Delay-induced periodic phenomenon in a diffusive regulated logistic model

The diffusive logistic growth model with time delay and feedback control is considered. First, the well-posedness and permanence of solutions are discussed by using some comparison techniques. Then, the sufficient conditions for stability of nonnegative constant steady states are established, and the occurrence of Hopf bifurcation at positive steady state is performed. Next, the bifurcation properties are derived by computing the normal form on center manifold. Our results not only supplement but also generalized some existing ones. Finally, some numerical simulations show the feasibility of our theoretical analyses.


Background
The classic logistic model was first proposed by Verhulst in 1838. It can be utilized to describe the single-species growth and has been the basis of varieties of models in population ecology and epidemiology. For system (1) and its generalized forms, the significant results involve the asymptotic properties (Berezansky et al. 2004;Röst 2011), permanence and stability (Fan and Wang 2010;Chen et al. 2006), periodicity (Sun and Chen 2007) and almost periodicity (Yang and Yuan 2008) of solutions, Hopf bifurcation Song and Yuan 2007;Song and Peng 2006;Chen and Shi 2012), traveling wave front (Zhang and Sun 2014), free boundary problem (Gu and Lin 2014), and so on. In addition, the Hopf bifurcation analyses for some diffusive predator-prey systems were also done (see Yang 2015;Yang and Zhang 2016a, b).
In particular, Gopalsamy (1993) considered the controlled delay system in the following form where all the coefficients and time delay τ are positive constants, N(t) is the number of individuals at time t, and variable u(t) denotes an indirect control variable (see Aizerman and Gantmacher 1964;Lefschetz 1965). They have derived the sufficient conditions to guarantee that the positive equilibrium solution is globally asymptotical stable. Strictly speaking, spatial diffusion can not be ignored in studying the natural biological system (Murray 2003;Ghergu and Radulescu 2012). In the real world, most populations are moving and the densities are dependent of time and space. Therefore, diffusion should be taken into account in studying the basic logistic equation. However, there have been very few results on the influence of time delay on the reaction-diffusion logistic model with feedback control.
The model (3) is considered with the initial value conditions as follows We also assume that the model (3) is closed and there is no emigration or immigration across the boundary. Hence, the boundary conditions are considered as where ∂/∂ν represents the outward normal derivative on the boundary ∂Ω.
In this paper, we develop a reaction-diffusion logistic model with time delay and diffusion, which makes up perfectly for the deficiencies of the previous literatures. The main objective is to explore the dynamics of system (3) by regarding τ as the bifurcation parameter. The structure of this paper is arranged as follows. In section "Preliminaries", we derive the well-posedness of solutions and the permanence of the system. In section "Occurrence of the Hopf bifurcation", we establish the existence of Hopf bifurcation. In section "Bifurcation properties", we get the formulae for determining the Hopf bifurcation properties. In section "Numerical simulations", we illustrate our theoretical results by some numerical simulations. Finally, we give some discussions and conclusions.

Preliminaries
As we know, spatial diffusion and time delay do not change the number and locations of constant equilibria because of no-flux boundary conditions. Then system (3) has two nonnegative equlibria E 0 = (0, 0) and E * = (N * , u * ), where N * = aK a(a 1 + a 2 ) + bcK , u * = b a N * = bK a(a 1 + a 2 ) + bcK .
By means of the comparison theorem, we can obtain that 0 ≤ N (x, t) ≤ M 1 and 0 ≤ u(x, t) ≤ M 2 for x ∈Ω and t ∈ [0, T ). It is obvious that the upper bound of solution is independent of the maximal existence interval [0, T). It follows from the standard theory for semilinear parabolic systems (Wu 1996;Henry 1993) that the solution globally exists. The proof is complete

Dissipativeness and permanence
In the following, we will show that system (3) is permanent, which means that any nonnegative solution of (3) is bounded as t → +∞ for all x ∈ Ω.
Theorem 2 (Dissipativeness) The nonnegative solution (N, u) of system (3) satisfies Proof Based on the first equation in system (3), we get Then from the standard comparison principle of parabolic equations, we can easily get For an arbitrary ε 1 > 0, we could get a positive constant T 1 such that for any t ≥ T 1 , Thus, for any T ∈ [T 1 + τ , +∞), we have This implies by comparison principle of parabolic equations and the arbitrariness of ε 1 .
Proof From Theorem 2, for an arbitrary ε 2 > 0, we can find a constant T > T 1 + T 2 , such that in Ω × [T 2 , +∞). Moreover, we can obtain the comparison principle shows that due to the continuity as ε 1 → 0 and ε 2 → 0. Similarly, we can also have Combining the results in Theorem 2, we can easily conclude that system (3) is permanent.

