Stability and Hopf bifurcation of a diffusive Gompertz population model with nonlocal delay effect

In this paper, we investigate the dynamics of a diffusive Gompertz population model with nonlocal delay effect and Dirichlet boundary condition. The stability of the positive spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcations with the change of the time delay are discussed by analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system. Then we derive the stability and bifurcation direction of Hopf bifurcating periodic orbits by using the normal form theory and the center manifold reduction. Finally, we give some numerical simulations.


Introduction
The Gompertz equation is one of the models that are often used to describe the dynamics of the populations, including cellular populations of tumour growth, see [18,26,[28][29][30]37].The basic Gompertz model has the following form where V is simply the number of cells/individuals and K is the plateau number of cells/individuals.It was proposed by Benjamin Gompertz in 1825 for the first time (see [18]).Since Laird et al. [30] showed that the Gompertz model could describe the normal growth of an organism such as the guinea pig over an incredible 10000-fold range of the growth in [26], the Gompertz equation is often used in the formulation of equations describing the population dynamics and to describe the inner growth of tumour.In order to better describe the investigated phenomena, the time delays are often introduced into models [1-4, 7, 12-17, 31, 33, 34, 36].Literature [35] introduced the discrete time delay to the classical Gompertz model in different ways and obtained the following four models with delays: and it also introduced another model with two delays in which it separated two right-hand side terms describing two different processes, namely, the term r ln KV(t) (with K = 1) describing the growth of the population and the term −rV(t) ln V(t) describing the competition between individuals, and by using such biological interpretation, it proposed the model with two delays : where τ 1 and τ 2 reflect the delay of growth and competition, respectively.In [35], it showed that the model's dynamics depend crucially on the place where the delay/delays are included.
As the placement of delays in the models reflects the delays of different biological processes to their stimuli, so this conclusion is not surprising from the biological point of view.The mathematical and numerical analysis presented in it could help researchers who want to incorporate the Gompertz equation with delays into their models to choose the most appropriate version of the equation.Moreover, in mathematical biology, many models of population dynamics can be described by the delayed reaction-diffusion equations [6,8,9,20].In recent years, some researchers [27,32,39,41] have worked on the following reaction-diffusion equations with delay effect: = d∆u + u f (u(x, t − τ), v(x, t − τ)), ∂v ∂t = d∆v + vg(u(x, t − τ), v(x, t − τ)).
In a reaction-diffusion model with time-delay effect, the individuals which located at x in previous times may not be at the same point in space presently.So the diffusion and time delay are always not independent of each other for a delayed reaction-diffusion model (see References [5,10,11,19,21,22,24,40]). Thus, it is more reasonable to consider the diffusive type model with nonlocal delay.For instance, Britton [5] introduced the following model: where and analyzed the traveling waves on unbounded domain.Then Gourley and Britton [19] proposed a predator-prey system with spatiotemporal delay.In [10], Chen and Yu analyzed the following reaction-diffusion equation with spatiotemporal delay and homogeneous Dirichlet boundary condition: where e −dk 2 t sin kx sin ky, and f (t) is the delay kernel, satisfying f (t) ≥ 0, for t ≥ 0, and It is shown that a positive spatially nonhomogeneous equilibrium can bifurcate from the trivial equilibrium.Moreover, the stability of the bifurcated positive equilibrium was investigated.And they also proved that, for the given spatiotemporal delay, the bifurcated equilibrium is stable under some conditions, and Hopf bifurcation cannot occur.Chen and Yu [11] studied the following general form: Guo and Yan [24] investigated the following diffusive Lotka-Volterra type population model with nonlocal delay effect: for all x ∈ Ω and t > 0, where A ij , i, j = 1, 2, are kernel functions and The existence and multiplicity of spatially nonhomogeneous steady-state solutions are obtained by using Lyapunov-Schmidt reduction.Through analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system, we show the stability of spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcation with the changes of the time delay.The stability and bifurcation direction of Hopf bifurcating periodic orbits are derived by the normal form theory and the center manifold reduction.
In this paper, we investigate the following diffusive Gompertz population model with nonlocal delay effect: where w(x, t) is the population density at time t and location x, d > 0 is the diffusion coefficient, τ ≥ 0 is the time delay, λ > 0 is a scaling constant, Ω is a connected bounded open domain in R n (n ≥ 1), with a smooth boundary ∂Ω, and Dirichlet boundary condition is imposed so the exterior environment is hostile, ρ(λ) is the function of λ, K(x, y) is a kernel function which describes the dispersal behavior of the population.The nonlocal growth rate per capita incorporates the possible dispersal of the individuals during the maturation period, hence it is a more realistic model.
We first introduce some notations.Denote X = H 2 (Ω) For a space Z, we also define the complexification of Z to be Z For a linear operator L : Z 1 → Z 2 , we denote the domain of L by D(L).For the complex-valued Hilbert space Y 2 C , we use the standard inner product u, v = Ω ūT (x)v(x)dx.Let λ * be the principal eigenvalue of the linear operator −d∆ subject to the homogeneous Dirichlet boundary condition w = 0 on ∂Ω, and let φ be the corresponding eigenfunction of λ * such that φ(x) > 0 for all x ∈ Ω.
Throughout the paper, we assume that the kernel function K(x, y) is a continuous and nonnegative function on Ω × Ω, and Ω K(x, y)ϕ(y)dy > 0 for all positive continuous functions ϕ on Ω, and ρ(λ) = λ − λ * .When ρ(λ) = λ − λ * , the above model becomes (1.1) We consider system (1.1) with the following initial condition: where η ∈ C([−τ, 0], Y).From [25], we know that the operator d∆ generates an analytic strongly positive semigroup T(t) on Y with the domain D(d∆) = X.The rest of the paper is organized as follows.In Section 2, we study the existence of the positive spatially nonhomogeneous equilibrium of system (1.1).In Section 3, we consider the eigenvalue problems.In Section 4, we show the stability of the bifurcated positive equilibrium and the occurrence of Hopf bifurcation.In Section 5, the direction of the Hopf bifurcation is given by using normal form theorem and the center manifold theorem.Some numerical simulations are given in Section 6.

