
In this study, we used 16S rRNA gene sequence analysis to describe the diversity of cultivable endophytic bacteria associated with fennel (Foeniculum vulgare Mill.) and determined their plant-beneficial traits. The bacterial isolates from the roots of fennel belonged to four phyla: Firmicutes (BRN1 and BRN3), Proteobacteria (BRN5, BRN6, and BRN7), Gammaproteobacteria (BRN2), and Actinobacteria (BRN4). The bacterial isolates from the shoot of fennel represented the phyla Proteobacteria (BSN1, BSN2, BSN3, BSN5, BSN6, BSN7, and BSN8), Firmicutes (BSN4, BRN1, and BRN3), and Actinobacteria (BRN4). The bacterial species Bacillus megaterium, Bacillus aryabhattai, and Brevibacterium frigoritolerans were found both in the roots and shoots of fennel. The bacterial isolates were found to produce siderophores, HCN, and indole-3-acetic acid (IAA), as well as hydrolytic enzymes such as chitinase, protease, glucanase, and lipase. Seven bacterial isolates showed antagonistic activity against Fusarium culmorum, Fusarium solani, and Rhizoctonia. solani. Our findings show that medicinal plants with antibacterial activity may serve as a source for the selection of microorganisms that exhibit antagonistic activity against plant fungal infections and may be considered as a viable option for the management of fungal diseases. They can also serve as an active part of biopreparation, improving plant growth.
Citation: Vyacheslav Shurigin, Li Li, Burak Alaylar, Dilfuza Egamberdieva, Yong-Hong Liu, Wen-Jun Li. Plant beneficial traits of endophytic bacteria associated with fennel (Foeniculum vulgare Mill.)[J]. AIMS Microbiology, 2024, 10(2): 449-467. doi: 10.3934/microbiol.2024022
[1] | Liqin Liu, Chunrui Zhang . A neural network model for goat gait. Mathematical Biosciences and Engineering, 2024, 21(8): 6898-6914. doi: 10.3934/mbe.2024302 |
[2] | Van Dong Nguyen, Dinh Quoc Vo, Van Tu Duong, Huy Hung Nguyen, Tan Tien Nguyen . Reinforcement learning-based optimization of locomotion controller using multiple coupled CPG oscillators for elongated undulating fin propulsion. Mathematical Biosciences and Engineering, 2022, 19(1): 738-758. doi: 10.3934/mbe.2022033 |
[3] | Jiacan Xu, Donglin Li, Peng Zhou, Chunsheng Li, Zinan Wang, Shenghao Tong . A multi-band centroid contrastive reconstruction fusion network for motor imagery electroencephalogram signal decoding. Mathematical Biosciences and Engineering, 2023, 20(12): 20624-20647. doi: 10.3934/mbe.2023912 |
[4] | Yong Yao . Dynamics of a delay turbidostat system with contois growth rate. Mathematical Biosciences and Engineering, 2019, 16(1): 56-77. doi: 10.3934/mbe.2019003 |
[5] | Changyong Xu, Qiang Li, Tonghua Zhang, Sanling Yuan . Stability and Hopf bifurcation for a delayed diffusive competition model with saturation effect. Mathematical Biosciences and Engineering, 2020, 17(6): 8037-8051. doi: 10.3934/mbe.2020407 |
[6] | Ranjit Kumar Upadhyay, Swati Mishra, Yueping Dong, Yasuhiro Takeuchi . Exploring the dynamics of a tritrophic food chain model with multiple gestation periods. Mathematical Biosciences and Engineering, 2019, 16(5): 4660-4691. doi: 10.3934/mbe.2019234 |
[7] | Qianqian Zheng, Jianwei Shen, Lingli Zhou, Linan Guan . Turing pattern induced by the directed ER network and delay. Mathematical Biosciences and Engineering, 2022, 19(12): 11854-11867. doi: 10.3934/mbe.2022553 |
[8] | Shunyi Li . Hopf bifurcation, stability switches and chaos in a prey-predator system with three stage structure and two time delays. Mathematical Biosciences and Engineering, 2019, 16(6): 6934-6961. doi: 10.3934/mbe.2019348 |
[9] | Jinhu Xu, Yicang Zhou . Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences and Engineering, 2016, 13(2): 343-367. doi: 10.3934/mbe.2015006 |
[10] | Ranjit Kumar Upadhyay, Swati Mishra . Population dynamic consequences of fearful prey in a spatiotemporal predator-prey system. Mathematical Biosciences and Engineering, 2019, 16(1): 338-372. doi: 10.3934/mbe.2019017 |
In this study, we used 16S rRNA gene sequence analysis to describe the diversity of cultivable endophytic bacteria associated with fennel (Foeniculum vulgare Mill.) and determined their plant-beneficial traits. The bacterial isolates from the roots of fennel belonged to four phyla: Firmicutes (BRN1 and BRN3), Proteobacteria (BRN5, BRN6, and BRN7), Gammaproteobacteria (BRN2), and Actinobacteria (BRN4). The bacterial isolates from the shoot of fennel represented the phyla Proteobacteria (BSN1, BSN2, BSN3, BSN5, BSN6, BSN7, and BSN8), Firmicutes (BSN4, BRN1, and BRN3), and Actinobacteria (BRN4). The bacterial species Bacillus megaterium, Bacillus aryabhattai, and Brevibacterium frigoritolerans were found both in the roots and shoots of fennel. The bacterial isolates were found to produce siderophores, HCN, and indole-3-acetic acid (IAA), as well as hydrolytic enzymes such as chitinase, protease, glucanase, and lipase. Seven bacterial isolates showed antagonistic activity against Fusarium culmorum, Fusarium solani, and Rhizoctonia. solani. Our findings show that medicinal plants with antibacterial activity may serve as a source for the selection of microorganisms that exhibit antagonistic activity against plant fungal infections and may be considered as a viable option for the management of fungal diseases. They can also serve as an active part of biopreparation, improving plant growth.
With the development of neuroscience, the controlling mechanism and mode of biological motion have been paid much attention by biologists [1,2,3,4], and rhythmic movement is a common mode of motion in biology. Rhythmic movement refers to periodic movement with symmetry of time and space, such as walking, running, jumping, flying, swimming and so on. Biologists have shown that rhythmic movement is not related to the consciousness of the brain, but to the self-excitation of the lower nerve centers. It is a spatiotemporal motion mode controlled by a central pattern generator located in the spinal cord of vertebrates or in the thoracic and abdominal ganglia of invertebrates [5]. They have the ability to automatically generate complex high dimensional control signals for the coordination of the muscles during rhythmic movements [6,7,8,9].