Occurrence of the Hopf bifurcation
For system (3), we shall study the local stability of two constant steady states and the occurrence of Hopf bifurcation phenomenon through discussing the distribution of characteristic values. Denote By defining the phase space C = C([−τ , 0], X), we can rewritten system (3) as the semilinear functional differential equation: The characteristic equation of (7) is 0]. We know that the operator ∆ in Ω with homogeneous Neumann boundary condition has the eigenvalues −n 2 /l 2 and the corresponding eigenfunctions cos(nx/l), n ∈ N 0 = {0, 1, 2, . . .}. By using the Fourier expansion in (8), where α n , γ n ∈ C. Therefore, the characteristic equation (8) can be transferred into We then obtain the characteristic values as follows It is obvious that 1,0 = r > 0, and we can establish the instability of E 0 .

Theorem 4
The trivial equilibrium E 0 of system (3) is always unstable.
Next, we will focus on the occurrence of Hopf bifurcation phenomenon. Linearizing system (3) at E * = (N * , u * ), we get where L : C → X is given by Similar to the previous discussion, we can obtain the characteristic equation where For τ = 0, Eq. (10) can be reduced to with A n + C > 0 and B n + D > 0. On the basis of Routh-Hurwitz stability criterion, we can obtain the local stability of E * when τ = 0.

Lemma 1
The positive equilibrium is always locally asymptotically stable without time delay.
Remark 1 From Lemma 1, we can find that there is no Turing instability without time delay.
For τ � = 0, let us suppose that = iω(ω > 0) satisfies Eq. (10). First, plugging = iω into Eq. (10) and then segregating the real and imaginary components with the help of Euler's formula, we can get the following two equations of ω Second, solving these equations, we can obtain Third, squaring both sides of those two equations and then adding them up, we get the following equation ω 2 − B n = D n cos ωτ + Cω sin ωτ , −ωA n = Cω cos ωτ − D n sin ωτ .
where Lemma 2 For τ > 0, we have (i) If a 1 > a 2 + bcK ar , then Eq. (10) does not have purely imaginary root. (ii) If a 2 < a 1 < a 2 + bcK ar , then there exists N 0 ∈ N 0 , such that Eq. (10) does not have purely imaginary root when n > N 0 , and has a pair of conjugate purely imaginary eigenvalues when 0 ≤ n ≤ N 0 .
Proof We can easily verify that A 2 n − 2B n − C 2 > 0 and B 2 n − D 2 n > 0 when a 1 > a 2 + bcK ar . This means that Eq. (12) has no positive root. In other words, there could be no purely imaginary root in Eq. (10) for any τ > 0.
On the contrary, if a 2 < a 1 < a 2 + bcK ar , then B 2 0 − D 2 0 < 0 and there exists N ∈ N 0 such that That is to say, Eq. (12) has no positive root when n > N 0 and has the unique positive root ω n when 0 ≤ n ≤ N 0 , where By direct computation, we have Moreover, Eq. (10) has characteristic values ±iω n with where A 2 n − 2B n − C 2 = d 2 1 + d 2 2 n 4 l 4 + 2 ra 1 K N * d 1 + ad 2 n 2 l 2 + r 2 a 2 1 − a 2 2 K 2 N * 2 , This completes the proof. We now check the transversality condition.
Proof By taking the derivatives on both sides of (10) with respect to τ, we can get and On the basis of (11) and (12), we get Further simplification will lead to The proof is complete.

Bifurcation properties
In Theorem 5, we have demonstrated that there exist some spatially homogeneous or inhomogeneous periodic solutions when time delay crosses through some particular values. We are now in the position to investigate the bifurcation properties.
In general, we use τ * to denote an arbitrary value of τ (n) j with j ∈ N 0 and n ∈ {0, 1, 2, . . . , N 0 }. And we also use ±iω * to denote the corresponding simply purely imaginary roots ±iω n .

Discussions and conclusions
In this paper, we considered the reaction-diffusion regulated logistic growth model. We have investigated the basic properties and Hopf bifurcation under the Neumann boundary conditions. It is shown that the logistic model may undergo Hopf bifurcation when time delay varies. We further give the formulae for determining the bifurcation properties, such as the direction of bifurcation, the stability of periodic solution and the monotonicity of period of periodic solution.
Here, we only discussed the single-species diffusive model with feedback control. In fact, how spatial diffusion and time delay affect the dynamic behaviors of multi-species controlled model remains unclear. We will focus on these novel and interesting models in the future.
Furthermore, from the numerical simulations in section "Numerical simulations", we conjecture that the Hopf bifurcation induced by time delay is global. This means that the periodic solutions due to Hopf bifurcation still exist even if the time delay is sufficiently large.