion is of great importance in the spatial pattern formation. Besides, time delay is another important component in constructing population models. Researchers investigated the effects of both diffusion and time delay, and have observed some interesting phenomena, such as Turing pattern [27][28][29], stability [30,31] and the occurrence of Hopf bifurcation induced by time delay [32][33][34].
Motivated by the above mentioned biological facts and the observed nonlinear phenomena, we propose the following diffusive competition model with delay: , where d 1 and d 2 are respectively the diffusion constants for the two species, and ∆ the usual Laplacian operator. The other symbols play similar roles as in model (1). The main objective of this paper is to study the effect of time delay on population dynamics of model (2). The conclusions show that delay may destabilize the stable positive equilibrium, resulting the spatially homogeneous or nonhomogeneous Hopf bifurcations. The organization of this paper is as follows. In Section 2, by analyzing the characteristic equation, the stability of positive equilibrium and the existence of Hopf bifurcations are studied. In Section 3, we obtain the formulae that determine the direction and stability of the spatial Hopf bifurcation by applying the normal form theory and the center manifold theorem. In Section 4, some numerical simulations are given to demonstrate the theoretical results, and finally, some conclusion and discussion are presented in Section 5.

Stability of the positive equilibrium and existence of Hopf bifurcations
We have shown in [21] that when [(a + b)d − a f ][(e + d)a − cd] > 0, model (1) has a unique positive constant equilibrium E * = (u * , v * ), which is global asymptotically stable provided otherwise it is unstable. Note that model (1) has the same constant equilibrium as that in model (2). Now we discuss the stability of E * and the existence of Hopf bifurcation for system (2). Under the hypothesis (H), we shift the equilibrium E * to the origin by lettingũ( . Therefore, we obtain the following system (dropping off the tildes) Then the linearized system corresponds to model where . We can rewrite model (4) as an abstract functional differential equation in C ([−τ, 0], X): where X = {u, v ∈ W 2,2 (0, lπ) : u x = v x = 0 for x = 0, lπ}, the linear operator L : C ([−τ, 0] , X) → X is given by with characteristic equation being where dom (∆) ⊂ X and (e λ· y)(θ) = e λθ y. Notice that the linear operator ∆ on X has the eigenvalues −k 2 (k ≥ 0). It follows from [35] that λ is a characteristic value of Eq (7) provided that λ satisfies Under condition (H), we have Let z = λ + k 2 e λτ , Eq (8) will be transformed into the following form: Because T 2 − 4D = (a 11 − a 22 ) 2 + 4a 12 a 21 > 0, Eq (9) has two negative real roots: When τ = 0, Eq (8) has two negative real eigenvalues λ 1 = −k 2 + z 1 , λ 2 = −k 2 + z 2 and λ 1 < λ 2 < 0 for any k. We can have the following lemma: Lemma 2.1. Suppose that condition (H) holds. Then all the eigenvalues of characteristic Eq (8) with τ = 0 have negative real parts for all k ≥ 0. Now we examine whether there is a positive constant τ * such that Re (λ n (τ * )) = 0. Let λ = ±ωi, ω > 0 be a pair of roots of Eq (10), then separating the real and imaginary parts, we obtain Since cos 2 ωτ + sin 2 ωτ = 1, the above equation is equivalent to then If k = 0, we can obtain that ω n = −z n , cos ωτ=0, sin ωτ = 0, which implies that Thus, when k = 0 and τ = τ n j , Eq (8) has a pair of pure imaginary roots ±iω n . If k 0, i.e. k ≥ 1. We have the following three cases.
In this case, since z 2 n < 1 ≤ k 4 , no positive ω satisfies the Eq (10). That is to say that all the eigenvalues of characteristic Eq (8) have negative real parts for all k ≥ 1. Noting that z 1 < z 2 < 0, therefore, τ 1 j < τ 2 j and τ n j < τ n j+1 . We have We have the following conclusion. (2) For τ = τ 0 , we only have a pair of pure imaginary roots ±iz 1 for Eq (8) that corresponds to k = 0, and all other roots have negative real parts; (3) For τ = τ n j (n = 1, 2; j = 0, 1, 2, · · · ), we only have a pair of pure imaginary roots ±iz n for each n = 1, 2corresponding to k = 0, and all other roots have negative real parts.
To summarize, we have the following result.

The linear system of Eq
By the Riesz representation theorem, there exists a 2 × 2 matrix function η (θ, µ), θ ∈ [−1, 0] with its elements being of bounded variation such that In fact, we can choose where δ(θ) is the Dirac function.

Numerical simulations
To support the theoretical results obtained in this paper, we illustrate some numerical simulations. We choose the parameters as a = 1.5, b = 1.1, c = 1.1, d = 1.1, e = 1.15, f = 0.5, d 1 = d 2 = 1.It is easy to check that condition (H) holds, which indicates that there exists a unique positive equilibrium and it is globally asymptotically stable.
By simple calculations, we can know that the positive equilibrium is E * = (0.9363, 0.7463) and z 1 = −1.1465 < −1 < z 2 = −0.7416. In this case, ω 0 = 1.1466 and τ 0 = 1.365. Based on Theorem 2.5, we know that the positive equilibrium E * is asymptotically stable when 0 ≤ τ < τ 0 and unstable when τ > τ 0 . And a spatially homogenous periodic solution emerges from the positive equilibrium E * when delay τ is increasing across the critical value τ 0 = 1.365. We can see that if τ = 1.2 < τ 0 , the positive equilibrium E * is stable and an uniform distribution can be observed (see Figure 1). If τ = 1.37 > τ 0 , the positive equilibrium E * lose its stability and a spatially homogeneous periodic distribution can be observed (see Figure 1). For the direction and stability of spatial Hopf bifurcation, we can obtain that µ 2 = 0.1691, β 2 = −0.4445 and T 2 = 0.1544 if τ = τ 0 . Due to µ 2 > 0, β 2 < 0, by Theorem 3.1, it is obvious that the directions of Hopf bifurcations are supercritical and the periodic solutions are stable. In addition, the period of periodic solutions increases with the increase of delay τ since T 2 > 0. These phenomenon also can be observed in Figure 2.

Conclusions and discussions
In this paper, we investigated the rich dynamics of a delayed diffusive competition model with saturation effect. It is known that the classical competition model does not have the Hopf bifurcation near the positive equilibrium E * . However, when the delay is included, our results indicate that the delayed diffusive competition model (2) show more complex dynamics, such as the existence of the Hopf bifurcation. For example, by taking delay τ as the bifurcation parameter, stability switches phenomenon of the positive equilibrium E * can be observed. We demonstrated that if τ < τ 0 , the positive equilibrium E * is asymptotically stable. It is unstable if delay τ > τ 0 . And we further proved that model (2) can generate spatially homogeneous and nonhomogeneous Hopf bifurcations when the delay τ passes through some critical values. Besides, by using the normal form theory and center manifold theorems, we analysed the directions of the Hopf bifurcations and the stability of the bifurcating periodic solutions and show the Hopf bifurcation of system (2) at the equilibrium E * when delay τ = τ 0 is supercritical and the periodic solution is asymptotically stable.
In fact, time delay can be found in various biological applications, such as infectious disease modeling, population modeling and the response of vegetation coverage to climate change so on. Some rich phenomenon can be induced by time delay. For example, time delay may induce the occurrence of spatiotemporal patterns [37,38]. In this paper, we only consider the occurrence of Hopf bifurcation induced by time delay. The global dynamics of the diffusive competition model still needs further study.