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Discrete and Continuous Dynamical Systems - B

September 2022 , Volume 27 , Issue 9

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Pseudo compact almost automorphy of neutral type Clifford-valued neural networks with mixed delays
Yongkun Li and Bing Li
2022, 27(9): 4703-4724 doi: 10.3934/dcdsb.2021248 +[Abstract](849) +[HTML](280) +[PDF](392.91KB)

We consider a class of neutral type Clifford-valued cellular neural networks with discrete delays and infinitely distributed delays. Unlike most previous studies on Clifford-valued neural networks, we assume that the self feedback connection weights of the networks are Clifford numbers rather than real numbers. In order to study the existence of \begin{document}$ (\mu, \nu) $\end{document}-pseudo compact almost automorphic solutions of the networks, we prove a composition theorem of \begin{document}$ (\mu, \nu) $\end{document}-pseudo compact almost automorphic functions with varying deviating arguments. Based on this composition theorem and the fixed point theorem, we establish the existence and the uniqueness of \begin{document}$ (\mu, \nu) $\end{document}-pseudo compact almost automorphic solutions of the networks. Then, we investigate the global exponential stability of the solution by employing differential inequality techniques. Finally, we give an example to illustrate our theoretical finding. Our results obtained in this paper are completely new, even when the considered networks are degenerated into real-valued, complex-valued or quaternion-valued networks.

Weak convergence of delay SDEs with applications to Carathéodory approximation
Ta Cong Son, Nguyen Tien Dung, Nguyen Van Tan, Tran Manh Cuong, Hoang Thi Phuong Thao and Pham Dinh Tung
2022, 27(9): 4725-4747 doi: 10.3934/dcdsb.2021249 +[Abstract](802) +[HTML](288) +[PDF](417.75KB)

In this paper, we consider a fundamental class of stochastic differential equations with time delays. Our aim is to investigate the weak convergence with respect to delay parameter of the solutions. Based on the techniques of Malliavin calculus, we obtain an explicit estimate for the rate of convergence. An application to the Carathéodory approximation scheme of stochastic differential equations is provided as well.

Reaction-advection-diffusion competition models under lethal boundary conditions
Kwangjoong Kim, Wonhyung Choi and Inkyung Ahn
2022, 27(9): 4749-4767 doi: 10.3934/dcdsb.2021250 +[Abstract](736) +[HTML](373) +[PDF](720.17KB)

In this study, we consider a Lotka–Volterra reaction–diffusion–advection model for two competing species under homogeneous Dirichlet boundary conditions, describing a hostile environment at the boundary. In particular, we deal with the case in which one species diffuses at a constant rate, whereas the other species has a constant rate diffusion rate with a directed movement toward a better habitat in a heterogeneous environment with a lethal boundary. By analyzing linearized eigenvalue problems from the system, we conclude that the species dispersion in the advection direction is not always beneficial, and survival may be determined by the convexity of the environment. Further, we obtain the coexistence of steady-states to the system under the instability conditions of two semi-trivial solutions and the uniqueness of the coexistence steady states, implying the global asymptotic stability of the positive steady-state.

Trajectory statistical solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines
Bo You
2022, 27(9): 4769-4785 doi: 10.3934/dcdsb.2021251 +[Abstract](603) +[HTML](276) +[PDF](382.98KB)

The objective of this paper is to consider the long-time behavior of solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines. As we know, it is very difficult to obtain the uniqueness of an energy solution for this system even in two dimensions caused by the presence of the strong coupling at the boundary. Thus, we first prove the existence of a trajectory attractor for such system, which is a minimal compact trajectory attracting set for the natural translation semigroup defined on the trajectory space. Furthermore, based on the abstract results (trajectory attractor approach) developed in [38], we construct trajectory statistical solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines.

