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.
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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 (Sun et al. 2007; 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 \(\tau\) 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.
Inspired by the previous discussions, we mainly consider the reaction–diffusion system as follows:
where \((x,t)\in \varOmega \times [0,+\infty )\), \(\varOmega =(0,l\pi )\).
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 \(\partial /\partial \nu\) represents the outward normal derivative on the boundary \(\partial \varOmega\).
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
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
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.
It is easy to find that \(\mathbf {0}=(0,0)\) and \(\mathbf {M}=(M_1,M_2)\) are a pair of coupled lower–upper solutions to problem (3)–(5), where
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
(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 \(\varepsilon _1>0\), we could get a positive constant \(T_1\) such that for any \(t\ge T_1\),
Thus, for any \(T\in [T_1+\tau ,+\infty )\), we have
This implies
by comparison principle of parabolic equations and the arbitrariness of \(\varepsilon _1\). \(\square\)
Theorem 3
If \(aa_1>aa_2+bcK\), then system (3) is permanent.
Proof
From Theorem 2, for an arbitrary \(\varepsilon _2>0\), we can find a constant \(T>T_1+T_2\), such that
in \(\varOmega \times [T_2,+\infty )\). Moreover, we can obtain
the comparison principle shows that
due to the continuity as \(\varepsilon _1 \rightarrow 0\) and \(\varepsilon _2 \rightarrow 0\).
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 \({\mathcal {C}}=C([-\tau ,0],X)\), we can rewritten system (3) as the semilinear functional differential equation:
where \(X=\{ (u,v)\in H^2(0,l\pi )\times H^2(0,l\pi )|u_x=v_x=0,x=0,l\pi \}\), \(U_t(\cdot )=U(t+\cdot )\), \(D=\text{ diag } \{d_1,d_2\}\), \(\varDelta =\text{ diag }\{ \partial ^2/\partial x^2, \partial ^2/\partial x^2 \}\), and \(G(U_t):{\mathcal {C}}\rightarrow X\) is defined by
The linear system of (6) at \(E_0(0,0)\) is
where
for \(\varphi (\theta )=U_t(\theta )\), \(\varphi =(\varphi _1,\varphi _2)^T\in {\mathcal {C}}\). The characteristic equation of (7) is
where \(y\in \text{ dom }(\varDelta )\backslash \{0\}\), \(\text{ dom }\varDelta \subset X\) and \(e^{\lambda \cdot }(\theta )y=e^{\lambda \theta }y\) for \(\theta \in [-\tau ,0]\). We know that the operator \(\varDelta\) in \(\varOmega\) with homogeneous Neumann boundary condition has the eigenvalues \(-n^2/l^2\) and the corresponding eigenfunctions \(\cos (nx/l)\), \(n\in {\mathbb {N}}_0=\{0,1,2,\ldots \}\). By using the Fourier expansion in (8),
where \(\alpha _n\), \(\gamma _n\in {\mathbb {C}}\). Therefore, the characteristic equation (8) can be transferred into
We then obtain the characteristic values as follows
It is obvious that \(\lambda _{1,0}=r>0\), 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 \(L:{\mathcal {C}}\rightarrow X\) is given by
with \(\varphi (\theta )=U_t(\theta )\), \(\varphi =(\varphi _1,\varphi _2)^T\in {\mathcal {C}}\). Similar to the previous discussion, we can obtain the characteristic equation
where
For \(\tau =0\), 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 \(\tau =0\).
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 \(\tau \ne 0\), let us suppose that \(\lambda =i\omega (\omega >0)\) satisfies Eq. (10).
First, plugging \(\lambda =i\omega\) 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 \(\omega\)
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
where
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 \(\tau >0\).
On the contrary, if \(a_2<a_1<a_2+\frac{bcK}{ar}\), then \(B_0^2-D_0^2<0\) and there exists \(N\in {\mathbb {N}}_0\) such that
That is to say, Eq. (12) has no positive root when \(n>N_0\) and has the unique positive root \(\omega _n\) when \(0\le n \le N_0\), where
By direct computation, we have
Moreover, Eq. (10) has characteristic values \(\pm i\omega _n\) with
where
This completes the proof. \(\square\)
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
By taking the derivatives on both sides of (10) with respect to \(\tau\), we can get
and
On the basis of (11) and (12), we get
Further simplification will lead to
The proof is complete. \(\square\)
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 \(\pm i\omega _n\).