In engineering, CPG can be regarded as a distributed system consisting of a group of coupled nonlinear oscillators. The generation of rhythmic signals can be realized by phase coupling. Changing the coupling relationship of oscillators can produce spatiotemporal sequence signals with different phase relations, and realize different movement modes. CPG of animals lays a foundation for the research of bionic robots. For example, in [10,11] the gait control of quadruped robots based on CPG is studied. Mathematically, there are several common types of CPG oscillators systems, such as Hopf oscillators systems [12,13], Kimura oscillators systems, Rayleigh oscillators systems, Matsuoa oscillators systems and VDP oscillator systems [14,15], etc.
Quadrupedal gait is a kind of gait that people are very concerned. The gait of quadruped is an important type described by a symmetrical system [16,17,18]. For example, in [17,18], base on the symmetry property, the primary and secondary gait modes of quadruped are described, respectively. In animal gait movement, the legs are coupled with each other, and the coupling strength affects the complexity of animal gait. In this paper, the delay of leg signal is considered according to CPG model, the basic gait CPG model of a class of quadruped is constructed by using VDP oscillators, and the ranges of coupling strength between legs under four basic gaits are given. This paper is organized as follows. Firstly, a kind of delay CPG network system is constructed by using VDP oscillator. Secondly, the conditions of Hopf bifurcation in VDP-CPG network corresponding to the four basic gaits are given, and the coupling ranges between legs in four basic gaits are given. Finally, the theoretical results are supported by numerical simulations.
CPG, as the control center of rhythmic motion, is a kind of neural network that can generate the output of rhythmic mode without sensor feedback. It sends out motion instructions from the high-level center to control the initial state of rhythmic motion, and integrates the feedback information and perception information of CPG to regulate the motion organically. The CPG network in this paper adopts the following network structure [14].
In Figure 1, LF, RF, LH and RH represent the animal's left foreleg, right foreleg, left hind leg and right hind leg, respectively. The black arrows represent the leg raising sequence, and the numbers in the circles are the phase difference between other legs and LF leg. In order to generate the rhythmic signal of each leg, the VDP oscillator used in this paper can refer to [14], the equation is as follows.
{˙x=y,˙y=α(p2−x2)˙x−w2x, |
where x is the output signal from oscillator, α, p and w are variable parameters which can influence the character of oscillators. Commonly, the shape of the wave is affected by parameter α, and the amplitude of an output counts on the parameter p mostly. The output frequency is mainly relying on the parameter w when the amplitude parameter p is fixed. But the alteration of parameter p can lightly change the frequency of the signal, and α also can effect the output frequency.
Four-legged muscle groups are regarded as VDP oscillators for feedback motion signals, respectively. The animal's left foreleg, right foreleg, right hind leg and left hind leg are recorded as oscillator x1,x2,x3 and x4, respectively.
Then the oscillator of the ith leg is as follows
{˙xi=yi,˙yi=αi(p2i−x2ki)yi−w2ixki,i=1,2,3,4, |
where xki=xi+4∑j=1,j≠iKijxj denotes the coupling variable. Here Kij is the coupling coefficient, which represents strength of coupling from j oscillator to i oscillator.
Because the motion state of each leg depends on the motion state of the other three legs in the past short time, the time delay is introduced as follows
xki=xi(t)+4∑j=1,j≠iKijxj(t−τ). |
Assuming that the biological mechanism of each leg is similar and the degree of excitation or inhibition is the same between legs, and the excitation is positive coupling, then the inhibition is negative coupling. Therefore,
α1=α2=α3=α4=α, |
p1=p2=p3=p4=p, |
w1=w2=w3=w4=w, |
Kij={K,whenthejlegexcitestheileg,−K,whenthejlegrestrainstheileg.K>0. |
Thus, we study the following VDP-CPG system
{˙xi=yi,˙yi=α(p2−(xi(t)+4∑j=1,j≠iKijxj(t−τ))2)yi−w2(xi(t)+4∑j=1,j≠iKijxj(t−τ)), | (1) |
where i=1,2,3,4. It is clear that the origin (0, 0, 0, 0, 0, 0, 0, 0) is an equilibrium of Eq (1).
In this section, we construct a VDP-CPG network which is used for generation four basic gaits patterns (walk, trot, pace and bound). Then we analyze the conditions for four gait systems to produce Hopf bifurcation.
In order to analyses the four basic gaits, we make the following assumptions.
(H1) h<0,
(H2) 2s−h2>0,19m<K2<19,
(H3) K2<m,
where h=αp2,s=w2,m=4h2s−h44s2.
In walking gait, one leg is inhibited by the other three legs, then there are
Kij=−K,i,j=1,2,3,4,i≠j. |
So the VDP-CPG network in walking gait is as follows
{˙xi=yi,˙yi=α(p2−(xi(t)+4∑j=1,j≠i(−K)xj(t−τ))2)yi−w2(xi(t)+4∑j=1,j≠i(−K)xj(t−τ)). | (2) |
This is a symmetric system. We first explore the symmetry of system (2), then study the existence of Hopf bifurcation of system (2).
Let Yi=(xiyi)∈R2,i=1,2,3,4, system (2) can be written in block form as follows
˙Yi=MYi(t)+NYi+1(t−τ)+NYi+2(t−τ)+NYi+3(t−τ)+g(Yi(t)),i=1,2,3,4(mod4), | (3) |
where
M=(01−w2αp2),N=(00Kw20), |
g(xiyi)=(0−α(xi−Kxi+1(t−τ)−Kxi+2(t−τ)−Kxi+3(t−τ))2yi). |
Let Γ be a compact Lie group. It follows from [19], system ˙u(t)=G(ut) is said to be Γ− equivariant if G(γut)=γG(ut) for all γ∈Γ. Let Γ=D4 be the dihedral group of order 8, which is generated by the cyclic group Z4 of order 4 together with the flip of order 2. Denote by ρ the generator of the cyclic subgroup Z4 and k the flip. Define the action of D4 on R8 by
(ρU)i=Ui+1,(kU)i=U6−i,Ui∈R2,i=1,2,3,4(mod4). |
Then it is easy to get the following lemma.
Lemma 3.1. System (3) is D4− equivariant.
