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Discrete & Continuous Dynamical Systems - S

October 2017 , Volume 10 , Issue 5

Issue on recent advances of differential equations with applications in life sciences

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Preface for special session entitled "Recent Advances of Differential Equations with Applications in Life Sciences"
Ping Liu, Ying Su and Fengqi Yi
2017, 10(5): i-i doi: 10.3934/dcdss.201705i +[Abstract](160) +[HTML](20) +[PDF](87.9KB)
Abstract:
Stability analysis of a model on varying domain with the Robin boundary condition
Xiaofei Cao and Guowei Dai
2017, 10(5): 935-942 doi: 10.3934/dcdss.2017048 +[Abstract](221) +[HTML](20) +[PDF](312.2KB)
Abstract:

In this paper we develop a non-autonomous reaction-diffusion model with the Robin boundary conditions to describe insect dispersal on an isotropically varying domain. We investigate the stability of the reaction-diffusion model. The stability results of the model describe either insect survival or vanishing.

Mathematical modeling about nonlinear delayed hydraulic cylinder system and its analysis on dynamical behaviors
Yuting Ding, Jinli Xu, Jun Cao and Dongyan Zhang
2017, 10(5): 943-958 doi: 10.3934/dcdss.2017049 +[Abstract](149) +[HTML](21) +[PDF](496.7KB)
Abstract:

In this paper, we study dynamics in delayed nonlinear hydraulic cylinder equation, with particular attention focused on several types of bifurcations. Firstly, basing on a series of original equations, we model a nonlinear delayed differential equations associated with hydraulic cylinder in glue dosing processes for particleboard. Secondly, we identify the critical values for fixed point, Hopf, Hopf-zero, double Hopf and tri-Hopf bifurcations using the method of bifurcation analysis. Thirdly, by applying the multiple time scales method, the normal form near the Hopf-zero bifurcation critical points is derived. Finally, two examples are presented to demonstrate the application of the theoretical results.

Existence of periodic solutions of dynamic equations on time scales by averaging
Ruichao Guo, Yong Li, Jiamin Xing and Xue Yang
2017, 10(5): 959-971 doi: 10.3934/dcdss.2017050 +[Abstract](236) +[HTML](21) +[PDF](366.3KB)
Abstract:

In this paper, we study the existence of periodic solutions for perturbed dynamic equations on time scales. Our approach is based on the averaging method. Further, we extend some averaging theorem to periodic solutions of dynamic equations on time scales to $k-$th order in $\varepsilon$. More precisely, results of higher order averaging for finding periodic solutions are given via the topological degree theory.

Global Hopf bifurcation of a population model with stage structure and strong Allee effect
Pengmiao Hao, Xuechen Wang and Junjie Wei
2017, 10(5): 973-993 doi: 10.3934/dcdss.2017051 +[Abstract](247) +[HTML](24) +[PDF](399.2KB)
Abstract:

This paper is devoted to the study of a single-species population model with stage structure and strong Allee effect. By taking $τ$ as a bifurcation parameter, we study the Hopf bifurcation and global existence of periodic solutions using Wu's theory on global Hopf bifurcation for FDEs and the Bendixson criterion for higher dimensional ODEs proposed by Li and Muldowney. Some numerical simulations are presented to illustrate our analytic results using MATLAB and DDE-BIFTOOL. In addition, interesting phenomenon can be observed such as two kinds of bistability.

A Perron-type theorem for nonautonomous differential equations with different growth rates
Yongxin Jiang, Can Zhang and Zhaosheng Feng
2017, 10(5): 995-1008 doi: 10.3934/dcdss.2017052 +[Abstract](238) +[HTML](15) +[PDF](381.2KB)
Abstract:

We show that if the Lyapunov exponents associated to a linear equation $x'=A(t)x$ are equal to the given limits, then this asymptotic behavior can be reproduced by the solutions of the nonlinear equation $x'=A(t)x+f(t, x)$ for any sufficiently small perturbation $f$. We consider the linear equation with a very general nonuniform behavior which has different growth rates.

Steady states of a Sel'kov-Schnakenberg reaction-diffusion system
Bo Li and Xiaoyan Zhang
2017, 10(5): 1009-1023 doi: 10.3934/dcdss.2017053 +[Abstract](171) +[HTML](18) +[PDF](408.2KB)
Abstract:

In this paper, we are concerned with a reaction-diffusion model, known as the Sel'kov-Schnakenberg system, and study the associated steady state problem. We obtain existence and nonexistence results of nonconstant steady states, which in turn imply the criteria for the formation of spatial pattern (especially, Turing pattern). Our results reveal the different roles of the diffusion rates of the two reactants in generating spatial pattern.