Set \(\tilde{N}(\cdot ,t)=N(\cdot ,\tau t)\), \(\tilde{u}(\cdot ,t)=u(\cdot ,\tau t)\), \(\tilde{U}(t)=(\tilde{N}(\cdot ,t),\tilde{u}(\cdot ,t))\), and \(\tau =\tau ^{*}+\alpha\) with \(\alpha \in {\mathbb {R}}\). For simplicity we drop the tilde and rewrite system (3) as follows,
where \(\varphi =(\varphi _1,\varphi _2)^T\in {\mathcal {C}}\), \(L(\alpha )(\cdot ): {\mathcal {C}}\rightarrow X\) and \(f: {\mathcal {C}}\times {\mathbb {R}}\rightarrow X\) are respectively denoted by
and
Note that \(\alpha =\tau -\tau ^{*}\), we can find that system (13) may causes a Hopf bifurcation when \(\alpha =0\).
For the following linear differential equation:
we can easily deduce that the corresponding characteristic equation has characteristic values \(\pm \text{ i }\omega ^{*}\tau ^{*}\) when \(\alpha =0\).
Next, we discuss the following differential equation:
We can use Riesz representation theorem here, which tells us that there is a \(2\times 2\) matrix function \(\eta (\theta , \alpha )\) \((-1\le \theta \le 0)\) with bounded variation elements satisfying
where
For \(\varPhi \in C^1([-1,0],{\mathbb {R}}^2)\), \(\varPsi \in C^1([0,1],{\mathbb {R}}^2)\), we define
Then the formal adjoint, \(A_1^{*}\), of \(A_1\) is given by
By calculation, we can find that \(q(\theta )=(1,\xi )^Te^{i\omega ^{*}\theta \tau ^{*}}\) and \(q^{*}(s)=M(1,\eta )e^{i\omega ^{*} s \tau ^{*}}\) are eigenvectors of \(A_1\) and \(A_1^{*}\) associated with \(i\omega ^{*}\tau ^{*}\), respectively, where
and
Then \(P=span\{ q(\theta ),\overline{q(\theta )} \}\), \(P^{*}=span\{ q^{*}(s),\overline{q^{*}(s)} \}\) are the center subspace of system (3).
Define \(h\cdot f_n=h_1\beta _n^1+h_2\beta _n^2\), \(f_n=\left( \beta _n^1,\beta _n^2\right)\) and \(\beta _n^1=\left( \cos \frac{nx}{l},0\right) ^T\), \(\beta _n^2=\left( 0,\cos \frac{nx}{l}\right) ^T\). The complex-valued \(L^2\) inner product on Hilbert space \(X_C\) are
for \(U_1=(u_1,u_2), U_2=(v_1,v_2)\in X_C\). And \(\langle \beta _0^i, \beta _0^i \rangle =1\), \(\langle \beta _n^i, \beta _n^i \rangle =\frac{1}{2}\), \(i=1,2\), \(n=1,2,\ldots\),
where \(\varPhi \in C([-1,0],X)\). We can establish the center subspace of system (14) at \(\alpha =0\) as follows
Based on the conclusions drawn by Wu (1996) and Hassard et al. (1981), the solutions of (13) are
where
Moreover, for \(U_t\in C_0\) of (13) at \(\tau =\tau ^{*}\), we have \(\dot{z}=i\omega ^{*}\tau ^{*} z+g(z,\bar{z})\), where
Then we should compute \(W_{20}(\theta )\) and \(W_{11}(\theta )\) to determine \(g_{21}\). Following the formulas in Wu (1996), We can obtain that
and
From the previous expressions of \(g_{20}\), \(g_{11}\), \(g_{02}\) and \(g_{21}\), we can further compute
On account of preceding calculations, we arrive at the following conclusion on the bifurcation properties.
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.