The linearization of Eq (3) at the origin is
˙Yi=MYi(t)+NYi+1(t−τ)+NYi+2(t−τ)+NYi+3(t−τ),i=1,2,3,4(mod4). | (4) |
The characteristic matrix of Eq (4) is given by
A(τ,λ)=(λI2−M−Ne−λτ−Ne−λτ−Ne−λτ−Ne−λτλI2−M−Ne−λτ−Ne−λτ−Ne−λτ−Ne−λτλI2−M−Ne−λτ−Ne−λτ−Ne−λτ−Ne−λτλI2−M), |
where I2 is a 2×2 identity matrix. This is a block circulant matrix, from [20], we have
det(A(τ,λ))=3∏j=0det(λI2−M−χjNe−λτ−(χj)2Ne−λτ−(χj)3Ne−λτ), |
where χj=eπj2i,i is the imaginary unit. The characteristic equation of Eq (4) at the zero solution is
Δ(τ,λ)=det(A(τ,λ))=Δ1(Δ2)3, | (5) |
with
Δ1=λ(λ−h)+s(1−3Ke−λτ)),Δ2=λ(λ−h)+s(1+Ke−λτ),h=αp2,s=w2. |
Lemma 3.2. If (H1) and (H2) hold, for the equation Δ1=0, we have the following results.
(1) when τ=0, all roots of equation Δ1=0 have negative real parts,
(2) when τ>0, there exist τj, such that when τ=τj(j=0,1,2,…), Δ1(±iβ)=0 holds,
(3) the transversality condition:
Re(dλdτ)|λ=iβ+,τ=τjwalk+>0,Re(dλdτ)|λ=iβ−,τ=τjwalk−<0, |
where
β=β±=√2s−h2±√(h2−2s)2−4s2(1−9K2)2, |
τj=τjwalk±=1β±(−arccoss−β2±3Ks+2jπ+2π),j=0,1,2,…. |
Proof. (1) When τ=0, equation Δ1=0 becomes λ(λ−h)+s(1−3K)=0, and the solution is obtained as follows
λ=h±√h2−4s(1−3K)2. |
By (H1) and (H2), the roots of equation Δ1=0 have negative real parts.
(2) When τ>0, let λ=iβ(β>0) be a root of Δ1=0. Substituting iβ into Δ1=0, then we have
−β2−iβh+s(1−3Ke−iβτ)=0. |
Separating the real and imaginary parts, we get the following form
{s−β2=3Kscos(βτ),βh=3Kssin(βτ). | (6) |
If (H2) holds, by solving the above equation, we have
β±=β=√2s−h2±√(h2−2s)2−4s2(1−9K2)2, | (7) |
τjwalk±=τj=1β(−arccoss−β23Ks+2jπ+2π),j=0,1,2,…. |
(3) Let λ(τ)=α(τ)+iβ(τ) be the root of equation Δ1=0, satisfying α(τj)=0 and β(τj)=β. Taking the derivative of the equation Δ1=0 with respect to τ, we can get
dλdτ=−3Ksλeλτ2λ−h+3Ksτeλτ. |
Then
Re(dλdτ)|λ=iβ,τ=τj=3Ksβhsin(βτj)−6Ksβ2cos(βτj)(−h+3Ksτcos(βτj))2+(2β−3Ksτjsin(βτj))2, |
by (6) and (7), we have
Re(dλdτ)|λ=iβ+,τ=τjwalk+>0,Re(dλdτ)|λ=iβ−,τ=τjwalk−<0, |
which means that the transversality condition holds at τjwalk±,j=0,1,2,….
The lemma 3.2 holds.
Lemma 3.3. For Δ2=0, we have the following results.
(1) if (H1) holds, when τ=0 all roots of equation Δ2=0 have negative real parts,
(2) if (H3) holds, when τ>0 equation Δ2=0 has no pure imaginary root.
Proof. (1) When τ=0, equation Δ2=0 becomes λ(λ−h)+s(1+K)=0, and the solution is obtained as follows
λ=h±√h2−4s(1+K)2. |
By (H1), the roots of equation Δ2=0 have negative real parts.
(2) When τ>0, let λ=iβ(β>0) be a root of Δ2=0. Substituting iβ into Δ2=0 then we have
−β2−iβh+s(1+Ke−iβτ)=0. |
The real and imaginary parts of the above equation are separated, then we obtain
{s−β2=−Kscos(βτ),βh=−Kssin(βτ). |
By solving the above equation, we have
β=√2s−h2±√(h2−2s)2−4s2(1−K2)2. |
By (H3), we obtain (h2−2s)2−4s2(1−K2)<0, then the formula above is not valid. So the lemma 3.3 holds.
From lemma 3.2 and 3.3, we have following theorem.
Theorem 3.1. If (H1), (H2) and (H3) hold, then we have the following results.
(1) all roots of Eq (5) have negative real parts for 0≤τ<τ0walk, and at least a pair of roots with positive real parts for τ∈(τ0walk,τ0walk+ε), for some ε>0,
(2) zero equilibrium of system (2) is asymptotically stable for 0≤τ<τ0walk, and unstable for τ∈(τ0walk,τ0walk+ε), for some ε>0,
(3) when τ=τ0walk, system (2) undergoes a Hopf bifurcation at zero equilibrium, where τ0walk=min{τ0walk+,τ0walk−}.
Remark 3.1. Near the critical value τ=τ0walk, the periodic solution of system (2) at the origin accords with walking gait.
In a trot, a leg on the same diagonal as the current leg stimulates the current leg, and two legs on the other diagonal suppress the current leg, thus
K12=−K,K13=K,K14=−K,K21=−K,K23=−K,K24=K,K31=K,K32=−K,K34=−K,K41=−K,K42=K,K43=−K. |
The VDP-CPG network for trotting is as follows.
{˙xi=yi,˙yi=αp2yi−w2(xi(t)+(−K)xi+1(t−τ)+Kxi+2(t−τ)+(−K)xi+3(t−τ))−α(xi(t)+(−K)xi+1(t−τ)+Kxi+2(t−τ)+(−K)xi+3(t−τ))2yi. | (8) |
This is also a symmetric system. Similarly, by lemma 3.1, we have
Lemma 3.4. System (8) is D4− equivariant.
The characteristic matrix of linearization of Eq (8) is given by
A1(τ,λ)=(λI2−M−Ne−λτNe−λτ−Ne−λτ−Ne−λτλI2−M−Ne−λτNe−λτNe−λτ−Ne−λτλI2−M−Ne−λτ−Ne−λτNe−λτ−Ne−λτλI2−M). |
This is a block circulant matrix, and we have
det(A1(τ,λ))=3∏j=0det(λI2−M−χjNe−λτ+(χj)2Ne−λτ−(χj)3Ne−λτ), |
with χj=eπj2i.
The characteristic equation of linearization of Eq (8) at zero solution is
Δ(τ,λ)=det(A1(τ,λ))=Δ3(Δ4)3, | (9) |
where
Δ3=λ(λ−h)+s(1+3Ke−λτ), |
Δ4=λ(λ−h)+s(1−Ke−λτ). |
Similarly, by lemma 3.2 and 3.3, we have following lemmas.