Pattern dynamics of a delayed eco-epidemiological model with disease in the predator
Jing Li, Zhen Jin, Gui-Quan Sun and Li-Peng Song
2017, 10(5): 1025-1042 doi: 10.3934/dcdss.2017054 +[Abstract](374) +[HTML](15) +[PDF](1008.7KB)
Abstract:

The eco-epidemiology, combining interacting species with epidemiology, can describe some complex phenomena in real ecosystem. Most diseases contain the latent stage in the process of disease transmission. In this paper, a spatial eco-epidemiological model with delay and disease in the predator is studied. By mathematical analysis, the characteristic equations are derived, then we give the conditions of diffusion-driven equilibrium instability and delay-driven equilibrium instability, and find the ranges of existence of Turing patterns in parameter space. Moreover, numerical results indicate that a parameter variation has influences on time and spatially averaged densities of pattern reaching stationary states when other parameters are fixed. The obtained results may explain some mechanisms of phenomena existing in real ecosystem.

A kind of generalized transversality theorem for $C^r$ mapping with parameter
Qiang Li
2017, 10(5): 1043-1050 doi: 10.3934/dcdss.2017055 +[Abstract](124) +[HTML](13) +[PDF](165.5KB)
Abstract:

The author considers a generalized transversality theorem of the mappings with parameter in infinite dimensional Banach space. If the mapping is generalized transversal to a single point set, and in the sense of exterior parameters, the mapping is a Fredholm operator, then there exists a residual set of parameter, such that the Fredholm operator is generalized transversal to the single point set.

Pattern formation of a coupled two-cell Schnakenberg model
Guanqi Liu and Yuwen Wang
2017, 10(5): 1051-1062 doi: 10.3934/dcdss.2017056 +[Abstract](203) +[HTML](15) +[PDF](416.7KB)
Abstract:

In this paper, we study a coupled two-cell Schnakenberg model with homogenous Neumann boundary condition, i.e.,

We give a priori estimate to the positive solution. Meanwhile, we obtain the non-existence and existence of positive non-constant solution as parameters \begin{document}$ d_1, d_2, a$\end{document} and b changes.

Traveling wave solutions of a reaction-diffusion predator-prey model
Jiang Liu, Xiaohui Shang and Zengji Du
2017, 10(5): 1063-1078 doi: 10.3934/dcdss.2017057 +[Abstract](193) +[HTML](21) +[PDF](441.0KB)
Abstract:

This paper is concerned with the dynamics of traveling wave solutions for a reaction-diffusion predator-prey model with a nonlocal delay. By using Schauder's fixed point theorem, we establish the existence result of a traveling wave solution connecting two steady states by constructing a pair of upper-lower solutions which are easy to construct in practice. We also investigate the asymptotic behavior of traveling wave solutions by employing the standard asymptotic theory.

Dynamical behavior of a new oncolytic virotherapy model based on gene variation
Zizi Wang, Zhiming Guo and Huaqin Peng
2017, 10(5): 1079-1093 doi: 10.3934/dcdss.2017058 +[Abstract](175) +[HTML](22) +[PDF](532.9KB)
Abstract:

Oncolytic virotherapy is an experimental treatment of cancer patients. This method is based on the administration of replication-competent viruses that selectively destroy tumor cells but remain healthy tissue unaffected. In order to obtain optimal dosage for complete tumor eradication, we derive and analyze a new oncolytic virotherapy model with a fixed time period \begin{document}$τ $\end{document} and non-local infection which is caused by the diffusion of the target cells in a continuous bounded domain, where \begin{document}$τ $\end{document} is assumed to be the duration that oncolytic viruses spend to destroy the target cells and to release new viruses since they enter into the target cells. This model is a delayed reaction diffusion system with nonlocal reaction term. By analyzing the global stability of tumor cell eradication equilibrium, we give different treatment strategies for cancer therapy according to the different genes mutations (oncogene and antioncogene).

On concentration of semi-classical solitary waves for a generalized Kadomtsev-Petviashvili equation
Yuanhong Wei, Yong Li and Xue Yang
2017, 10(5): 1095-1106 doi: 10.3934/dcdss.2017059 +[Abstract](154) +[HTML](21) +[PDF](420.9KB)
Abstract:

The present paper is concerned with semi-classical solitary wave solutions of a generalized Kadomtsev-Petviashvili equation in \begin{document} $\mathbb{R}^{2}$ \end{document}. Parameter \begin{document} $\varepsilon$ \end{document} and potential \begin{document} $V(x,y)$ \end{document} are included in the problem. The existence of the least energy solution is established for all \begin{document} $\varepsilon>0$ \end{document} small. Moreover, we point out that these solutions converge to a least energy solution of the associated limit problem and concentrate to the minimum point of the potential as \begin{document} $\varepsilon \to 0$ \end{document}.