The existence of the positive spatially nonhomogeneous equilibrium
In this section, we study the existence of the spatially nonhomogeneous positive steady state solutions of system (1.1), which satisfies the following boundary value problem: (2.1) Firstly, we have the following decompositions: where Then we can obtain the following theorem about the existence of the positive equilibrium solutions of Eq. (2.1) by using the implicit function theorem.
The proof is similar to Theorem 2.5 of [8] and we omit it here.
To summarise, we have the following result about the eigenvalue problem.Corollary 3.2.For λ ∈ (λ * , λ * ], the eigenvalue problem has a solution if and only if and where c is a nonzero constant, and z λ , α λ , h λ , θ λ are defined as in Theorem 3.1. Next, we consider the adjoint operator of B τ (λ) for later application.Similar as in [8], we see that the adjoint operator is Its point spectrum is the same as that of ∆(λ, iω, τ): We conclude that if the corresponding adjoint equation Similarly, for λ ∈ (λ * , λ * ], there is a unique ( ω, θ, ψ) which is the solution to (3.7), ψ( = 0) ∈ X C .ψ can be represented as (3.9) Similarly to (3.5), we obtain , and zλ * ∈ (X 1 ) C is the unique solution of the following equation ).Then we have the following result which can be proved similarly as in Theorem 3.1 and Corollary 3.2.

Stability and Hopf bifurcations
In this section, we first give the stability of the positive equilibrium w λ of (1.1) when τ = 0 and then discuss the existence of Hopf bifurcations.
For n ≥ 1, we write ψ λ n as ψ λ n = c λ n w λ n + φ λ n , where c λ n ∈ C and c λ n = w λ n , ψ λ n / w λ n , w λ n .w λ n is the positive solution of (1.1) when λ = λ n , and φ λ n ∈ X C satisfies φ λ n , w λ n = 0.If φ λ n ≡ 0, then we substitute ψ λ n = c λ n w λ n and µ = µ λ n into the first equation of (4.1) and obtain which is a contradiction.Hence φ λ n ≡ 0 for each n ≥ 1.Since multiplying the first equation of (4.1) by ψ λ n = c λ n w λ n + φ λ n when µ = µ λ n , we can get As w λ n is the principal eigenfunction of B λ n with principal eigenvalue 0, so as n → ∞.Similar to the proof of Lemma 2.3 of [8], we get where and lim n→∞ φ λ n Y C = 0, then there exists N * ∈ N such that for each n ≥ N * , Re(E λ n ) > 0, which implies that Re(µ This is a contradiction with Re(µ λ n ) ≥ 0 for n ≥ 1.So all the eigenvalues of B τ (λ) have negative real parts when τ = 0.