Lemma 3.5. For the equation Δ3=0, we have the following results.
(1) if (H1) holds, when τ=0, all roots of equation Δ3=0 have negative real parts,
(2) if (H2) holds, when τ>0, there exist τj, such that when τ=τj(j=0,1,2,…), Δ3(±iβ)=0 holds,
(3) the transversality condition:
Re(dλdτ)|λ=iβ+,τ=τjtrot+>0,Re(dλdτ)|λ=iβ−,τ=τjtrot−<0, |
where
β=β±=√2s−h2±√(h2−2s)2−4s2(1−9K2)2, |
τj=τjtrot±=1β±(arccoss−β2±−3Ks+2jπ),j=0,1,2,… |
Lemma 3.6. For Δ4=0, we have the following results.
(1) if (H1)and K<1 hold, when τ=0, all roots of equation Δ4=0 have negative real parts,
(2) if (H3) holds, when τ>0, equation Δ4=0 has no pure imaginary root.
From lemma 3.5 and 3.6, we have following theorem.
Theorem 3.2. If (H1), (H2) and (H3) hold, we have the following results.
(1) all roots of Eq (9) have negative real parts for 0≤τ<τ0trot, and at least a pair of roots with positive real parts for τ∈(τ0trot,τ0trot+ε), for some ε>0,
(2) zero equilibrium of Eq (8) is asymptotically stable for 0≤τ<τ0trot, and unstable for τ∈(τ0trot,τ0trot+ε), for some ε>0,
(3) when τ=τ0trot, system (8) undergoes a Hopf bifurcation at zero equilibrium, where τ0trot=min{τ0trot+,τ0trot−}
Remark 3.2. Near the critical value τ=τ0trot, the periodic solution of system (8) at the origin accords with trotting gait.
In a pace, the leg on the same side (left or right) of the current leg stimulates the current leg, and the other two legs inhibit the current leg, thus
K12=−K,K13=−K,K14=K,K21=−K,K23=K,K24=−K,K31=−K,K32=K,K34=−K,K41=K,K42=−K,K43=−K. |
Thus Eq (1) becomes the following VDP-CPG pacing system.
{˙xi=yi,˙yi=α(p2−(xi(t)+(−K)xi+1(t−τ)+(−K)xi+2(t−τ)+Kxi+3(t−τ))2)yi−w2(xi(t)+(−K)xi+1(t−τ)+(−K)xi+2(t−τ)+Kxi+3(t−τ)),i=1,3(mod4) | (10) |
{˙xi=yi,˙yi=α(p2−(xi(t)+(−K)xi−1(t−τ)+Kxi+1(t−τ)+(−K)xi+2(t−τ))2)yi−w2(xi(t)+(−K)xi−1(t−τ)+Kxi+1(t−τ)+(−K)xi+2(t−τ)),i=2,4(mod4) |
and the linearization of Eq (10) at the origin is
{˙xi=yi,˙yi=αp2yi−w2(xi(t)+(−K)xi+1(t−τ)+(−K)xi+2(t−τ)+Kxi+3(t−τ))i=1,3(mod4) | (11) |
{˙xi=yi,˙yi=αp2yi−w2(xi(t)+(−K)xi−1(t−τ)+Kxi+1(t−τ)+(−K)xi+2(t−τ)),i=2,4(mod4) |
the characteristic equation of system (11) is
|Rm−m−m+m−Rm+m−m−m+Rm−m+m−m−R|=Δ5(Δ6)3=0, | (12) |
where
Δ5=λ(λ−h)+s(1+3Ke−λτ),Δ6=λ(λ−h)+s(1−Ke−λτ). |
R=(λ−1w2λ−αp2),m+=(00Kw2e−λτ0),m−=(00−Kw2e−λτ0), |
Similarly, by theorem 3.1, we have following theorem.
Theorem 3.3. If (H1), (H2) and (H3) hold, we have the following results.
(1) all roots of Eq (12) have negative real parts for 0≤τ<τ0pace, and at least a pair of roots with positive real parts for τ∈(τ0pace,τ0pace+ε), for some ε>0,
(2) zero equilibrium of system (10) is asymptotically stable for 0≤τ<τ0pace, and unstable for τ∈(τ0pace,τ0pace+ε), for some ε>0,
(3) when τ=τ0pace, system (10) undergoes a Hopf bifurcation at zero equilibrium,
where
τ0pace=min{τ0pace+,τ0pace−}, |
τjpace±=1β±(arccoss−β2±−3Ks+2jπ),j=0,1,2,…, |
β±=√2s−h2±√(h2−2s)2−4s2(1−9K2)2. |
Remark 3.3. Near the critical value τ=τ0pace, the periodic solution of system (10) at the origin accords with pacing gait.
In a bound, legs on the same side (front or hind) as the current leg stimulate the current leg, and the other two legs inhibit the current leg, thus
K12=K,K13=−K,K14=−K,K21=K,K23=−K,K24=−K, |
K31=−K,K32=−K,K34=K,K41=−K,K42=−K,K43=K. |
Eq (1) becomes the following bounding VDP-CPG system.
{˙xi=yi,˙yi=α(p2−(xi(t)+Kxi+1(t−τ)+(−K)xi+2(t−τ)+(−K)xi+3(t−τ))2)yi−w2(xi(t)+Kxi+1(t−τ)+(−K)xi+2(t−τ)+(−K)xi+3(t−τ)),i=1,3(mod4) | (13) |
{˙xi=yi,˙yi=α(p2−(xi(t)+Kxi−1(t−τ)+(−K)xi+1(t−τ)+(−K)xi+2(t−τ))2)yi−w2(xi(t)+Kxi−1(t−τ)+(−K)xi+1(t−τ)+(−K)xi+2(t−τ)),i=2,4(mod4) |
and the linearization of Eq (13) at the origin is
{˙xi=yi,˙yi=αp2yi−w2(xi(t)+Kxi+1(t−τ)+(−K)xi+2(t−τ)+(−K)xi+3(t−τ)),i=1,3(mod4) | (14) |
{˙xi=yi,˙yi=αp2yi−w2(xi(t)+Kxi−1(t−τ)+(−K)xi+1(t−τ)+(−K)xi+2(t−τ)),i=2,4(mod4) |
the characteristic equation of system (14) is
|Rm+m−m−m+Rm−m−m−m−Rm+m−m−m+R|=Δ7(Δ8)3=0, | (15) |
where
Δ7=λ(λ−h)+s(1+3Ke−λτ),Δ8=λ(λ−h)+s(1−Ke−λτ). |
Similarly, by theorem 3.1, we have following theorem.
