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## Control measures of pine wilt disease

### Computational and Applied Mathematics (2016-07-01) 35: 519-531 , July 01, 2016

In this paper, we study a vector–host model of pine wilt disease with vital dynamics to determine the equilibria and their stability by considering standard incidence rates and horizontal transmission. The complete global analysis for the equilibria of the model is analyzed. The explicit formula for the reproductive number is obtained, and it is shown that the “disease-free” equilibrium always exists and is globally asymptotically stable whenever $$R_{0}\le 1$$ . Furthermore, the disease persists at an “ endemic” level when the reproductive number exceeds unity. It will be very helpful in providing a theoretical basis for the prevention and control of the disease.

## Spikes for the Gierer–Meinhardt System with Discontinuous Diffusion Coefficients

### Journal of Nonlinear Science (2009-06-01) 19: 301-339 , June 01, 2009

We rigorously prove results on spiky patterns for the Gierer–Meinhardt system (Kybernetik (Berlin) 12:30–39, 1972) with a jump discontinuity in the diffusion coefficient of the inhibitor. Using numerical computations in combination with a Turing-type instability analysis, this system has been investigated by Benson, Maini, and Sherratt (Math. Comput. Model. 17:29–34, 1993a; Bull. Math. Biol. 55:365–384, 1993b; IMA J. Math. Appl. Med. Biol. 9:197–213, 1992).

Firstly, we show the existence of an interior spike located away from the jump discontinuity, deriving a necessary condition for the position of the spike. In particular, we show that the spike is located in one-and-only-one of the two subintervals created by the jump discontinuity of the inhibitor diffusivity. This *localization principle* for a spike is a *new effect* which does not occur for homogeneous diffusion coefficients. Further, we show that this interior spike is stable.

Secondly, we establish the existence of a spike whose distance from the jump discontinuity is of the same order as its spatial extent. The existence of such a *spike near the jump discontinuity* is the second *new effect* presented in this paper.

To derive these new effects in a mathematically rigorous way, we use analytical tools like Liapunov–Schmidt reduction and nonlocal eigenvalue problems which have been developed in our previous work (J. Nonlinear Sci. 11:415–458, 2001).

Finally, we confirm our results by numerical computations for the dynamical behavior of the system. We observe a moving spike which converges to a stationary spike located in the interior of one of the subintervals or near the jump discontinuity.

## Short-time existence theory toward stability for nonlinear parabolic systems

### Journal of Evolution Equations (2015-06-01) 15: 403-456 , June 01, 2015

We establish existence of classical solutions for nonlinear parabolic systems in divergence form on $${\mathbb{R}^n}$$ , under mild regularity assumptions on coefficients in the problem, and under the assumption of Hölder continuous initial conditions. Our analysis is motivated by the study of stability for stationary and traveling wave solutions arising in such systems. In this setting, large time bounds obtained by pointwise semigroup techniques are often coupled with appropriate short time bounds in order to close an iteration based on Duhamel-type integral equations, and our analysis gives precisely the required short time bounds. This development both clarifies previous applications of this idea (by Zumbrun and Howard) and establishes a general result that covers many additional cases.

## Stability of the filter with Poisson observations

### Statistical Inference for Stochastic Processes (2015-10-01) 18: 293-313 , October 01, 2015

The short interest rate process is modeled by a diffusion process $$X(t)$$ . With the counting process observations, a filtering problem is formulated and its exponential stability is derived when the process $$X(t)$$ is asymptotically stationary.

## Dynamics of a rational difference equation

### Chinese Annals of Mathematics, Series B (2009-02-18) 30: 187-198 , February 18, 2009

The authors investigate the global behavior of the solutions of the difference equation
$$
x_{n + 1} = \frac{{ax_{n - l} x_{n - k} }}
{{bx_{n - p} + cx_{n - q} }}, n = 0, 1, \cdots ,
$$
where the initial conditions *x*_{−r}, *x*_{−r+1}, x_{−r+2}, …, *x*_{0} are arbitrary positive real numbers, *r* = max{*l*, *k*, *p*, *q*} is a nonnegative integer and *a*, *b*, *c* are positive constants. Some special cases of this equation are also studied in this paper.