Dynamics of a Leslie-Gower predator-prey system with hunting cooperation and prey harvesting
Yong Yao and Lingling Liu
2022, 27(9): 4787-4815 doi: 10.3934/dcdsb.2021252 +[Abstract](789) +[HTML](348) +[PDF](958.69KB)

In this paper, we study the dynamics of a Leslie-Gower predator-prey system with hunting cooperation among predator population and constant-rate harvesting for prey population. It is shown that there are a weak focus of multiplicity up to three and a cusp of codimension at most two for various parameter values, and the system exhibits two saddle-node bifurcations, a Bogdanov-Takens bifurcation of codimension two and a Hopf bifurcation as the bifurcation parameters vary. The results developed in this article reveal far more complex dynamics compared to the Leslie-Gower system and show how the prey harvesting and the hunting cooperation affect the dynamics of the system. In particular, there exist some critical values of prey harvesting and hunting cooperation such that the predator and prey populations are at risk of extinction if the intensities of harvesting and hunting cooperation are greater than these critical values. Moreover, numerical simulations are presented to illustrate our theoretical results.

The existence of time-dependent attractor for wave equation with fractional damping and lower regular forcing term
Xudong Luo and Qiaozhen Ma
2022, 27(9): 4817-4835 doi: 10.3934/dcdsb.2021253 +[Abstract](850) +[HTML](275) +[PDF](409.18KB)

We investigate the well-posedness and longtime dynamics of fractional damping wave equation whose coefficient \begin{document}$ \varepsilon $\end{document} depends explicitly on time. First of all, when \begin{document}$ 1\leq p\leq p^{\ast\ast} = \frac{N+2}{N-2}\; (N\geq3) $\end{document}, we obtain existence of solution for the fractional damping wave equation with time-dependent decay coefficient in \begin{document}$ H_{0}^{1}(\Omega)\times L^{2}(\Omega) $\end{document}. Furthermore, when \begin{document}$ 1\leq p<p^{*} = \frac{N+4\alpha}{N-2} $\end{document}, \begin{document}$ u_{t} $\end{document} is proved to be of higher regularity in \begin{document}$ H^{1-\alpha}\; (t>\tau) $\end{document} and show that the solution is quasi-stable in weaker space \begin{document}$ H^{1-\alpha}\times H^{-\alpha} $\end{document}. Finally, we get the existence and regularity of time-dependent attractor.

Fractional $ 1 $-Laplacian evolution equations to remove multiplicative noise
Tianling Gao, Qiang Liu and Zhiguang Zhang
2022, 27(9): 4837-4854 doi: 10.3934/dcdsb.2021254 +[Abstract](671) +[HTML](276) +[PDF](1172.46KB)

In this paper, we propose a new image denosing model to remove the multiplicative noise by a maximum a posteriori estimation and an inhomogeneous fractional \begin{document}$ 1 $\end{document}-Laplace evolution equation. The main difficulty of the problem is the equation will become very singular when \begin{document}$ u(x) = u(y) $\end{document}. The existence and uniqueness of the weak positive solution are proved. Numerical examples demonstrate the better capability of our model on some heavy multiplicative noised images.

Boundedness and asymptotic behavior in a predator-prey model with indirect pursuit-evasion interaction
Chao Liu and Bin Liu
2022, 27(9): 4855-4874 doi: 10.3934/dcdsb.2021255 +[Abstract](764) +[HTML](253) +[PDF](635.18KB)

In this paper, we study the prey-predator model with indirect pursuit-evasion interaction defined on a smooth bounded domain with homogeneous Neumann boundary conditions. We obtain the globa existence and boundedness of the classical solution of the model by estimating \begin{document}$ L^{p} $\end{document}-norm of \begin{document}$ u $\end{document} and \begin{document}$ v $\end{document}, and we also show the large time behavior and convergence rate of the solution.

More traveling waves in the Holling-Tanner model with weak diffusion
Vahagn Manukian and Stephen Schecter
2022, 27(9): 4875-4890 doi: 10.3934/dcdsb.2021256 +[Abstract](598) +[HTML](241) +[PDF](1011.68KB)

We identify two new traveling waves of the Holling-Tanner model with weak diffusion. One connects two constant states; at one of them, the model is undefined. The other connects a constant state to a periodic wave train. We exploit the multi-scale structure of the Holling-Tanner model in the weak diffusion limit. Our analysis uses geometric singular perturbation theory, compactification and the blow-up method.