For system (3), let \(\varOmega =(0,2\pi )\) and choose
and the initial values \(N(x,0)=0.5\) and \(u(x,0)=0.9\). Then we can get the positive equilibrium \(E^{*}=(0.2,0.2)\). By direct computation, we have \(N_0=0\), \(\omega _0\approx 0.348266\), and \(\tau _0^{(0)}\approx 5.81966\), then the Hopf bifurcation values are given by
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).
In the light of these simulations, we can find that spatially periodic solutions still exist even when \(\tau =50\in (\tau _2^{(0)}, \tau _3^{(0)})\) and \(\tau =130\in (\tau _6^{(0)},\tau _7^{(0)})\) (see Figs. 3, 4).
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.
References
Aizerman MA, Gantmacher FR (1964) Absolute stability of refgulator systems. Holden Day, San Francisco
Berezansky L, Braverman E, Idels L (2004) Delay differential logistic equations with harvesting. Math Comput Model 39:1243–1259
Chen F, Xie X, Chen X (2006) Permanence and global attractivity of a delayed periodic logistic equation. Appl Math Comput 177:118–127
Chen S, Shi J (2012) Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect. J Diff Eq 253:3440–3470
Fan Y, Wang L (2010) Global asymptotical stability of a Logistic model with feedback control. Nonlinear Anal Real World Appl 11:2686–2697
Ghergu M, Radulescu VD (2012) Nonlinear PDEs: mathematical models in biology. Chemistry and population genetics. Springer, Heidelberg
Gopalsamy K (1993) Peixuan weng, feedback regulation of logistic growth. Intern J Math Math Sci 16:177–192
Hassard B, Kazarinoff N, Yan Y (1981) Theory and applications of Hopf bifurcation. Cambridge University Press, Cambridge
Hattaf K, Yousfi N (2015) A generalized HBV model with diffusion and two delays. Comput Math Appl 69:31–40
Hattaf K, Yousfi N (2015) Global dynamics of a delay reaction-diffusion model for viral infection with specific functional response. Comput Appl Math 34:807–818
Henry D (1993) Geometric theory of semilinear parabolic equations. Springer, Berlin
Gu H, Lin Z (2014) Long time behavior of solutions of a diffusion-advection logistic model with free boundaries. Appl Math Lett 37:49–53
Lefschetz S (1965) Stability of nonlinear control systems. Academic Press, New York
Murray JD (2003) Mathematical biology II: spatial models and biomedical applications. Springer, New York
Röst G (2011) On an approximate method for the delay logistic equation. Commun Nonlinear Sci Numer Simul 16:3470–3474
Song Y, Peng Y (2006) Stability and bifurcation analysis on a Logistic model with discrete and distributed delays. Appl Math Comput 181:1745–1757
Song Y, Yuan S (2007) Bifurcation analysis for a regulated logistic growth model. Appl Math Model 31:1729–1738
Sun C, Han M, Lin Y (2007) Analysis of stability and Hopf bifurcation for a delayed logistic equation. Chaos Solitons Fractals 31:672–682
Sun S, Chen L (2007) Existence of positive periodic solution of an impulsive delay Logistic model. Appl Math Comput 184:617–623
Wu J (1996) Theory and applications of partial functional differrential equations. Springer, New York
Yang R (2015) Hopf bifurcation analysis of a delayed diffusive predator-prey system with nonconstant death rate. Chaos Solitons and Fractals 81:224–232
Yang X, Yuan R (2008) Global attractivity and positive almost periodic solution for delay logistic differential equation. Nonlinear Anal 68:54–72
Yang R, Zhang C (2016a) The effect of prey refuge and time delay on a diffusive predator-prey system with hyperbolic mortality. Complexity 50:105–113
Yang R, Zhang C (2016b) Dynamics in a diffusive predator-prey system with a constant prey refuge and delay. Nonlinear Anal Real World Appl 31:1–22
Zhang J, Sun Y (2014) Dynamical analysis of a logistic equation with spatio-temporal delay. Appl Math Comput 247:996–1002
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).
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The authors declare that they have no competing interests.
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Zhuang, K., Jia, G. Delay-induced periodic phenomenon in a diffusive regulated logistic model. SpringerPlus 5, 1303 (2016). https://doi.org/10.1186/s40064-016-2958-y
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DOI: https://doi.org/10.1186/s40064-016-2958-y