## Dynamic formation of oriented patches in chondrocyte cell cultures

### Journal of Mathematical Biology (2011-10-01) 63: 757-777 , October 01, 2011

Growth factors have a significant impact not only on the growth dynamics but also on the phenotype of chondrocytes (Barbero et al. in J. Cell. Phys. 204:830–838, 2005). In particular, as chondrocytes approach confluence, the cells tend to align and form coherent patches. Starting from a mathematical model for fibroblast populations at equilibrium (Mogilner et al. in Physica D 89:346–367, 1996), a dynamic continuum model with logistic growth is developed. Both linear stability analysis and numerical solutions of the time-dependent nonlinear integro-partial differential equation are used to identify the key parameters that lead to pattern formation in the model. The numerical results are compared quantitatively to experimental data by extracting statistical information on orientation, density and patch size through Gabor filters.

## Stable equilibria of a singularly perturbed reaction–diffusion equation when the roots of the degenerate equation contact or intersect along a non-smooth hypersurface

### Journal of Evolution Equations (2016-06-01) 16: 317-339 , June 01, 2016

We use the variational concept of
$${\Gamma}$$
-convergence to prove existence, stability and exhibit the geometric structure of four families of stationary solutions to the singularly perturbed parabolic equation
$${u_t=\epsilon^2 {\rm div}(k\nabla u)+f(u,x)}$$
, for
$${(t,x)\in \mathbb{R}^+\times\Omega}$$
, where
$${\Omega\subset\mathbb{R}^n}$$
,
$${n\geq 1}$$
, supplied with no-flux boundary condition. The novelty here lies in the fact that the roots of the bistable function *f* are not isolated, meaning that the graphs of its roots are allowed to have contact or intersect each other along a Lipschitz-continuous (*n* − 1)-dimensional hypersurface
$${\gamma \subset \Omega}$$
; across this hypersurface, the stable equilibria may have corners. The case of intersecting roots stems from the phenomenon known as exchange of stability which is characterized by
$${f(\cdot,x)}$$
having only two roots.

## Localized Patterns in a Three-Component FitzHugh–Nagumo Model Revisited Via an Action Functional

### Journal of Dynamics and Differential Equations (2016-10-25): 1-35 , October 25, 2016

In this manuscript, we combine geometrical singular perturbation techniques and an action functional to revisit—and further study—the existence and stability of stationary localized structures in a singularly perturbed three-component FitzHugh–Nagumo model. In particular, the action functional replaces the Melnikov integral approach used in Doelman et al. (J Dyn Differ Equ 21:73–115, 2009) to explicitly derive existence conditions for stationary localized structures. In addition, the action functional also gives critical information on the stability of these stationary localized structures, thus circumventing a tedious Evans function computation. This highlights the strength of the action functional approach.

## Non-local heat flows and gradient estimates on closed manifolds

### Journal of Evolution Equations (2009-08-26) 9: 787-807 , August 26, 2009

In this paper, we study two kinds of *L*^{2} norm preserved non-local heat flows on closed manifolds. We first study the global existence, stability, and asymptotic behavior of such non-local heat flows. Next we give the gradient estimates of positive solutions to these heat flows.

## Stability of radial symmetry for a Monge-Ampère overdetermined problem

### Annali di Matematica Pura ed Applicata (2008-09-17) 188: 445-453 , September 17, 2008

Recently the symmetry of solutions to overdetermined problems has been established for the class of Hessian operators, including the Monge-Ampère operator. In this paper we prove that the radial symmetry of the domain and of the solution to an overdetermined Dirichlet problem for the Monge-Ampère equation is stable under suitable perturbations of the data.