Global existence of weak solutions to inhomogeneous Doi-Onsager equations
Wenji Chen and Jianfeng Zhou
2022, 27(9): 4891-4921 doi: 10.3934/dcdsb.2021257 +[Abstract](661) +[HTML](299) +[PDF](441.65KB)

In this paper, we study the inhomogeneous Doi-Onsager equations with a special viscous stress. We prove the global existence of weak solutions in the case of periodic regions without considering the effect of the constraint force arising from the rigidity of the rods. The key ingredient is to show the convergence of the nonlinear terms, which can be reduced to proving the strong compactness of the moment of the family of number density functions. The proof is based on the propagation of strong compactness by studying a transport equation for some defect measure, \begin{document}$ L^2 $\end{document}-estimates for a family of number density functions, and energy dissipation estimates.

Robust uniform persistence for structured models of delay differential equations
Paul L. Salceanu
2022, 27(9): 4923-4939 doi: 10.3934/dcdsb.2021258 +[Abstract](635) +[HTML](317) +[PDF](310.49KB)

We consider a general class of delay differential equations systems, typically used to model the dynamics of structured biological populations, and establish necessary conditions for the part of the attractor contained in the boundary of the state space to repel the complementary dynamics contained in the interior of the state space. The conditions are formulated in terms of Lyapunov exponents and invariant probability measures and we use them to prove a robust uniform persistence result.

Inverse scattering transform for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation
Wei-Kang Xun, Shou-Fu Tian and Tian-Tian Zhang
2022, 27(9): 4941-4967 doi: 10.3934/dcdsb.2021259 +[Abstract](873) +[HTML](247) +[PDF](1300.0KB)

In this work, a generalized nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation is introduced, and its integrability as an infinite dimensional Hamilton dynamic system is established. We successfully derive the inverse scattering transform (IST) of the nonlocal LPD equation. The direct scattering problem of the equation is first constructed, and some important symmetries of the eigenfunctions and the scattering data are discussed. By using a novel Left-Right Riemann-Hilbert (RH) problem, the inverse scattering problem is analyzed, and the potential function is recovered. By introducing the special conditions of reflectionless case, the time-periodic soliton solutions formula of the equation is derived successfully. Take \begin{document}$ J = \overline{J} = 1,2,3 $\end{document} and \begin{document}$ 4 $\end{document} for example, we obtain some interesting phenomenon such as breather-type solitons, arc solitons, three soliton and four soliton. Furthermore, the influence of parameter \begin{document}$ \delta $\end{document} on these solutions is further considered via the graphical analysis. Finally, the eigenvalues and conserved quantities are investigated under a few special initial conditions.

Threshold dynamics of a reaction-diffusion-advection Leslie-Gower predator-prey system
Baifeng Zhang, Guohong Zhang and Xiaoli Wang
2022, 27(9): 4969-4993 doi: 10.3934/dcdsb.2021260 +[Abstract](755) +[HTML](218) +[PDF](1147.43KB)

In this paper, we investigate the global dynamics of a Leslie-Gower predator-prey system in advective homogeneous environments. We discuss the existence and uniqueness of positive steady-state solutions. We study the large time behavior of solutions and establish threshold conditions for persistence and extinction of two species when they live in open advective environments. Numerical simulations indicate that the introduction of advection leads to the evolution of spatial distribution patterns of species and specially it may induce spatial separation of the prey and predator under some conditions.

Attractors for a class of perturbed nonclassical diffusion equations with memory
Jianbo Yuan, Shixuan Zhang, Yongqin Xie and Jiangwei Zhang
2022, 27(9): 4995-5007 doi: 10.3934/dcdsb.2021261 +[Abstract](731) +[HTML](258) +[PDF](343.01KB)

In this paper, using a new operator decomposition method (or framework), we establish the existence, regularity and upper semi-continuity of global attractors for a perturbed nonclassical diffusion equation with fading memory. It is worth noting that we get the same conclusion in [7,14] as the perturbed parameters \begin{document}$ \nu = 0 $\end{document}, but the nonlinearity \begin{document}$ f $\end{document} satisfies arbitrary polynomial growth condition rather than critical exponential growth condition.


Correction: “College of Arts and Sciences” has been changed to "College of Science"; “Postgraduate scientific research innovation project of Hunan Province (CX20210751)” has been added to Fund Project. We apologize for any inconvenience this may cause.