Theorem 3.4. If (H1), (H2) and (H3) hold, we have the following results.
(1) all roots of Eq (15) have negative real parts for 0≤τ<τ0bound, and at least a pair of roots with positive real parts for τ∈(τ0bound,τ0bound+ε), for some ε>0,
(2) zero equilibrium of system (13) is asymptotically stable for 0≤τ<τ0bound, and unstable for τ∈(τ0bound,τ0bound+ε), for some ε>0,
(3) when τ=τ0bound, system (13) undergoes a Hopf bifurcation at zero equilibrium,
where
τ0bound=min{τ0bound+,τ0bound−}, |
τjbound±=1β±(arccoss−β2±−3Ks+2jπ),j=0,1,2,…, |
β±=√2s−h2±√(h2−2s)2−4s2(1−9K2)2. |
Remark 3.4. Near the critical value τ=τ0bound , the periodic solution of system (13) at the origin accords with bounding gait.
In this section, the numerical simulation of model is carried out to verify the results obtained in the previous sections. Let α=−1.5, p=1,w=4, K=0.3, according to the calculation, we obtain the h=−1.5,s=16,m=0.1357,K2=0.09,19m=0.0151. Thus 2s−h2=29.7500>0,19m<K2<min{m,19} and the critical value τ0walk=0.7039, τ0trot=τ0pace=τ0bound=0.1103 are obtained. Basing on Theorem 3.2, we know the zero equilibrium is asymptotically stable when τ<τ0trot (shown in Figure 2a), when τ>τ0trot, the zero equilibrium of system (8) is unstable, and the periodic solution corresponding to the trot gait occurs (see Figure 2b). From theorem 3.3, we know the zero equilibrium is asymptotically stable when τ<τ0pace (shown in Figure 3a), when τ>τ0pace, the zero equilibrium of system (10) is unstable, and the periodic solution corresponding to the pace gait occurs (see Figure 3b). From theorem 3.4, we know the zero equilibrium is asymptotically stable when τ<τ0bound (shown in Figure 4a), when τ>τ0bound, the zero equilibrium of system (13) is unstable, and the periodic solution corresponding to the bound gait occurs (see Figure 4b).
In this paper, a kind of CPG network system is constructed by using VDP oscillators, and a VDP-CPG network system with four basic gaits (walk, trot, pace and bound) is presented. By studying the corresponding characteristic equations of four gaits systems, it is found that the conditions for the periodic solutions of four gaits systems are h<0,2s−h2>0 and 19m<K2<min{m,19} and the critical values τjwalk,τjtrot,τjpace andτjbound,j=0,1,2⋯. Thus, the range of coupling strength between legs in four gaits is 19m<K2<min{m,19}. Finally, the numerical simulations show that the gait systems (trot, pace and bound) produce corresponding gaits near the corresponding critical value.
This research is supported by the Fundamental Research Funds for the Central Universities (No.2572019BC12). The authors wish to express their gratitude to the editors and the reviewers for the helpful comments.
The authors declare there is no conflict of interest
[1] | Hui Xiang (2005) Flora of China. Tropicos Flora of China Checklist project 14: 134. |
[2] |
Ozbek H, Ugras S, Dulger H, et al. (2003) Hepatoprotective effect of Foeniculum vulgare essential oil. Fitoterapia 74: 317-319. https://doi.org/10.1016/s0367-326x(03)00028-5 ![]() |
[3] |
Faudale M, Viladomat F, Bastida J, et al. (2008) Antioxidant activity and phenolic composition of wild, edible, and medicinal fennel from different mediterranean countries. J Agric Food Chem 56: 1912-1920. https://doi.org/10.1021/jf073083c ![]() |
[4] |
Mohsenzadeh M (2007) Evaluation of antibacterial activity of selected Iranian essential oils against Staphylococcus aureus and Escherichia coli in nutrient broth medium. Pak J Biol Sci 10: 3693-3697. https://doi.org/10.3923/pjbs.2007.3693.3697 ![]() |
[5] |
Kaur GJ, Arora DS (2008) In-vitro antibacterial activity of three plants belonging to the family Umbelliferae. Int J Antimicrob Agents 31: 393-395. https://doi.org/10.1016/j.ijantimicag.2007.11.007 ![]() |
[6] | Abed KF (2007) Antimicrobial activity of essential oils of some medicinal plants from Saudi Arabia. Saudi J Biol Sci 14: 53-60. |
[7] |
Choi EM, Hwang JK (2004) Anti-inflammatory, analgesic and antioxidant activities of the fruit of Foeniculum vulgare. Fitoterapia 75: 557-565. https://doi.org/10.1016/j.fitote.2004.05.005 ![]() |
[8] |
Tognolini M, Ballabeni V, Bertoni S, et al. (2007) Protective effect of Foeniculum vulgare essential oil and anethole in an experimental model of thrombosis. Pharmacol Res 56: 254-260. https://doi.org/10.1016/j.phrs.2007.07.002 ![]() |
[9] | El-Soud NA, El-Laithy N, El-Saeed G, et al. (2011) Antidiabetic activities of Foeniculum vulgare Mill. essential oil in streptozotocin induced diabetic rats. Macedonian J Med Sci 173: 1857-5773. https://doi.org/10.3889/MJMS.1857-5773.2011.0173 |
[10] | Pradhan M, Sribhuwaneswari S, Karthikeyan D, et al. (2008) In-vitro cytoprotection activity of Foeniculum vulgare and Helicteres isora in cultured human blood lymphocytes and antitumour activity against B16F10 melanoma cell line. Res J Pharm Technol 1: 450-452. |
[11] | Reynolds JEF (1980) Essential oils and aromatic carminatives, Martindale-The Extra. Pharmacopeia, Royal Pharmaceutical Society, London . |
[12] | Shaker GA, Alhamadany HS (2015) Isolation and identification of fungi which infect fennel Foeniculum vulgare Mill. and its impact as antifungal agent. Bulletin of the Iraq Natural History Museum 13: 31-38. |
[13] |
Cacciola SO, Pane A, Cooke DEL, et al. (2006) First report of brown rot and wilt of fennel caused by Phytophthora megasperma in Italy. Plant Dis 90: 110. https://doi.org/10.1094/PD-90-0110A ![]() |
[14] |
