Email: info@jnma.ca

ISSN 2562-2854 (print)

ISSN 2562-2862 (online)

# Category: 2019, Vol. 1, No. 1

### From the Editors

Maoan Han and Junling Ma

The publishers, Zhejiang Normal University, China, and Pacific Edilite Academic Inc, Canada, and the editorial board, are very proud to announce the inauguration of the Journal of Nonlinear Modeling and Analysis.

Mathematical modeling is a thriving field of research, with applications to engineering, computer science, physics, chemistry, earth science and geography, biology, medicine, public health, economics, management sciences, and many others. In fact, the list to too long to enumerate here. The research on modeling relies on, and drives the research of, almost all fields of mathematics and statistics, such as nonlinear analysis, differential equations and dynamical systems, optimization, operation research, probability theory, graph theory, combinatory, topology, experimental design, data analysis, parameter estimation, and many others. As we know, models widely used in different applications may resemble striking similarities, and can be studied with similar theories and methods.

There have been numerous journals on mathematical modeling and analysis, ranging from very theoretical to very applied aspects. However, there is a strong need for an easy-to-access journal for modelers and mathematicians to present their new models, powerful methods, and innovative analysis, so that researchers from different fields in both pure and applied mathematics can inspire and learn from each other. This journal aims to provide such a platform.

The editorial board welcome quality contributions in original research, reviews, and communications. We invite the readers, authors, and reviewers to work closely with the editors, to help us fulfill our vision.

Sincerely,

Maoan Han and Junling Ma, Editors in Chief

On behalf of the editorial board

### Geometric properties and exact travelling wave solutions for the generalized Burger-Fisher equation and the Sharma-Tasso-Olver equation

Jibin Li

In this paper, we study the dynamical behavior and exact parametric representations of the traveling wave  solutions for the generalized Burger-Fisher  equation and the Sharma-Tasso-Olver equation under different parametric conditions, the exact monotonic and non-monotonic kink wave solutions, two-peak solitary wave solutions, periodic wave solutions, as well as  unbounded traveling wave solutions  are obtained.

### Periodic solutions of the Duffing differential equation revisited via the averaging theory

Rebiha Benterki and Jaume Llibre

We use three different results of the averaging theory of first order for studying the existence of new periodic solutions in the two Duffing differential equations $\ddot y+ a \sin y= b \sin t$ and $\ddot y+a y-c y^3=b\sin t$, where $a$, $b$ and $c$ are real parameters.

### Numerical method for homoclinic and heteroclinic orbits of neuron models

Bo Deng

A twisted heteroclinic cycle was proved to exist more than twenty-five years ago for the reaction-diffusion FitzHugh-Nagumo equations in their traveling wave moving frame. The result implies the existence of infinitely many traveling front waves and infinitely many traveling back waves for the system. However efforts to numerically render the twisted cycle were not fruitful for the main reason that such orbits are structurally unstable. Presented here is a bisectional search method for the primary types of traveling wave solutions for the type of bistable reaction-diffusion systems the FitzHugh-Nagumo equations represent. The algorithm converges at a geometric rate and the wave speed can be approximated to significant precision in principle. The method is then applied for a recently obtained axon model with the conclusion that twisted heteroclinic cycle maybe more of a theoretical artifact.

### Nonexistence of nonconstant positive steady states of a diffusive predator-prey model with fear effect

Shanshan Chen, Zonghao Liu and Junping Shi

In this paper, we investigate a diffusive predator-prey model with fear effect. It is shown that, for the linear predator functional response case, the positive constant steady state is globally asymptotically stable if it exists. On the other hand, for the Holling type II predator functional response case, it is proved that there exist no nonconstant positive steady states for large conversion rate. Our results limit the parameters range where complex spatiotemporal pattern formation can occur.

### Dynamics of a predator-prey model with delay and fear effect

Weiwei Gao and Binxiang Dai

Recent manipulations on vertebrates showed that the fear of predators, caused by prey after they perceived predation risk, could reduce the prey’s reproduction greatly. And it’s known that predator-prey systems with fear effect exhibit very rich dynamics. On the other hand, incorporating the time delay into predator-prey models could also induce instability and oscillations via Hopf bifurcation. In this paper, we are interested in studying the combined effects of the fear effect and time delay on the dynamics of the classic Lotka-Volterra predator-prey model. It’s shown that the time delay can cause the stable equilibrium to become unstable, while the fear effect has a stabilizing effect on the equilibrium. In particular, the model loses stability when the delay varies and then regains its stability when the fear effect is stronger. At last, by using the normal form theory and center manifold argument, we derive explicit formulas which determine the stability and direction of periodic solutions bifurcating from Hopf bifurcation. Numerical simulations are carried to explain the mathematical conclusions.

### Bifurcation of a modified Leslie-Gower system with discrete and distributed delays

Zhongkai Guo,  Haifeng Huo, Qiuyan Ren  and Hong Xiang

A modified Leslie-Gower predator-prey system with  discrete and distributed delays is introduced. By analyzing the associated characteristic equation,  stability and local Hopf bifurcation of the model  are studied. It is found that the positive equilibrium is asymptotically stable when $\tau$ is less than a critical value and unstable when $\tau$ is greater than this critical value and the system can also undergo Hopf bifurcation at the positive equilibrium when $\tau$ crosses this critical value. Furthermore, using the normal form theory and center manifold theorem, the formulae for determining the direction of periodic solutions bifurcating from positive equilibrium are derived. Some numerical simulations are also carried out to illustrate our results.

### The complete biorthogonal expansion theorem and its application to a class of rectangular plate equations

Jianbo Zhu and Xianlong Fu

In this paper, we first establish the separable $Hamiltonian$ system of rectangular  cantilever thin plate bending problems by choosing proper dual vectors. Then using the characteristics of off-diagonal infinite-dimensional $Hamiltonian$ operator matrix, we derive the biorthogonal relationships of the eigenfunction systems and based on it we further obtain the complete biorthogonal expansion theorem. Finally, applying this theorem we obtain the general solutions of rectangular cantilever thin plate bending problems with two opposite edges slidingly supported.