Threshold of a stochastic SIQS epidemic model with isolation
Nguyen Thanh Dieu, Vu Hai Sam and Nguyen Huu Du
2022, 27(9): 5009-5028 doi: 10.3934/dcdsb.2021262 +[Abstract](831) +[HTML](274) +[PDF](2985.46KB)

The aim of this paper is to give sufficient conditions, very close to the necessary one, to classify the stochastic permanence of SIQS epidemic model with isolation via a threshold value \begin{document}$ \widehat R $\end{document}. Precisely, we show that if \begin{document}$ \widehat R<1 $\end{document} then the stochastic SIQS system goes to the disease free case in sense the density of infected \begin{document}$ I_z(t) $\end{document} and quarantined \begin{document}$ Q_z(t) $\end{document} classes extincts to \begin{document}$ 0 $\end{document} at exponential rate and the density of susceptible class \begin{document}$ S_z(t) $\end{document} converges almost surely at exponential rate to the solution of boundary equation. In the case \begin{document}$ \widehat R>1 $\end{document}, the model is permanent. We show the existence of a unique invariant probability measure and prove the convergence in total variation norm of transition probability to this invariant measure. Some numerical examples are also provided to illustrate our findings.

Jacobi stability analysis and impulsive control of a 5D self-exciting homopolar disc dynamo
Zhouchao Wei, Fanrui Wang, Huijuan Li and Wei Zhang
2022, 27(9): 5029-5045 doi: 10.3934/dcdsb.2021263 +[Abstract](868) +[HTML](277) +[PDF](1892.66KB)

In this paper, we make a thorough inquiry about the Jacobi stability of 5D self-exciting homopolar disc dynamo system on the basis of differential geometric methods namely Kosambi-Cartan-Chern theory. The Jacobi stability of the equilibria under specific parameter values are discussed through the characteristic value of the matrix of second KCC invariants. Periodic orbit is proved to be Jacobi unstable. Then we make use of the deviation vector to analyze the trajectories behaviors in the neighborhood of the equilibria. Instability exponent is applicable for predicting the onset of chaos quantitatively. In addition, we also consider impulsive control problem and suppress hidden attractor effectively in the 5D self-exciting homopolar disc dynamo.

Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones
Fang Wu, Lihong Huang and Jiafu Wang
2022, 27(9): 5047-5083 doi: 10.3934/dcdsb.2021264 +[Abstract](741) +[HTML](302) +[PDF](3598.8KB)

In this paper, we consider the nonsmooth bifurcation around a class of critical crossing cycles, which are codimension-2 closed orbits composed of tangency singularities and regular orbits, for a two-parameter family of planar piecewise smooth system with two zones. By the construction of suitable displacement function (equivalently, Poincar\begin{document}$ {\rm\acute{e}} $\end{document} map), the stability and the existence of periodic solutions under the variation of the parameters inside this system are characterized. More precisely, we obtain some parameter regions on the existence of crossing cycles and sliding cycles near those loops. As applications, several examples are given to illustrate our main conclusions.

The minimal wave speed of the Lotka-Volterra competition model with seasonal succession
Wentao Meng, Yuanxi Yue and Manjun Ma
2022, 27(9): 5085-5100 doi: 10.3934/dcdsb.2021265 +[Abstract](689) +[HTML](369) +[PDF](410.53KB)

This paper focuses on the minimal wave speed of time-periodic traveling waves to a Lotka-Volterra competition model with seasonal succession. It is the first time the general conditions of linear selection and nonlinear selection have been derived by the comparison principle and the upper-lower solution method. Based on the decay characteristics of traveling waves, we obtain some explicit conditions for determining the selection mechanism of the minimal wave speed by constructing upper/lower solutions, which include the first explicit condition for the nonlinear selection and the explicit conditions for the linear selection that greatly improve the result in the reference.

Recent developments on spatial propagation for diffusion equations in shifting environments
Jia-Bing Wang, Wan-Tong Li, Fang-Di Dong and Shao-Xia Qiao
2022, 27(9): 5101-5127 doi: 10.3934/dcdsb.2021266 +[Abstract](890) +[HTML](393) +[PDF](420.04KB)

In this short review, we describe some recent developments on the spatial propagation for diffusion problems in shifting environments, including single species models, competition/cooperative models and chemotaxis models submitted to classical reaction-diffusion equations (with or without free boundaries), integro-difference equations, lattice differential equations and nonlocal dispersal equations. The considered topics may typically come from modeling the threats associated with global climate change and the worsening of the environment resulting from industrialization which lead to the shifting or translating of the habitat ranges, and also arise indirectly in studying the pathophoresis as well as some multi-stage invasion processes. Some open problems and potential research directions are also presented.