Proof.
Suppose that there exists So there exists a constant a such that From the first equation of (4.4), we have Then from Eqs. (4.4) and (4.5), we can obtain By induction, we obtain Then we have the following transversality condition.
From the above conclusions, we have the following theorem.
Then we obtain the following theorem.

The direction of the Hopf bifurcation
Let W(t) = w(•, t) − w λ , τ = τ n + γ, then γ = 0 is the Hopf bifurcation value of system (1.1).Let t → t τ , then system (1.1) can be written in the following form dW(t) dt = τ n (d∆W(t) + L 0 (W(t))) + J(W t , γ) ( where Denote B τ n to be the infinitesimal generator of the linearized equation where Hence (5.1) can be written in the following abstract form where We know that B τ n has only one pair of purely imaginary eigenvalues ±iω λ τ n which are simple.The corresponding eigenfunction with respect to iω λ τ n (or −iω λ τ n ) is ψ λ (x)e iω λ τ n θ (or ψλ (x)e −iω λ τ n θ ) for θ ∈ [−1, 0].We introduce the formal duality where The operator B * τ n has only one pair of purely imaginary eigenvalues ±iω λ τ n which are simple, and the corresponding eigenfunction with respect to iω λ τ n (or −iω λ τ n ) is ψλ (x)e −iω λ τ n s (or ψλ (x)e iω λ τ n s ) for s ∈ [0, 1].So B * τ n and B τ n are adjoint operators under the bilinear form (5.3).The center subspace of (5.1) is P = span{p(θ), p(θ)}, where p(θ) = ψ λ e iω λ τ n θ is the eigenfunction of B τ n with respect to iω λ τ n .Similarly, the formal adjoint subspace of P with respect to the bilinear form (5.3) is P * = span{q(s), q(s)}, where q(s) = ψλ e iω λ τ n s is the eigenfunction of B * τ n with respect to −iω λ τ n .Then C C can be decomposed as where Id 2 is the identity matrix in R 2×2 .Define z(t) = Dq(s), W t , and denote then H(t, θ) ∈ Q.
On the center manifold, we have H(t, θ) = H(z, z, θ), where H(z, z, θ) can be expanded the power series of z and z, where z and z are local coordinates for the center manifold in the direction of q and q.The flow of (5.1) on the center manifold can be written as: (5.12) When θ ∈ [−1, 0), we have Solving this equation, we obtain Substituting E 1 into (5.13)gives H 20 (θ).Then we obtain We know that µ 2 determines the directions of the Hopf bifurcation: if µ 2 > 0 (µ 2 < 0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for τ > τ n (τ < τ n ); β 2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if β 2 < 0 (β 2 > 0); and T 2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if T 2 > 0 (T 2 < 0).
We avoid the complex computation and only show two numerical simulations of system (1.1).The numerical simulations with a homogeneous kernel K(x, y) = sin y and a nonhomogeneous kernel K(x, y) = (x − y) 2 are shown in Figure 6.1 and in Figure 6.2, respectively.In each figure, λ = 1.2, Ω = (0, π), d = 1, and the initial value is w(x, t) = 0.5 sin 2 x.In each case, the convergence to the spatially nonhomogeneous equilibrium w λ occurs when τ is less than the first Hopf bifurcation point τ 0 and an oscillatory pattern emerges for τ > τ 0 .Each simulation verifies the occurrence of spatially nonhomogeneous temporal oscillation and the spatial profiles of the periodic solutions are different due to the different dispersal kernels.