Choi IY, Kim JH, Kim BS, et al. (2016) First report of Sclerotinia stem rot of fennel caused by Sclerotinia sclerotiorum in Korea. Plant Dis 100: 223. https://doi.org/10.1094/PDIS-05-15-0512-PDN ![]() |
[15] |
D'Amico M, Frisullo S, Cirulli M (2008) Endophytic fungi occurring in fennel, lettuce, chicory, and celery-commercial crops in southern Italy. Mycol Res 112: 100-107. https://doi.org/10.1016/j.mycres.2007.11.007 ![]() |
[16] | Egamberdieva D, Wirth S, Behrendt U, et al. (2017a) Antimicrobial activity of medicinal plants correlates with the proportion of antagonistic endophytes. Front Microbiol 8: 199. https://doi.org/10.3389/fmicb.2017.00199 |
[17] |
Egamberdieva D, Wirth S, Alqarawi AA, et al. (2017b) Phytohormones and beneficial microbes: Essential components for plants to balance stress and fitness. Front Microbiol 8: 2104. https://doi.org/10.3389/fmicb.2017.02104 ![]() |
[18] |
Rezaei-Chiyaneh E, Battaglia ML, Sadeghpour A, et al. (2021) Optimizing intercropping systems of black cumin (Nigella sativa L.) and fenugreek (Trigonella foenum-graecum L.) through inoculation with bacteria and mycorrhizal fungi. Adv Sustainable Syst 5: 2000269. https://doi.org/10.1002/adsu.202000269 ![]() |
[19] |
Pawlik M, Cania B, Thijs S, et al. (2017) Hydrocarbon degradation potential and plant growth-promoting activity of culturable endophytic bacteria of Lotus corniculatus and Oenothera biennis from a long-term polluted site. Environ Sci Pollut Res 24: 19640-19652. https://doi.org/10.1007/s11356-017-9496-1 ![]() |
[20] |
Egamberdieva D, Shurigin V, Alaylar B, et al. (2020a) Bacterial endophytes from horseradish (Armoracia rusticana G. Gaertn., B. Mey. & Scherb.) with antimicrobial efficacy against pathogens. Plant Soil Environ 66: 309-316. https://doi.org/10.17221/137/2020-PSE ![]() |
[21] |
Egamberdieva D, Shurigin V, Alaylar B, et al. (2020b) The effect of biochars and endophytic bacteria on growth and root rot disease incidence of Fusarium infested narrow-leafed lupin (Lupinus angustifolius L.). Microorganisms 8: 496. https://doi.org/10.3390/microorganisms8040496 ![]() |
[22] |
Nejatzadeh-Barandozi F (2013) Antibacterial activities and antioxidant capacity of Aloe vera. Bioorganic Med Chem Lett 3: 1-8. https://doi.org/10.1186/2191-2858-3-5 ![]() |
[23] |
Bafana A, Lohiya R (2013) Diversity and metabolic potential of culturable root-associated bacteria from Origanum vulgare in sub-Himalayan region. World J Microbiol Biotechnol 29: 63-74. https://doi.org/10.1007/s11274-012-1158-3 ![]() |
[24] |
Phetcharat P, Duangpaeng A (2012) Screening of endophytic bacteria from organic rice tissue for indole acetic acid production. Procedia Eng 32: 177-183. https://doi.org/10.1016/j.proeng.2012.01.1254 ![]() |
[25] |
Shurigin V, Egamberdieva D, Samadiy S, et al. (2020) Endophytes from medicinal plants as biocontrol agents against Fusarium caused diseases. Mikrobiolohichnyi Zh 82: 41-52. https://doi.org/10.15407/microbiolj82.04.041 ![]() |
[26] | Shurigin V, Alikulov B, Davranov K, et al. (2022) Bacterial endophytes from halophyte black saxaul (Haloxylon aphyllum Minkw.) and their plant growth-promoting properties. J Appl Biol Biotech 10: 45-53. https://doi.org/10.7324/JABB.2021.100106 |
[27] |
Koberl M, Ramadan EM, Adam M, et al. (2013) Bacillus and Streptomyces were selected as broad-spectrum antagonists against soilborne pathogens from arid areas in Egypt. FEMS Microbiol Lett 342: 168-178. https://doi.org/10.1111/1574-6968.12089 ![]() |
[28] |
Katoch M, Pull S (2017) Endophytic fungi associated with Monarda citriodora, an aromatic and medicinal plant and their biocontrol potential. Pharm Biol 55: 1528-1535. https://doi.org/10.1080/13880209.2017.1309054 ![]() |
[29] | Tamilarasi S, Nanthakumar K, Karthikeyan K, et al. (2008) Diversity of root associated microorganisms of selected medicinal plants and influence of rhizomicroorganisms on the antimicrobial property of Coriandrum sativum. J Environ Biol 29: 127-134. |
[30] | Salam N, Khieu TN, Liu MJ, et al. (2017) Endophytic actinobacteria associated with Dracaena cochinchinensis Lour.: isolation, diversity, and their cytotoxic activities. Biomed Res Int 1308563. https://doi.org/10.1155/2017/1308563 |
[31] |
Rustamova N, Wubulikasimu A, Mukhamedov N, et al. (2020) Endophytic bacteria associated with medicinal plant Baccharoides anthelmintica diversity and characterization. Curr Microbiol 77: 1457-1465. https://doi.org/10.1007/s00284-020-01924-5 ![]() |
[32] |
Shurigin V, Alaylar B, Davranov K, et al. (2021) Diversity and biological activity of culturable endophytic bacteria associated with marigold (Calendula officinalis L.). AIMS Microbiol 7: 336-353. https://doi.org/10.3934/microbiol.2021021 ![]() |
[33] |
Mora-Ruiz MDR, Font-Verdera F, Díaz-Gil C, et al. (2015) Moderate halophilic bacteria colonizing the phylloplane of halophytes of the subfamily Salicornioideae (Amaranthaceae). Syst Appl Microbiol 38: 406-416. https://doi.org/10.1016/j.syapm.2015.05.004 ![]() |
[34] | Dashti AA, Jadaon MM, Abdulsamad AM, et al. (2009) Heat treatment of bacteria: a simple method of DNA extraction for molecular techniques. Kuwait Med J 41: 117-122. |
[35] | Lane DJ (1991) 16S/23S rRNA Sequencing. Nucleic acid techniques in bacterial systematic. New York: John Wiley and Sons 115-175. |
[36] | Jinneman KC, Wetherington JH, Adams AM, et al. (1996) Differentiation of Cyclospora sp. and Eimeria spp. by using the polymerase chain reaction amplification products and restriction fragment length polymorphisms. Food and Drug Administration Laboratory Information Bulletin LIB no 4044 . |