Dynamics of fractional nonclassical diffusion equations with delay driven by additive noise on $ \mathbb{R}^n $
Pengyu Chen, Bixiang Wang and Xuping Zhang
2022, 27(9): 5129-5159 doi: 10.3934/dcdsb.2021267 +[Abstract](647) +[HTML](307) +[PDF](505.51KB)

In this paper, we study the asymptotic behavior of solutions of fractional nonclassical diffusion equations with delay driven by additive noise defined on unbounded domains. We first prove the uniform compactness of pullback random attractors of the equation with respect to noise intensity and time delay, and then establish the upper semi-continuity of these attractors as either noise intensity or time delay approaches zero.

Analysis of a model of the Calvin cycle with diffusion of ATP
Burcu Gürbüz and Alan D. Rendall
2022, 27(9): 5161-5177 doi: 10.3934/dcdsb.2021268 +[Abstract](700) +[HTML](260) +[PDF](433.13KB)

The dynamics of a mathematical model of the Calvin cycle, which is part of photosynthesis, is analysed. Since diffusion of ATP is included in the model a system of reaction-diffusion equations is obtained. It is proved that for a suitable choice of parameters there exist spatially inhomogeneous positive steady states, in fact infinitely many of them. It is also shown that all positive steady states, homogeneous and inhomogeneous, are nonlinearly unstable. The only smooth steady state which could be stable is a trivial one, where all concentrations except that of ATP are zero. It is found that in the spatially homogeneous case there are steady states with the property that the linearization about that state has eigenvalues which are not real, indicating the presence of oscillations. Numerical simulations exhibit solutions for which the concentrations are not monotone functions of time.

Erratum: Existence and uniqueness of solutions of free boundary problems in heterogeneous environments
Mingxin Wang
2022, 27(9): 5179-5180 doi: 10.3934/dcdsb.2021269 +[Abstract](1319) +[HTML](210) +[PDF](237.48KB)

The proof of Theorem 1.1 in [Mingxin Wang, Discrete Cont. Dyn. Syst. B. 24(2)(2019), 415-421] contains a mistake. In this erratum, we point out the correct version of this estimate.

On the zeroth law of turbulence for the stochastically forced Navier-Stokes equations
Yat Tin Chow and Ali Pakzad
2022, 27(9): 5181-5203 doi: 10.3934/dcdsb.2021270 +[Abstract](620) +[HTML](198) +[PDF](388.23KB)

We consider the three-dimensional stochastically forced Navier–Stokes equations subjected to white-in-time (colored-in-space) forcing in the absence of boundaries. Upper bounds of the mean value of the time-averaged energy dissipation rate are derived directly from the equations for weak (martingale) solutions. This estimate is consistent with the Kolmogorov dissipation law. Moreover, an additional hypothesis of energy balance implies the zeroth law of turbulence in the absence of a deterministic force.

Asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise
Yiju Chen and Xiaohu Wang
2022, 27(9): 5205-5224 doi: 10.3934/dcdsb.2021271 +[Abstract](617) +[HTML](230) +[PDF](408.41KB)

In this paper, we study the asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise. The considered systems are driven by the fractional discrete Laplacian, which features the infinite-range interactions. We first prove the existence of pullback random attractor in \begin{document}$ \ell^2 $\end{document} for stochastic lattice systems. The upper semicontinuity of random attractors is also established when the intensity of noise approaches zero.

Existence and approximation of attractors for nonlinear coupled lattice wave equations
Lianbing She, Mirelson M. Freitas, Mauricio S. Vinhote and Renhai Wang
2022, 27(9): 5225-5253 doi: 10.3934/dcdsb.2021272 +[Abstract](648) +[HTML](261) +[PDF](434.0KB)

This paper is concerned with the asymptotic behavior of solutions to a class of nonlinear coupled discrete wave equations defined on the whole integer set. We first establish the well-posedness of the systems in \begin{document}$ E: = \ell^2\times\ell^2\times\ell^2\times\ell^2 $\end{document}. We then prove that the solution semigroup has a unique global attractor in \begin{document}$ E $\end{document}. We finally prove that this attractor can be approximated in terms of upper semicontinuity of \begin{document}$ E $\end{document} by a finite-dimensional global attractor of a \begin{document}$ 2(2n+1) $\end{document}-dimensional truncation system as \begin{document}$ n $\end{document} goes to infinity. The idea of uniform tail-estimates developed by Wang (Phys. D, 128 (1999) 41-52) is employed to prove the asymptotic compactness of the solution semigroups in order to overcome the lack of compactness in infinite lattices.

