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# Delay-induced periodic phenomenon in a diffusive regulated logistic model

- Kejun Zhuang
^{1, 2}Email author and - Gao Jia
^{3}

**Received:**10 May 2016**Accepted:**29 July 2016**Published:**9 August 2016

## Abstract

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.

## Keywords

- Logistic model
- Feedback control
- Hopf bifurcation
- Reaction–diffusion system
- Delay

## Mathematics Subject Classification

- 35K57
- 35B10
- 35B32

## Background

*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.

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 \(\tau\) 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

### Well–posedness of solutions

Here, for problem (3)–(5), we devote ourselves to the existence, uniqueness, nonnegativity and boundedness of solutions.

###
**Theorem 1**

*For any given initial data satisfying the conditions* (4) *and boundary conditions* (5), *system* (3) *has a unique global solution of system and the solution maintains nonnegative and uniformly bounded for all*
\(t\ge 0\).

###
*Proof*

Using the similar methods in Hattaf and Yousfi (2015), Hattaf and Yousfi (2015), we can get the local existence and uniqueness of solution (*N*(*x*, *t*), *u*(*x*, *t*)) with \(x\in \bar{\varOmega }\) and \(t\in [0,T)\), where *T* is the maximal existence time of solution.

By means of the comparison theorem, we can obtain that \(0\le N(x,t)\le M_1\) and \(0\le u(x,t)\le M_2\) for \(x\in \bar{\varOmega }\) and \(t\in [0, T)\). 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. \(\square\)

### 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 \(t\rightarrow +\infty\) for all \(x\in \varOmega\).

###
**Theorem 2**

*The nonnegative solution*(

*N*,

*u*)

*of system*(3)

*satisfies*

###
*Proof*

###
**Theorem 3**

*If*
\(aa_1>aa_2+bcK\), *then system* (3) *is permanent.*

###
*Proof*

## 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.

###
**Theorem 4**

*The trivial equilibrium*
\(E_0\)
*of system* (3) *is always unstable.*

Next, we will focus on the occurrence of Hopf bifurcation phenomenon.

###
**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 \(\tau \ne 0\), let us suppose that \(\lambda =i\omega (\omega >0)\) satisfies Eq. (10).

###
**Lemma 2**

*For*
\(\tau >0\), *we have*

*(i) If*
\(a_1>a_2+\frac{bcK}{ar}\), *then Eq.* (10) *does not have purely imaginary root.*

*(ii) If*
\(a_2<a_1<a_2+\frac{bcK}{ar}\), *then there exists*
\(N_0 \in {\mathbb {N}}_0\), *such that Eq.* (10) *does not have purely imaginary root when*
\(n>N_0\), *and has a pair of conjugate purely imaginary eigenvalues when*
\(0\le n\le N_0\).

###
*Proof*

We can easily verify that \(A_n^2-2B_n-C^2>0\) and \(B_n^2-D_n^2>0\) when \(a_1>a_2+\frac{bcK}{ar}\). This means that Eq. (12) has no positive root. In other words, there could be no purely imaginary root in Eq. (10) for any \(\tau >0\).

We now check the transversality condition.

###
**Lemma 3**

*If*
\(a_2<a_1<a_2+\frac{bcK}{ar}\), *then*
\(\left. \frac{\text{ dRe }(\lambda ) }{\text{d}\tau }\right| _{\tau =\tau _j^{(n)}}>0\)
*for*
\(j\in {\mathbb {N}}_0\)
*and*
\(n\in \{ 0, 1, 2, \ldots , N_0 \}\) .

###
*Proof*

According to Lemmas 1–3 and the Hopf bifurcation theory developed by Wu (1996), the following conclusions can be drawn.

###
**Theorem 5**

*Define*

*(i) If*\(a_1>a_2+\frac{bcK}{ar}\),

*then for any*\(\tau >0\),

*the positive equilibrium*\(E^{*}\)

*is always locally asymptotically stable.*

*(ii) If*
\(a_2< a_1< a_2+\frac{bcK}{ar}\), *then*
\(E^{*}\)
*is locally asymptotically stable when*
\(\tau \in [0,\tau _0)\), *and is unstable when*
\(\tau \in (\tau _0,+\infty )\).

*(iii) System* (3) *has a Hopf bifurcation from*
\(E^{*}\)
*at*
\(\tau _j^{(n)}\)
*with*
\(n\in \{ 0,1,2,\ldots ,N_0 \}\)
*and*
\(j\in {\mathbb {N}}_0\). *If*
\(n=0\), *the periodic solutions bifurcating positive equilibrium are all spatially homogeneous. Otherwise, these bifurcating periodic solutions are spatially inhomogeneous.*

## 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 \(\tau ^{*}\) to denote an arbitrary value of \(\tau _j^{(n)}\) with \(j\in {\mathbb {N}}_0\) and \(n\in \{0,1,2,\ldots ,N_0\}\). And we also use \(\pm i\omega ^{*}\) to denote the corresponding simply purely imaginary roots \(\pm i\omega _n\).

Note that \(\alpha =\tau -\tau ^{*}\), we can find that system (13) may causes a Hopf bifurcation when \(\alpha =0\).

Then \(P=span\{ q(\theta ),\overline{q(\theta )} \}\), \(P^{*}=span\{ q^{*}(s),\overline{q^{*}(s)} \}\) are the center subspace of system (3).

###
**Theorem 6**

*The bifurcation direction is supercritical if*
\(\mu _2>0\), *which means that the periodic solution exists for*
\(\tau >\tau _0\). *On the contrary, the bifurcation direction is subcritical if*
\(\mu _2<0\), *which means that the periodic solution exists for*
\(\tau <\tau _0\).

*Moreover, the periodic solution is orbitally asymptotically stable if*
\(\beta _2<0\), *or unstable if*
\(\beta _2>0\). *The period of periodic solution is monotonically increasing at the time delay*
\(\tau\)
*when*
\(T_2>0\), *or is monotonically decreasing at the time delay*
\(\tau\)
*when*
\(T_2<0\).

## Numerical simulations

In this section, we give some numerical examples to test the preceding results with assistance of MATLAB.

Concretely, \(\tau _0=\tau _0^{(0)}\approx 5.81966\), \(\tau _1^{(0)}\approx 23.861\), \(\tau _2^{(0)}\approx 41.9024\), ... From Fig. 1, we can see the asymptotical stability of positive equilibrium \(E^{*}\) when time delay is slightly smaller than the first bifurcation value \(\tau _0\).

Moreover, we can obtain \(c_1(0)\approx -1.4328+1.53343i\). From Theorem 6, the Hopf bifurcation is supercritical, that is, the periodic solutions exist for \(\tau >\tau _0\), and they are orbitally asymptotically stable (see Fig. 2).

## 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.

## Declarations

### Authors' contributions

KZ carried out the genetic studies and drafted the manusctipt. GJ designed the structure of this paper and helped to draft the manuscript. Both authors read and approved the final manuscript.

### Acknowledgements

This work is supported by the National Natural Science Foundation of China (11171220) and the Hujiang Foundation of China (B14005). This work is also sponsored by Key Project for Excellent Young Talents Fund Program of Higher Education Institutions of Anhui Province (gxyqZD2016100).

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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