[37] |
Tamura K, Nei M, Kumar S (2004) Prospects for inferring very large phylogenies by using the neighbor-joining method. Proc Natl Acad Sci USA 101: 11030-11035. https://doi.org/10.1073/pnas.0404206101 ![]() |
[38] |
Kumar S, Stecher G, Li M, et al. (2018) MEGA X: Molecular Evolutionary Genetics Analysis across computing platforms. Mol Biol Evol 35: 1547-1549. https://doi.org/10.1093/molbev/msy096 ![]() |
[39] |
Egamberdieva D, Wirth SJ, Shurigin VV, et al. (2017d) Endophytic bacteria improve plant growth, symbiotic performance of chickpea (Cicer arietinum L.) and induce suppression of root rot caused by Fusarium solani under salt stress. Front Microbiol 8: 1887. https://doi.org/10.3389/fmicb.2017.01887 ![]() |
[40] |
Castric PA (1975) Hydrogen cyanide, a secondary metabolite of Pseudomonas aeruginosa. Can J Microbiol 21: 613-618. https://doi.org/10.1139/m75-088 ![]() |
[41] |
Schwyn B, Neilands JB (1987) Universal chemical assay for the detection and determination of siderophores. Anal Biochem 160: 47-56. https://doi.org/10.1016/0003-2697(87)90612-9 ![]() |
[42] |
Brown MRW, Foster JHS (1970) A simple diagnostic milk medium for Pseudomonas aeruginosa. J Clin Pathol 23: 172-177. https://doi.org/10.1136/jcp.23.2.172 ![]() |
[43] |
Walsh GA, Murphy RA, Killeen GF, et al. (1995) Technical note: Detection and quantification of supplemental fungal b-glucanase activity in animal feed. J Anim Sci 73: 1074-1076. https://doi.org/10.2527/1995.7341074x ![]() |
[44] | Malleswari D, Bagyanarayan G (2017) In vitro screening of rhizobacteria isolated from the rhizosphere of medicinal and aromatic plants for multiple plant growth promoting activities. J Microbiol Biotechnol Res 3: 84-91. |
[45] |
Howe TG, Ward JM (1976) The utilization of tween 80 as carbon source by Pseudomonas. J Gen Microbiol 92: 234-235. https://doi.org/10.1099/00221287-92-1-234 ![]() |
[46] |
Bano N, Musarrat J (2003) Characterization of a new Pseudomonas aeruginosa strain NJ-15 as a potential biocontrol agent. Curr Microbiol 46: 324-328. https://doi.org/10.1007/s00284-002-3857-8 ![]() |
[47] |
Egamberdieva D, Kucharova Z (2009) Selection for root colonising bacteria stimulating wheat growth in saline soils. Biol Fertil Soils 45: 561-573. https://doi.org/10.1007/s00374-009-0366-y ![]() |
[48] | Chen Q, Liu S, Bai Y, et al. (2014) Screening and identification of phosphate-solubilizing bacteria from reclaimed soil in Shanxi mining area. Plant Nutr Fertilizer Sci 20: 1505-1516. |
[49] |
Egamberdieva D, Wirth S, Li L, et al. (2017c) Microbial cooperation in the rhizosphere improves liquorice growth under salt stress. Bioengineered 8: 433-438. https://doi.org/10.1080/21655979.2016.1250983 ![]() |
[50] |
Ali S, Duan J, Charles TC, et al. (2014) A bioinformatics approach to the determination of genes involved in endophytic behavior in Burkholderia spp. J Theor Biol 343: 193-198. https://doi.org/10.1016/j.jtbi.2013.10.007 ![]() |
[51] |
Cho ST, Chang HH, Egamberdieva D, et al. (2015) Genome analysis of Pseudomonas fluorescens PCL1751: a rhizobacterium that controls root diseases and alleviates salt stress for its plant host. PLoS ONE 10: e0140231. https://doi.org/10.1371/journal.pone.0140231 ![]() |
[52] |
Zhao L, Xu Y, Lai XH, et al. (2015) Screening and characterization of endophytic Bacillus and Paenibacillus strains from medicinal plant Lonicera japonica for use as potential plant growth promoters. Braz J Microbiol 46: 977-989. https://doi.org/10.1590/S1517-838246420140024 ![]() |
[53] |
Preveena J, Bhore SJ (2013) Identification of bacterial endophytes associated with traditional medicinal plant Tridax procumbens Linn. Anc Sci Life 32: 173-177. https://doi.org/10.4103/0257-7941.123002 ![]() |
[54] |
Webster G, Mullins AJ, Cunningham-Oakes E, et al. (2020) Culturable diversity of bacterial endophytes associated with medicinal plants of the Western Ghats, India. FEMS Microbiol Ecol 96: fiaa147. https://doi.org/10.1093/femsec/fiaa147 ![]() |
[55] |
Liu YH, Guo JW, Salam N, et al. (2016) Culturable endophytic bacteria associated with medicinal plant Ferula songorica: molecular phylogeny, distribution and screening for industrially important traits. 3 Biotech 6: 209. https://doi.org/10.1007/s13205-016-0522-7 ![]() |
[56] |
Chi F, Shen S, Cheng H, et al. (2005) Ascending migration of endophytic rhizobia, from roots to leaves, inside rice plants and assessment of benefits to rice growth physiology. Appl Environ Microbiol 71: 7271-7278. https://doi.org/10.1128/AEM.71.11.7271-7278.2005 ![]() |
[57] |
Shi W, Su G, Li M, et al. (2021) Distribution of bacterial endophytes in the non-lesion tissues of potato and their response to potato common scab. Front Microbiol 12: 616013. https://doi.org/10.3389/fmicb.2021.616013 ![]() |
[58] | Goryluk A, Rekosz-Burlaga H, Blaszczyk M (2009) Isolation and characterization of bacterial endophytes of Chelidonium majus L. Pol J Microbiol 58: 355-361. |
[59] |
Mehanni MM, Safwat MS (2010) Endophytes of medicinal plants. Acta Hortic 854: 31-40. https://doi.org/10.17660/ActaHortic.2010.854.3 ![]() |
[60] |
Egamberdieva D, Kucharova Z, Davranov K, et al. (2011) Bacteria able to control foot and root rot and to promote growth of cucumber in salinated soils. Biol Fertil Soils 47: 197-205. https://doi.org/10.1007/s00374-010-0523-3 ![]() |
[61] |
Nongkhlaw FMW, Joshi SR (2014) Epiphytic and endophytic bacteria that promote growth of ethnomedicinal plants in the subtropical forests of Meghalaya, India. Rev Biol Trop 62: 1295-1308. https://doi.org/10.15517/rbt.v62i4.12138 ![]() |