Invasion traveling wave solutions in temporally discrete random-diffusion systems with delays
Hui Xue, Jianhua Huang and Zhixian Yu
2017, 10(5): 1107-1131 doi: 10.3934/dcdss.2017060 +[Abstract](143) +[HTML](16) +[PDF](537.3KB)
Abstract:

This paper is devoted to the invasion traveling wave solutions for a temporally discrete delayed reaction-diffusion competitive system. The existence of invasion traveling wave solutions is established by using Schauder's fixed point Theorem. Ikeharaś theorem is applied to show the asymptotic behaviors. We further investigate the monotonicity and uniqueness invasion traveling waves with the help of sliding method and strong maximum principle.

Lyapunov-type inequalities and solvability of second-order ODEs across multi-resonance
He Zhang, Xue Yang and Yong Li
2017, 10(5): 1133-1148 doi: 10.3934/dcdss.2017061 +[Abstract](199) +[HTML](26) +[PDF](511.3KB)
Abstract:

We present some new Lyapunov-type inequalities for boundary value problems of the form \begin{document}$y''+u(x)y=0$\end{document}, \begin{document}$y(0)=0=y(1)$\end{document}, where \begin{document}$-A≤ u(x)≤ B$\end{document} and there are many resonance points lying inside the interval \begin{document}$[-A, B]$\end{document}. The classical Lyapunov's inequality and its reverse are improved by using Pontryagin's maximum principle. As applications, we establish two readily verifiable unique solvability criteria for general \begin{document}$u(x)$\end{document}. Some relevant examples are given to illustrate our results. Variants of Lyapunov-type inequalities for nonlinear BVPs are discussed at the end of the paper.

Dynamics and spatiotemporal pattern formations of a homogeneous reaction-diffusion Thomas model
Hongyan Zhang, Siyu Liu and Yue Zhang
2017, 10(5): 1149-1164 doi: 10.3934/dcdss.2017062 +[Abstract](196) +[HTML](17) +[PDF](402.3KB)
Abstract:

In this paper, we are mainly considered with a kind of homogeneous diffusive Thomas model arising from biochemical reaction. Firstly, we use the invariant rectangle technique to prove the global existence and uniqueness of the positive solutions of the parabolic system, and then use the maximum principle to show the existence of attraction region which attracts all the solutions of the system regardless of the initial values. Secondly, we consider the long time behaviors of the solutions of the system; Thirdly, we derive precise parameter ranges where the system does not have non-constant steady states by using use some useful inequalities and a priori estimates; Finally, we prove the existence of Turing patterns by using the steady state bifurcation theory.

Stability and bifurcation analysis in a chemotaxis bistable growth system
Shubo Zhao, Ping Liu and Mingchao Jiang
2017, 10(5): 1165-1174 doi: 10.3934/dcdss.2017063 +[Abstract](200) +[HTML](21) +[PDF](396.2KB)
Abstract:

The stability analysis of a chemotaxis model with a bistable growth term in both unbounded and bounded domains is studied analytically. By the global bifurcation theorem, we identify the full parameter regimes in which the steady state bifurcation occurs.

Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term
Jun Zhou
2017, 10(5): 1175-1185 doi: 10.3934/dcdss.2017064 +[Abstract](198) +[HTML](17) +[PDF](364.4KB)
Abstract:

This paper deals with a higher-order wave equation with general nonlinear dissipation and source term

which was studied extensively when \begin{document}$m=1, 2$\end{document} and the nonlinear dissipative term \begin{document}$g(u')$\end{document} is a polynomial, i.e., \begin{document}$g(u')=a|u'|^{q-2}u'$\end{document}. We obtain the global existence of solutions and show the energy decay estimate when \begin{document}$m≥1$\end{document} is a positive integer and the nonlinear dissipative term \begin{document}$g$\end{document} does not necessarily have a polynomial grow near the origin.

Bifurcations analysis of Leslie-Gower predator-prey models with nonlinear predator-harvesting
Changrong Zhu and Lei Kong
2017, 10(5): 1187-1206 doi: 10.3934/dcdss.2017065 +[Abstract](199) +[HTML](20) +[PDF](1575.2KB)
Abstract:

In the present paper the dynamics of a Leslie-Gower predator-prey model with Michaelis-Menten type predator harvesting is studied. We give out all the possible ranges of parameters for which the model has up to five equilibria. We prove that these equilibria can be topological saddles, nodes, foci, centers, saddle-nodes, cusps of codimension 2 or 3. Numerous kinds of bifurcations also occur, such as the transcritical bifurcation, pitchfork bifurcation, Bogdanov-Takens bifurcation and homoclinic bifurcation. Several numerical simulations are carried out to illustrate the validity of our results.

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