Global generalized solutions to the forager-exploiter model with logistic growth
Qian Zhao and Bin Liu
2022, 27(9): 5255-5282 doi: 10.3934/dcdsb.2021273 +[Abstract](714) +[HTML](286) +[PDF](398.8KB)

This paper presents the global existence of the generalized solutions for the forager-exploiter model with logistic growth under appropriate regularity assumption on the initial value. This result partially generalizes previously known ones.

Stability and applications of multi-order fractional systems
Javier Gallegos
2022, 27(9): 5283-5296 doi: 10.3934/dcdsb.2021274 +[Abstract](674) +[HTML](296) +[PDF](380.39KB)

This paper establishes conditions for global/local robust asymptotic stability for a class of multi-order nonlinear fractional systems consisting of a linear part plus a global/local Lipschitz nonlinear term. The derivation order can be different in each coordinate and take values in \begin{document}$ (0, 2) $\end{document}. As a consequence, a linearized stability theorem for multi-order systems is also obtained. The stability conditions are order-dependent, reducing the conservatism of order-independent ones. Detailed examples in robust control and population dynamics show the applicability of our results. Simulations are attached, showing the distinctive features that justify multi-order modelling.

Tempered fractional order compartment models and applications in biology
Yejuan Wang, Lijuan Zhang and Yuan Yuan
2022, 27(9): 5297-5316 doi: 10.3934/dcdsb.2021275 +[Abstract](617) +[HTML](271) +[PDF](1113.85KB)

Compartment models with classical derivatives have diverse applications and attracted a lot of interest among scientists. To model the dynamical behavior of the particles that existed in the system for a long period of time with little chance to be removed, a power-law waiting time technique was introduced in the most recent work of Angstmann et al. [2]. The divergent first moment makes the power-law waiting time distribution less physical because of the finite lifespan of the particles. In this work, we take the tempered power-law function as the waiting time distribution, which has finite first moment while keeping the power-law properties. From the underlying physical stochastic process with the exponentially truncated power-law waiting time distribution, we build the tempered fractional compartment model. As an application, the tempered fractional SEIR epidemic model is proposed to simulate the real data of confirmed cases of pandemic AH1N1/09 influenza from Bogotá D.C. (Colombia). Some analysis and numerical simulations are carried out around the equilibrium behavior.

Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems
Radosław Czaja
2022, 27(9): 5317-5342 doi: 10.3934/dcdsb.2021276 +[Abstract](651) +[HTML](211) +[PDF](474.65KB)

In this paper we study long-time behavior of first-order non-autono-mous lattice dynamical systems in square summable space of double-sided sequences using the cooperation between the discretized diffusion operator and the discretized reaction term. We obtain existence of a pullback global attractor and construct pullback exponential attractor applying the introduced notion of quasi-stability of the corresponding evolution process.

Dynamic transitions for the S-K-T competition system
Ruikuan Liu and Dongpei Zhang
2022, 27(9): 5343-5365 doi: 10.3934/dcdsb.2021277 +[Abstract](757) +[HTML](219) +[PDF](417.25KB)

This paper is concerned with dynamical transition for biological competition system modeled by the S-K-T equations. We study the dynamical behaviour of the S-K-T equations with two different boundary conditions. For the system under non-homogeneous Dirichlet boundary condition, we show that the system undergoes a mixed dynamic transition from the homogeneous state to steady state solutions when the bifurcation parameter cross the critical surface. For the system with Neumann boundary condition, we prove that the system undergoes a mixed dynamic transition, a jump transition and a continuous transition when the bifurcation parameter cross the critical number. Finally, two examples are provided to validate the effectiveness of the theoretical results.

2021 Impact Factor: 1.497
5 Year Impact Factor: 1.527
2021 CiteScore: 2.3




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