[62] |
Liu Y, Mohamad OAA, Salam N, et al. (2019) Diversity, community distribution and growth promotion activities of endophytes associated with halophyte Lycium ruthenicum Murr. 3 Biotech 9: 144. https://doi.org/10.1007/s13205-019-1678-8 ![]() |
[63] |
Ferchichi N, Toukabri W, Boularess M, et al. (2019) Isolation, identification and plant growth promotion ability of endophytic bacteria associated with lupine root nodule grown in Tunisian soil. Arch Microbiol 201: 1333-1349. https://doi.org/10.1007/s00203-019-01702-3 ![]() |
[64] | Fernando TC, Cruz JA (2019) Profiling and biochemical identification of potential plant growth-promoting endophytic bacteria from Nypa fruticans. Philipp J Crop Sci 44: 77-85. https://doi.org/10.13140/RG.2.2.15641.98408 |
[65] | Rana KL, Kour D, Yadav AH (2019) Endophytic microbiomes: Biodiversity, ecological significance and biotechnological applications. Res J Biotechnol 14: 142-162. |
[66] | Siddiqui ZA (2005) PGPR: prospective biocontrol agents of plant pathogens. PGPR: biocontrol and biofertilization. Dordrecht: Springer 111-142. https://doi.org/10.1007/1-4020-4152-7_4 |
[67] |
Michelsen CF, Stougaard P (2012) Hydrogen cyanide synthesis and antifungal activity of the biocontrol strain Pseudomonas fluorescens In5 from Greenland is highly dependent on growth medium. Can J Microbiol 58: 381-390. https://doi.org/10.1139/w2012-004 ![]() |
[68] |
Ahmed EA, Hassan EA, El Tobgy KMK, et al. (2014) Evaluation of rhizobacteria of some medicinal plants for plant growth promotion and biological control. Ann Agric Sci 59: 273-280. https://doi.org/10.1016/j.aoas.2014.11.016 ![]() |
[69] |
Arun B, Gopinath B, Sharma S (2012) Plant growth promoting potential of bacteria isolated on N free media from rhizosphere of Cassia occidentalis. World J Microbiol Biotechnol 28: 2849-2857. https://doi.org/10.1007/s11274-012-1095-1 ![]() |
[70] |
Ray S, Singh S, Sarma BK, et al. (2016) Endophytic alcaligenes isolated from horticultural and medicinal crops promotes growth in Okra (Abelmoschus esculentus). J Plant Growth Regul 35: 401-412. https://doi.org/10.1007/s00344-015-9548-z ![]() |
[71] |
Chowdhury EK, Jeon J, Rim SK, et al. (2017) Composition, diversity and bioactivity of culturable bacterial endophytes in mountain-cultivated ginseng in Korea. Sci Rep 7: 1-10. https://doi.org/10.1038/s41598-017-10280-7 ![]() |
[72] |
Wozniak M, Gałazka A, Tyskiewicz R, et al. (2019) Endophytic bacteria potentially promote plant growth by synthesizing different metabolites and their phenotypic/physiological profiles in the Biolog GEN III MicroPlateTM Test. Int J Mol Sci 20: 1-24. https://doi.org/10.3390/ijms20215283 ![]() |
[73] |
Musa Z, Ma J, Egamberdieva D, et al. (2020) Diversity and antimicrobial potential of cultivable endophytic actinobacteria associated with medicinal plant Thymus roseus. Front Microbiol 11: 191. https://doi.org/10.3389/fmicb.2020.00191 ![]() |
[74] |
Glick BR (2014) Bacteria with ACC deaminase can promote plant growth and help to feed the world. Microbiol Res 169: 30-39. https://doi.org/10.1016/j.micres.2013.09.009 ![]() |
[75] |
Leong J (1986) Siderophores: their biochemistry and possible role in the biocontrol of plant pathogens. Annu Rev Phytopathol 24: 187-209. https://doi.org/10.1146/annurev.py.24.090186.001155 ![]() |
[76] |
Neilands JB, Leong SA (1986) Siderophores in relation to plant growth and disease. Annu Rev Plant Physiol 37: 187-208. https://doi.org/10.1146/annurev.pp.37.060186.001155 ![]() |
[77] |
Goldstein AH (1986) Bacterial solubilization of mineral phosphates: Historical perspective and future prospects. Amer J Alternat Agric 1: 51-57. https://doi.org/10.1017/S0889189300000886 ![]() |
[78] |
Sudarshna, Sharma N (2024) Endophytic bacteria associated with critically endangered medicinal plant Trillium govanianum (Wall ex. Royle) and their potential in soil nutrition alleviation. Plant Stress 11: 100349. https://doi.org/10.1016/j.stress.2024.100349 ![]() |
[79] |
Deepa N, Chauhan Sh, Singh A (2024) Unraveling the functional characteristics of endophytic bacterial diversity for plant growth promotion and enhanced secondary metabolite production in Pelargonium graveolens. Microbiol Res 283: 127673. https://doi.org/10.1016/j.micres.2024.127673 ![]() |
1. | Liqin Liu, Xiaoxiao Liu, Chunrui Zhang, REALIZATION OF NEURAL NETWORK FOR GAIT CHARACTERIZATION OF QUADRUPED LOCOMOTION, 2022, 12, 2156-907X, 455, 10.11948/20210005 | |
2. | Mingfang Chen, Kangkang Hu, Yongxia Zhang, Fengping Qi, Motion coordination control of planar 5R parallel quadruped robot based on SCPL-CPG, 2022, 14, 1687-8140, 168781402110709, 10.1177/16878140211070910 | |
3. | Zigen Song, Jiayi Zhu, Jian Xu, Gaits generation of quadruped locomotion for the CPG controller by the delay-coupled VDP oscillators, 2023, 111, 0924-090X, 18461, 10.1007/s11071-023-08783-2 | |
4. | Yangyang Han, Guoping Liu, Zhenyu Lu, Huaizhi Zong, Junhui Zhang, Feifei Zhong, Liyu Gao, A stability locomotion-control strategy for quadruped robots with center-of-mass dynamic planning, 2023, 24, 1673-565X, 516, 10.1631/jzus.A2200310 | |
5. | Zigen Song, Fengchao Ji, Jian Xu, Is there a user-friendly building unit to replicate rhythmic patterns of CPG systems? Synchrony transition and application of the delayed bursting-HCO model, 2024, 182, 09600779, 114820, 10.1016/j.chaos.2024.114820 | |
6. | Liqin Liu, Chunrui Zhang, A neural network model for goat gait, 2024, 21, 1551-0018, 6898, 10.3934/mbe.2024302 |